Pacific Journal of Mathematics IN THIS ISSUE— Martin Bartelt, Strongly unique best approximates to a function on a set, and a finite subset thereof . . . . . . . . . . . . . . . . . . S. J. Bernau, Theorems of Korovkin type for L p -spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. J. Bernau and Howard E. Lacey, The range of a contractive projection on an L p -space . . . . . . . . . . . . . . . . . . . . . . . . . . . Marilyn Breen, Decomposition theorems for 3-convex subsets of the plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ronald Elroy Bruck, Jr., A common fixed point theorem for a commuting family of nonexpansive mappings . . . . . . . . . . . Aiden A. Bruen and J. C. Fisher, Blocking sets and complete k-arcs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R. Creighton Buck, Approximation properties of vector valued functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mary Rodriguez Embry and Marvin Rosenblum, Spectra, tensor products, and linear operator equations . . . . . . . . . . . . Edward William Formanek, Maximal quotient rings of group rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Barry J. Gardner, Some aspects of T -nilpotence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Juan A. Gatica and William A. Kirk, A fixed point theorem for k-set-contractions defined in a cone . . . . . . . . . . . . . . . . . . Kenneth R. Goodearl, Localization and splitting in hereditary noetherian prime rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . James Victor Herod, Generators for evolution systems with quasi continuous trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . C. V. Hinkle, The extended centralizer of an S-set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I. Martin (Irving) Isaacs, Lifting Brauer characters of p-solvable groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bruce R. Johnson, Generalized Lerch zeta function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Erwin Kleinfeld, A generalization of (−1, 1) rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Horst Leptin, On symmetry of some Banach algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Paul Weldon Lewis, Strongly bounded operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Arthur Larry Lieberman, Spectral distribution of the sum of self-adjoint operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I. J. Maddox and Michael A. L. Willey, Continuous operators on paranormed spaces and matrix transformations . . . . . James Dolan Reid, On rings on groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Richard Miles Schori and James Edward West, Hyperspaces of graphs are Hilbert cubes . . . . . . . . . . . . . . . . . . . . . . . . . . . William H. Specht, A factorization theorem for p-constrained groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Robert L Thele, Iterative techniques for approximation of fixed points of certain nonlinear mappings in Banach spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tim Eden Traynor, An elementary proof of the lifting theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Charles Irvin Vinsonhaler and William Jennings Wickless, Completely decomposable groups which admit only nilpotent multiplications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Raymond O’Neil Wells, Jr, Comparison of de Rham and Dolbeault cohomology for proper surjective mappings . . . . . . David Lee Wright, The non-minimality of induced central representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bertram Yood, Commutativity properties in Banach ∗ -algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Vol. 53, No. 1

1 11 21 43 59 73 85 95 109 117 131 137 153 163 171 189 195 203 207 211 217 229 239 253 259 267 273 281 301 307

March, 1974

PACIFIC JOURNAL OF MATHEMATICS EDITORS RICHARD ARENS (Managing Editor)

University of California Los Angeles, California 90024

R. A.

J. DUGUNDJI Department of Mathematics University of Southern California Los Angeles, California 90007 D. GlLBARG AND J. MlLGRAM Stanford University Stanford, California 94305

BEAUMONT

University of Washington Seattle, Washington 98105

ASSOCIATE EDITORS E. F. BECKENBACH

B. H. NEUMANN

F. WOLF

K. YOSHIDA

SUPPORTING INSTITUTIONS UNIVERSITY OF BRITISH COLUMBIA CALIFORNIA INSTITUTE OF TECHNOLOGY UNIVERSITY OF CALIFORNIA MONTANA STATE UNIVERSITY UNIVERSITY OF NEVADA NEW MEXICO STATE UNIVERSITY OREGON STATE UNIVERSITY UNIVERSITY OF OREGON OSAKA UNIVERSITY

UNIVERSITY OF SOUTHERN CALIFORNIA STANFORD UNIVERSITY UNIVERSITY OF TOKYO UNIVERSITY OF UTAH WASHINGTON STATE UNIVERSITY UNIVERSITY OF WASHINGTON * * * AMERICAN MATHEMATICAL SOCIETY NAVAL WEAPONS CENTER

Printed in Japan by Intarnational Academic Printing Co., Ltd., Tokyo, Japan

PACIFIC JOURNAL OF MATHEMATICS Vol. 53, No. 1, 1974

STRONGLY UNIQUE BEST APPROXIMATES TO A FUNCTION ON A SET, AND A FINITE SUBSET THEREOF M. W.

BARTELT

Let X be a compact Hausdorff space and let C(X) denote the space of continuous real valued functions defined on X, normed by the supremum norm ||/|| = m3Lxχeχ\f(x) |. Let M be a finite dimensional subspace of C(X). This note examines the problem of whether every best (unique best, strongly unique best) approximate to / on X is also a best (respectively: unique best, strongly unique best) approximate to / on some finite subset of X. Appropriate converse results are also considered.

The Kolmogorov criterion for best approximates shows that π e M is a best approximate to / on X if and only if it is a best approximate to / on a finite subset of E π = { x e X: \f(x) - π(x)\ = \\f - π\\} .

Example 1 shows that the corresponding result does not hold for unique best approximates. It can easily be shown that when π is a strongly unique best approximate to / in C[a, b] from a Haar subspace then there is a finite subset A of [α, 6] such that π is a strongly unique best approximate to / on A. In Theorem 2 the latter result is extended to an arbitrary finite dimensional subspace M of C(X) and in Theorem 3 a converse is proven in this general setting. The second algorithm of Remez [11] is an important method for the computation of the best approximate to a function / in C[a9 b] from a finite dimensional Haar subspace. This algorithm depends on the fact that a best approximate to / on [a, b] is a best approximate to / on some finite subset of [α, 6]. (One can think of the algorithm as a search for this subset.) In fact, the proof of the convergence of the algorithm given by E. W. Cheney [3] indicates that the algorithm depends more precisely on the facts that the best approximate π to / on [α, b] is strongly unique and that π is also a strongly unique best approximate to / on some finite subset of [α, b]. It would also be natural to consider In Lp[a, b] for 1 :g p < co the relationship between strongly unique best approximates on [a, b] and on finite subsets of [α, 6]. However, D. E. Wulbert ([15], [16]) has shown that strong unicity does not occur (nontrivially) in any smooth space and Lp[a, b] for 1 ^ p < co is smooth. In the last section a different proof of Wulbert's result is given because the

2

M. W. BARTELT

method of the proof enables one to study strong unicity in IΛ It should be observed (see Example 3) that even though there are no finite dimensional subspaces of &[a, 6] containing a unique best approximate to every / in U[a, 6], a given / in &[a, b] may have a strongly unique best approximate. The result mentioned above on the relationship between the best approximates to / on X and the best approximates to / on a finite subset of X can be found in [8], [13], and [18]. The results of this note hold with obvious modifications for the complex case. 2* DEFINITIONS. An element π in M is a best approximate to / in C(X) if | | / - m\\ ^ \\f - π\\ for all m in M; π is a unique best approximate if the inequality is strict for all m in M, m Φ π; and π is a strongly unique best approximate to / if there exists a real number r > 0 such that \\f - m\\ ^ | | / — π\\ + r\\π — m\\ for all m in M. Let M have dimension n. The subspace M is called a Haar (Chebyshev) subspace if no nonzero function in M has more than n — 1 zeros in X. If X is the finite interval [α, 6], then Mis called a weak Chebyshev subspace if no nonzero function in M has more than n — 1 sign changes on [α, b]. (For properties of Haar and weak Chebyshev systems, see e.g. [4], [5], [6], and [17].) In particular it is known that if If is a Haar subspace of C[a, b] then π is a best approximate to / on a closed set X in [a, b] (where X contains at least n + 1 points) if and only if there exists an equioscillation set for / — π, i.e., a subset A of X containing n + 1 points xί < x2 < • < xn+ί such that f(xi+ι) - π(xi+1) = - [f(xτ) - π(xt)], i = 1, 2, ,n and 1/(0?,) - π(x 0 such that THEOREM.

There exists a real

11/ - m|| ^ | | / - π\\ + r\\π - m\\VmeM if and only if max [f(x) — π(x)]m(x) > 0 Vm e M,

m ^ 0.

In proofs we assume without loss of generality that the best approximate to / is 0.

STRONGLY UNIQUE BEST APPROXIMATES

3

3* Results* The relationship between a strongly unique best approximate to a given / on [a, b] and on a finite subset A of [a, b] is especially simple when M is a Haar subspace. Recall that when M is a Haar subspace of C[a, b] every / in C{X), where X is a compact subset of [α, 6], has a strongly unique best approximate from M [9]. Hence by the strong Kolmogorov criterion we have the following, result: THEOREM 1. Let π be a best approximate from the Haar subspace M of C[a, b] to a given f in C[α, 6]. Then for every equioscillation set A S Eπ,

max [f(x) — π(x)]m(x) > 0 Vm e M, m =£ 0 . xe A

If we only assume that π is a strongly unique best approximate from a weak Chebyshev subspace, then the conclusion of the previous theorem does not hold. For example, in C[0, Aπ] let f(x) = sin x and let M be the linear span of ίSπ/2 - α 0 g x g 3ττ/2 flr(α) - I 0 3ττ/2 ^ α? ^ 5τr/2 (5τr/2 - a; 5ττ/2 ^ a; ^ 4ττ . Then 0 is strongly unique to / since md»xxeEQf(x)m(x) > 0, VnieM, m Φ 0, but m&xxeA f(x)(- g(x)) = 0 where A = {5ττ/2, 7ττ/2} is an equioscillation set for / — 0. However, we now show that when π is a strongly unique best approximate from an arbitrary subspace M in C(X), it follows that there does exist some finite subset A of Eπ such that π is a strongly unique best approximate to f on A. THEOREM 2. Let π be a strongly unique best approximate from a subspace M of C(X) to an element f in C(X). Then there exists a finite subset A of Eπ with ^ 2n points such that

max [f(x) - π(x)]m(x) > 0 Vm e M, m/A =£ 0 . xeA

Proof

Let M be the span of {glf

, gn}.

Let Eo = {(f(x)gί(x)9

--, f(x)9n(oή):xeE0}. Then it follows ([2], Theorem 6) that 0 is in the interior of the convex hull of Eo. Hence (see e.g. Theorem 3.13 in [14]) 0 is in the interior of the convex hull of A, where A is a finite subset of EQ consisting of g 2n points. It follows ([2], Theorem 6) that 0 is a strongly unique best approximate to / on A. By the strong Kolmogorov criterion max, e i f(x)m(x) > 0 for all m in M with m/A =£ 0.

4

M. W. BARTELT

It is not known in general whether it is possible to find a finite set A satisfying the conditions of the previous theorem such that if m is in M and m/A = 0, then m = 0. However, if Eπ is finite then by setting A — Eπ one can add to the conclusion of Theorem 2 that m/Eπ == 0 implies m = 0. This follows from the strong KoImogorov criterion. Also if Eπ is not finite but it is known that any nonzero function in M has at most N — 1 zeros for some integer N (for example N = n when M is a Haar set), then one can just add to the set A of the previous theorem enough points of Eπ so that A has N or more points. It would be of interest to determine whether the 2n of the theorem is in general best possible. If π is a unique best approximate to / on X> then it does not follow that π is a unique best approximate to f on Ez. This can be seen in the next example which will also be used later. EXAMPLE

1.

Let M be the subspace of C[0, 3π ] spanned by

g,(x) = 1 and 'π — x

0

0

π ^ x g5τr/2

βπ/2 - x

5τr/2

^ 3π .

Let f(x) = sin x. Then M is a weak Chebyshev system, but it is not a Haar set on [0, 3τr], Because f(x) has a horizontal tangent at x = 5ττ/2, the function —g2(%) is not as good an approximate to f(x) as 0 is. Clearly then, 0 is a unique best approximate to / on [0, 3ττ], Now EQ = {τr/2, 3τr/2, 5ττ/2}. Since M has dimension 2, 2?0 is an equioscillation set for / — 0 on [0, 3ττ]. Now 0 is not a unique best approximate on EQ = A since g2(x) is also a best approximate. Also observe that 0 is not a strongly unique best approximate to / on [0, 3ττ] since maxxe^0 /(«)[-flr2(»)] = 0. In fact even more holds. Let a? --π/2

0

x -- π

0 ^ a? ^ π/2

τr/2 ^ α; ^ π 7Γ ^ a; ^ 3τr/2

2(7ττ/4 - α?)

3ττ/2 ^ a? ^ 7π/4

,χ -- Ίπ/A

7τr/4 ^ a; ^ 3τr .

Then let M be the subspace of C[0, 3π] spanned by &(&) and gz(x), and let /(x) = sin x. Then by consideration of the values of any me M at points τr/2, 3ττ/2, and 5π/2, it is easy to verify that zero is a unique best approximate to / on [0, 3τr] and Eo = {π/2, 3π/2, 5ττ/2}.

STRONGLY UNIQUE BEST APPROXIMATES

5

Moreover on each subset A of E09 there is a function g e M such that g/A =£ 0 and g is a best approximate to / on A. Thus zero is not a unique best approximate to / on any finite subset A of Eo. The next proposition summarizes the results for an arbitrary subspace M of C(X). For the result on best approximates see [8], [13], and [18]. PROPOSITION. If π is a best (strongly unique best) approximate to f on X, then there exists a finite subset A of X with less than or equal to n + 1 (resp. 2ri) points such that π is a best (strongly unique best) approximate on A.

The Kolmogorov and strong Kolmogorov criteria and Example 1 also yield the relationship between the best approximate to / on X and on all of Eπ. As expected, π is a best (strongly unique best) approximate to / on X if and only if it has the same property on Eπ. This does not hold for a unique best approximate. REMARK.

4* Converse results* of the next theorem.

The Kolmogorov criterion shows part (i)

THEOREM 3. (i) If π is a best approximate to f on a finite subset of Eπ, then π is a best approximate to f on X. (ii) If π is a unique (strongly unique) best approximate to f on a finite subset A of Eπ, then π is a unique (strongly unique) best approximate to f on X, except possibly for those m in M with ml A = 0. In fact more than this holds. The following result says that if π is a unique best approximate to / on a finite subset A of X, then π is also a strongly unique best approximate to / on A. THEOREM 4. Let π be a unique best approximate to f on a finite subset A of X. Assume f(x) — π(x) Ξ£ 0 on A. Then

max [f(x) — π(x)]m(x) > 0

Vm Ξ£ 0 on

A.

xeΛ

Proof. (We show that if m a x ί 6 i f(x)q(x) ^ 0 for some qeM, then there exists a real number λ > 0 such that — Xq is a best approximate to / on A.) Let A' = {xeA: f(x)q(x) < 0}. Let λ > 0 be such that both the following hold: ( 1 ) λmax. β ^|?(a?)|2

^ Σ l/(i)l + r{\π(ί) - m(l)| + |τr(2) where one can choose r = 1 to be the strong unicity constant. 1 The space L contains a finite dimensional subspace M which contains a strongly unique best approximate to every element fe 1 L - M if and only if (Γ, Σ, μ) contains an atom ([1], [10]). To obtain further information about strong unicity in L\ let f e L\ \\f\\ = 1 and f £ M. Assume without loss of generality that 0 is a best approximate to / and let £?Q - {L e *: Lf = 1 = ||L||}. For a

8

M. W. BARTELT

given L e ^f0, there exists by the Riesz Representation Theorem a function he L°° such that 1

and

Lg =[ hgdμVgeL Thus for a given L e ^ (1)

1 = \hfdμ

we have £ \\h\ \f\dμ

£

L = 1.

The condition for equality in Holders inequality implies that \h\ \f\ = || AIU I/I = | / | a . e . Also (1) shows that hf = \h\\f\ a.e. Thus jδ^ can be identified with {heL00:

\f\(\h\ - 1) = 0 a.e.

and

(Λ/)(l-sgnΛsgn/) - 0 a.e.} .

This characterization of ^ can be used to study strong unicity in L 1 . For example if μ{x: f(x) = 0} = 0, then \h\ = 1 a.e., sgn A sgn / = 1 a.e. and therefore h is uniquely determined a.e. Since J*fo contains a unique element it follows as before that 0 is not a strongly unique best approximate to / . We have shown the following: THEOREM 6. Let f in L\T, Σ, μ) have a strongly unique best approximate π from a subspace M. Then μ{x: f(x) — π(x) = 0} > 0.

It should be pointed out that it is possible for an element f e L1 to have a strongly unique best approximate from a subspace M even when (T, Σ, μ) does not have an atom. It is not known whether a result like Theorem 2 exists for Lι[a, 6], 3. Let

EXAMPLE

Z/[-2, 2].

Let M be the constant functions, a subspace of 'x + 1 0

- 2 g . t g -1

ί» — 1

1 ^ x^ 2.

Then one can verify that

11/ — cjf α =

(\c+lY 4|c|

Thus 0 is a best approximate to / and also

11/ - c l l ^ 11/11, + 1/2Hell,.

STRONGLY UNIQUE BEST APPROXIMATES

9

REFERENCES 1. D. A. Ault, F. R. Deutsch, P. D. Morris, and J. E. Olson, Interpolating subspaces in approximation theory, J. Approx. Theory, 3 (1970), 164-182. 2. M. W. Bartelt and H. W. McLaughlin, Characterizations of strong unicity in approximation theory, J. Approx. Theory, 9 (1973), 255-266. 3. E. W. Cheney, Introduction to Approximation Theory, McGraw Hill, New York, 1966. 4. A. Haar, Die Minkowschische Geometrie und die Annaherung an stetige Funktionen, Math. Annalen, 7 8 (1918), 294-311. 5. R. C. Jones and L A. Karlovitz, Equioscillation under nonuniqueness in the approximation of continuous functions, J. Approx. Theory, 3 (1970), 138-145. 6. S. Karlin and W. J. Studden, Tschebycheff Systems: With Application in Analysis and Statistics, John Wiley and Sons, Inc., New York, 1966. 7. A. N. Kolmogorov, A remark on the polynomials of P. L. Cebysev deviating the least from a given function, Uspehi Mat. Nauk., 3 (1948), 216-221 (Russian). 8. G. G. Lorentz, Approximation of Functions, Holt, Rinehart, and Winston, New York, 1966. 9. D. J. Newman and H. S. Shapiro, Some theorems on Cebysev approximation, Duke Math. J., 30 (1963), 673-681. 10. R. R. Phelps, Uniqueness of Hahn-Banach extensions and unique best approximation, Trans. Amer. Math. Soc, 95 (1960), 238-255. 11. E. Ya. Remez, General computational methods of Tchebycheff approximation, Kiev (Russian), (1957), AECT No. 4491. 12. T. J. Rivlin and E. W. Cheney, A comparison of uniform approximations on an interval and a finite subset thereof, J. SIAM Number Anal., 3 No. 2 (1966), 311-320. 13. T. J. Rivlin and H. S. Shapiro, A unified approach to certain problems of approximation and minimization, J. SIAM, (9) 670-699. 14. C. P. Valentine, Convex Sets, McGraw Hill, New York, 1964. 15. D. E. Wulbert, Uniqueness and differential characterization of approximation from manifolds of functions, Bull. Amer. Math. Soc, 7 7 (1971), 88-91. 16. _ , Uniqueness and differential characterization of approximation from manifolds of functions, Amer. J. Math., 9 3 (1971), 350-366. 17. J. W. Young, General theory of approximation by functions involving a given number of arbitrary parameters, Trans. Amer. Math. Soc, 8 (1907), 331-344. 18. S. I. Zuhovickii, On approximation of real functions in the sense of P. L. Chebysev, Uspehi Mat. Nauk., 11 (1956), 125-159 (Russian), AMS Translation, Series 2, 19 221-252. Received July 13, 1973 and in revised form September 21, 1973. This research was partially supported by N. S. F. Grant GU-2605. RENSSELAER POLYTECHNIC INSTITUTE

PACIFIC JOURNAL OF MATHEMATICS Vol. 53, No. 1, 1974

THEOREMS OF KOROVKIN TYPE FOR ^-SPACES S. J. BERNAU

Suppose (X, Σ, μ) is a measure space, 1 < p < oo, p Φ2, and that (Tn) is a net of linear contractions on (real or

complex) LP(X, Σ, μ).

Let M = {xeLp:

Tnx->x]

(M is the

convergence set for (Tn)). It is obvious that M is a closed siabspace of Lp; indeed this would be true for an arbitrary normed space. In this paper we shall show that M is the range of a contractive projection on Lp and hence is itself isometrically isomorphic to an Lp-space. If S c LP(X, Σ, μ) we can define the shadow, x for every net of linear contractions (Tn) such that Tny -> y for all yeS. We shall also give a complete description of ^(S) (for p Φ 1, 2, oo).

Our results are new for finite p not equal to 1 or 2. In the case p — 2 the assertions about Mare trivial and S^(S) is the closed subspace spanned by S. The case p = 1 was first considered by Wulbert [9] for Lebesgue measure on [0, 1], He showed that if S — {1, x, x2} then S^{S) = LJO, 1]. (Actually he considered sequences of contractions and required only Tnl —> 1 and Tnf weakly convergent to / for / = x and / = x2.) Wulbert's results were inspired by and generalized the classical theorem of Korovkin [7] which contains the result that if S = {1, x, of) then the shadow of S in C[0, 1] is C[0, 1]. In [8] Lorentz considered separable Lλ spaces on finite measure spaces. He showed that for sequences of contractions such that Tnl —* 1 the convergence set is a closed sublattice of Lx. A corollary of this, which he noted, is that for LJO, 1], S^(S) = L, if S = {1, x}. This last result and some further discussion of LX{X, Σ, μ), with μ(X) = 1 is also contained in [1], The methods we use are suggested by the methods used in [3] in discussing contractive projections. I am very grateful to Professors Lorentz and Berens for discussions of this material and for supplying me with preprints of [1], [2], [8]. My first introduction to this circle of ideas was a colloquium lecture by Professor Lorentz in which some of the results from [2] and [8] were presented. 2» The convergence set* We shall fix notation as in the first paragraph of the introduction. It does not seem to matter whether our measure space is taken over a cr-ring, σ-algebra or δ-ring. For definiteness we shall assume that Σ is a cr-ring and measurability is as defined by Halmos [5]. We shall let q be the conjugate index to p, defined by 1/p + 1/q = 1. Since p Φ 1, 2, o, the same is true 11

12

S. J. BERNAU

for q and Lq(X, Σ, μ) is the topological dual of LP(X, Σ, μ) with the usual identifications. We shall consider the complex case; i.e., Lp (and Lq) are (equivalence classes of) complex valued functions. The real case is a little easier, but the methods are the same. If T is a bounded linear operator on Lp, the conjugate operator T* is defined on Lq by the identity x (T*y)dμ = j (Γa?) ydμ (x eLP,ye

Lq) .

DEFINITION. The conjugate convergence set M* for the net of contractions (Tn) is defined by M* = {yeLq: T*y~>y}.

2.1. (Compare [3, Lemma 2.2].) Let xeLp, if and only if | x \v~ι sgn x e M*. LEMMA

then

xeM

Proof. Suppose xe M and write u — | x I2'"1 sgn x. Then \\u\\q ~ \\x\\pq and (T*u) is a bounded net in Lq. Let w be a weak-* limit point of this net. We have I x-(w — T*u)dμ = \ x-wdμ — \ (Tnx)-udμ = \x

> \ x wdμ — \x udμ

(w — u)dμ .

Taking a subnet such that T*u —> w (weak — *), we conclude that

Since the T* are contractions, || T * ^ | | g ^ \\u\\q = ||^||p / ? and hence II w||? ^ ll^l!?/9 Holders' inequality now gives 11 χ 11 v _ [ x . wdμ < 11 α; 11 ll^li ^ l l ^ l l llα l ^ ^ ^ l l α ll27. This gives equality throughout so [6, § 13.5] we have 1

ιv = j x l^" sgn x = u . Thus u is the unique weak-* limit point of the net (T*u). Since every subnet of (T*u) has a convergent subnet (by weak-* compactness of the unit ball in Lq), we see that T*u is weak-* convergent to u. Hence || u \\q S lim inf || T^u \\q ^ lim sup || T%u \\q £ \\ u \\q, because the T* are contractions; and we also have \\u\\ = lim || T*u ||. Because Lq is uniformly convex [4; 6, § 15.17] it follows that T%u—>u

THEOREMS OF KOROVKIN TYPE FOR LP-SPACES p

13

ι

in the norm of Lq, which gives | x \ ~ sgn x e M* as required. The same argument applied to Lq shows that if u = I x \p~ι sgn x e M* ,

x = \ u \q'x sgn ΰ e M** = M,

so we are done. We now apply differentiation arguments like those in [3, Lemma 2.3]. Recall that if z, w are complex, X is real and h(X) — \z + Xw\ then, if z + Xw Φ 0, h is differentiable at λ with h'(X) = Re [w sgn (z + λw)] . LEMMA 2.2. I x I sgn yeM.

(Compare [3, Lemma 2.3(i)].) // x,yeM,

then

Proof. Assume first that p > 2 and define, for λ e iϋ, and 0 < λ < 1, Zx = λ-χ[| a? + Xy \p-' sgn (a; + Xy) - \ x I*"1 sgn ^] - λ-χ[(| x + Xy \v~2 - I a? \p~2)(x + λ»)] + | a? | p - 2 ^ . Now, make a fixed choice of functions from the equivalence classes determined by x, y and observe that, except for the null-set where x or y is infinite, our differentiation result quoted above shows that as λ—>0, tλ —• (p - 2) I a; p" 3 Re [T/ sgn ^] x + | a; | p " 2 ^ at all points where x Φ 0. Also, since #> > 2, ^-->0(λ—>0) at points where x = 0. Let ^ denote this (almost everywhere) pointwise limit of ^ as λ->0. At points where 2 | Xy \ < |α?|, the mean value theorem gives a θ, 0 < 0 < 1 such that s* = (p - 2) I a? + θXy \p~3 Re fo sgn (x + 0λi/)](a? + λy) + | a? l Since | x |/2 < | a? + 0λi/1 < 2 | a? |, w e h a v e \zx\ ^ (p - 2)2l2?~31 |a?| p - 8 |2/| 2 | a ? | + p

^(2 (p-2)

\x\*-ι\v\

2

+

l)\xr \y\eLq.

At points where | x | ^ 2 | Xy |, we have

21 1

31 1

1

= (3 - + 2 " ) I y p" 1 λ Γ ^(S*-1 +

2

2p-1)\y\p-1eLq.

14

S. J. BERNAU

Thus the pointwise convergence of zλ to p

Zl

3

p

2

= (p — 2) I x \ ~~ x Re (y sgn x) + | x \ ~ y

is dominated by an element of Lq. The dominated convergence theorem then shows that \\zλ — z11| —•> 0. Now, by Lemma 2.1, ^ e M* and ilf* is closed. Hence zιeM*. Apply this result to a; and — ί?/ to see that p

3

p

2

z2 — (p — 2) |a?| " ^Re ( — ίy sgnx) + i \x\ ~ yeM* . Thus ^ — iz2 e M* and zλ — iz2 — (p — 2) j x \p~3x [Re y sgn x — ίlmy 2

p

sgnx]

2

+ 2 I # p~ i/ — p j α; \ ~ y . Use Lemma 2.1 again to conclude t h a t 11 x \v~2y \q~ι sgn (I x \p~2y) - | x l 1 "^-^ 12/ Γ 1 sgn y e M . L e t kn = \x\ι-{q-1)n\y\{q-1)n$gny. Observe t h a t for each n, kn+1 = |α;| 1 ~ ( 9 " 1 ) [ kn I 9 " 1 sgn kn. The argument we have just provided shows inductively that kne M for all n. Since 0 < q — 1 < 1, kn—*\x\ sgn 2/, //-almost everywhere, and clearly I kn I ^ max (| x [, 12/1) e L P . By dominated convergence again, \\kn — | x \ sgny \\p —>0 and hence, [ x I sgn ^/ G M as required. If 1 < p < 2, then by Lemma 2.1, x1 = {x^sgnx and yι = p 1 |^/| ~ sgn^ are in Λί*. Since g > 2, our proof above shows that x11 sgny^ = \χ Y'1 sgn y e M*. Apply Lemma 2.1 again to get I x I sgn y = \\χ p" 1 sgn y\q~ι sgn (|a; j^^sgn y) e M. For our next result we need some terminology from [3]. A

subspace N of LP(X, Σ, μ) is a vector suhlattice if for each xe N, (Re x)+ G N; this means that N is closed under taking real (or imaginary) parts and that the set of real functions in N is a real subspace and a sublattice of Lp. For a nonempty subset K of L^, the polar K- = {xe Lp: \x\ Λ\y\ = 0(^/ e K)}. A δα^d in L^ is a subset JSΓ such 11 that K = K (a band is necessarily a solid vector sublattice). If K is a band in Lp there is a natural direct sum decomposition Lv = K@KL and the associated projections are positive and contractive. We write Jκ for the band projection on K. If K — y11 (the only case we need) we write Jy for the band projection and note that Jy is multiplication by the characteristic function of the set on which y is nonzero.

THEOREMS OF KOROVKIN TYPE FOR LP-SPACES

2.3. If yeM and Jy is the associated JyMdM.

LEMMA

then

hand

15

projection,

Proof. Let x e M, then Jyx = 11 x | sgn y | sgn x and this element is in M by two applications of Lemma 2.2. LEMMA

+

2.4. If x,ye M, then (Re (x sgn y)) sgn yeM.

Proof. Suppose λ e R, λ Φ 0, then by Lemma 2.2, vλ = λ -1 (| ?/ + Xx I — I y I) sgn yeM. Since ^ = 0 at points where y — 0 we see that as λ —» 0, vλ converges pointwise (/^-almost everywhere) to (Re (x sgn y)) sgn y. Since I vλ I ^ I λ I"1 j I y + Xx I - 12/11 ^ I λ I' 1 1 y + λ α - y | = | x | , dominated convergence shows t h a t [| vλ — (Re (x sgn ^7)) sgn y \\p —> 0; so t h a t (Re (# sgn y)) sgn yeM. Another application of Lemma 2.2 gives I Re (x sgn y) \ sgn yeM and our results follows. THEOREM 2.5. The convergence set M is the range of a contractive projection on Lp.

Proof. For yeM, Lemma 2.4 shows that the map Uy: LP—>L P9 defined by Uyx — x sgn y, is norm decreasing, linear, and maps M onto a closed vector sublattice of Lp. Such a map was called a unitary multiplication operator in [3]. Choose, by Zorn's lemma, a maximal subset Y of M such that | y1 \ Λ | y21 = 0 if 2/i, 2/2 ^ Y a n ( ^ 2/i ^ 2/2 (a maximal pairwise disjoint subset of M). If f e Lp the set (ίGZ:/(ί) ^ 0} is cr-finite so the set {ye Y: Uyf ψ 0} is countable. Thus we can define the direct sum U of the unitary multiplications Uy(yeY) by Uf = Σver Uyf, and the defining sum has at most countably many nonzero terms and is convergent in Lp norm. Clearly UM is a closed vector sublattice of Lp. We show that U is isometric on M. Suppose x eM.xΦ 0. Let yl9 - - -, yn— - be an enumeration of the countable set of y e Y such that \y \ A \ x \ Φ 0. Let y0 = Σu^~n WVnW^Vn' Then yoeM and, by Lemma 2.3, JVoxeM. Hence x — Jyox e M and | x — JyQx \ A \ y \ = 0(2/ e Γ ) . By maximality of Y x — JyQx. Hence

« = Σ ^ and II .τ | p - Σ I I Λ * IP = Σ I! ^ IΓ = II K8II' yeY

yeY

yeY

It now follows by Theorem 4.1 of [3] that M is the range of a contractive projection on Lp and, which is an equivalent condition, that M is isometrically isomorphic to some LP(XO, Σo, μ0).

16

S. J. BERNAU

3* The shadow of subset S. In a certain sense characterization of shadows is trivial. Call a subspace M of Lp an exchange suhspace if \x\ sgn y e M for any x, y e M. Clearly an intersection of (closed) exchange subspaces is again a (closed) exchange subspace. Hence for a subset S of Lp we can determine the closed exchange subspace of Lp generated by S as the intersection of all closed exchange subspaces of Lp which contains S. THEOREM 3.1. If SaLP then the shadoiv, S^(S), of S is the closed exchange suhspace of Lp generated by S.

Proof. Lemma 2.2 shows that 6^{S) is a closed exchange subspace of Lp which contains S. If M denotes the closed exchange subspace of Lp generated by S a careful check of the proofs of Lemmas 2.3 and 2.4 and Theorem 2.5 show that these are valid for any closed exchange subspace of Lp. Hence M is the range of a contractive projection, say P, on Lp. Define a sequence (Tn) of linear contractions oΐ Lp by Tn = P(n = 1, 2 •)• Then M is the convergence set for (Tn) and hence, y ( S ) c I . This proves our theorem. As a corollary of the proof of Theorem 3.1 we digress to state the following result. THEOREM 3.2. Suppose l ^ p < ^ , pφ2, a subspace M of LP(X9 Σ, μ) is the range of a contractive projection if and only if M is a closed exchange subspace of LP(X, Σ, μ).

Proof. If M is a closed exchange subspace of Lp then just as in Theorem 3.1, Theorem 2.5 is valid for M. (This is equally true for p = 1 and p = 2 as can easily be checked.) By [3, Theorem 4.1J it follows that M is the range of a contractive projection on Lp. The converse result is the statement of [3 ? Lemma 2.3(i)] if p Φ 1 and an easy consequence of [3, Lemma 3.3] if p = 1. Returning to shadows we note that Theorem 3.1 is difficult to apply in practice. The following alternative seems a little more useful. Let έ%? be the smallest sub σ-ring of Σ such that the functions JyX/y are ^-measurable for all x, y e S. (To be precise here, we consider all choices of functions x, y in equivalence classes in S. The ratios are zero, by definition, wherever the denominators are zero.) Choose, by Zorn's lemma, a subset 3ίΓ of ^ x S which is maximal with respect to the properties: (i) if (A, y) e SΓ, then Aa{te X:y(t)Φ0}; (ii) if (A, y) e SΓ, μA > 0; (iii) if (Au yj, (A2, y2) are distinct elements of J?7 then μ{Aι Π A2) = 0. Define a measure λ on & by

THEOREMS OF KOROVKIN TYPE FOR L P -SPACES

AΠB

17

\y\'dμ.

Also define a map V: LP(X, ^ λ)-*LP(X, Σ, μ) by Vf = Σu,y)^f yχA (/ G LP(X, &, λ)). We note that the sets in & all have σ-finite ^-measure; that the sum defining Vf has, therefore, only countably many nonzero terms and is convergent in the norm of LP(X, Σ, μ). Furthermore, V is an isometry of LP(X, &, λ). THEOREM

3.3.

The shadow £S(S) = VLP(X, &, λ).

Proof. Let U be a direct sum of unitary multiplications such that U is an isometry of &*(&) and US^(S) is a closed vector sublattice of LP(X, Σ, μ). For f e Lp write T(f) = {t e X: f(t) Φ 0}, (we allow the ambiguity of sets of measure zero here); and let &s — {T(f):fe^(S)}. I claim that ^ is a sub σ-ring of Σ. For this, observe that T(f) = T(Uf) so we may assume that Sf(S) is a closed vector sublattice of Lp. Now T(f) = T(\ /1) = Γ(| Re / I) U Γ(| Im /1) and I / I 6 (S) so T(g) ~ T(f) = T(\ g | - / , | ^ [) e &B.

This

proves our claim about α > 0} = T((Re (Jyx - ay)) ) e &s. Hence, every Jyx/y (x, yeS) is ^-measurable and &s 3 &. Suppose B e & then B = T(fB) for some fB e £f{β). If λ(J5) < oo, then VχB = Έj(A,y)e.^y 'XBΠA- Since J5 is ^-finite we can enumerate the countable set of pairs (A, y) in 3ίΓ such that μ(B Π A) ^ 0 as (An, yn) and choose fn e S^{S) such that T(f%) = AnΓ\B. Then = Σ=-i Σ ~ i ^/l/* Since the An are disjoint and Vn\vdμ

= XB

w

n

the series for VχB converges in LP(X, Σ9 μ). Since each yneS and each f%e£S(S)9 each JfnyneS^(S) and F χ * G ^ ( S ) . Extending linearly to simple functions in LP{X, ^ μ) and then taking closure we conclude that VLP(X, ^ μ)a S*(S). If x e S and {A, y) e ^Γ then χAxjy is ^-measurable and

18

S. J. BERNAU

Hence z = Σ u ^ e j r XAΦ e LP(X, &, λ) and x = Vze VLP(X, &, λ).

This shows that VLP(X, ^ λ) z> S. If/, g e LP(X, ^ λ), then | f\ sgn g e LP(X, sup {||χS(/)fe||p: / e l ) ,

We omit the remaining details. 1.2. This lemma can be strengthened, in case M is closed, to say that for each he Lp there exists f e M such that Jh = Jfh = %s(f)h. This depends essentially on the fact that the set of supports of functions whose equivalence classes are in M is closed under countable union. This is proved by Ando [1, Lemma 3] for finite μ, and we give a rather easier alternative proof in our appendix. REMARK

2* Preliminary results* In this section the cases p = 1, and 1 < p < co, p Φ 2, are treated separately. Our first lemma is based on an argument of Douglas [2, p. 452]. LEMMA 2.1. Let P be a contractive projection on Lt(X, Σ, μ) and suppose f e &(P); then ( i ) PJf = JfPJ/, (ii) P(h sgn /) - I P(h sgn /) | sgn / (0 ^ h e LL); (iii) ||P(fcsgn/)|| =

Proof. Suppose 0 2, let λ e U, 0 < | λ | < 1, and let ge&(P). By Lemma 2.2, p

gλ — λ Wf + λ^| * sgn (/ + λ#) — Since p > 2,

0* = λ-^d/ + λflr|M - |/Γ 2 )(T + λsrΓ2 - l/I^XZ + Xfif)] + l/l"-2^ . Recall, that for real λ and complex w, z, d/dX | w + Xz 11; = Re [z sgn {w + λ«)], provided w + λz ^ 0. It follows that as X —> 0, Λ

3

(P - 2) I /1"" Re (fir sgn / ) • / + ! / Γ*9

at all points of X where / Φ 0. If 2|λff| < I/I we have |/|/2 < 1/ + ^λ^| < 2 | / | if 0 < θ < 1; and, by the mean value theorem there exists θ, 0 < θ < 1 such that | Re(firsgn(/ 2 ^ (p - 2)2i'-'|/|'-|flr| 2 I/I + | / Γ | ί / | If 2|λflr| ^ I/I, | / + λflr| ^ 3|λflr| and lfif;|^λ- 1 [(3|λflr|r i = (3"-1 + 2 p - ι )|flΊ'~ 1 |λ| The penultimate line above shows that gλ —> 0(λ —• 0) if / = 0. This shows that gr^ converges to g0 = (p - 2 ) | / r 2 s g n / R e ( s r s g n / ) + \f\'~'g , pointwise almost everywhere on X and that the convergence is dominated by an element of Lq. Hence \\gλ — g0\\g —* 0 and gr0ε &(P*) because ^ ( P * ) is closed. By the same argument, applied to — ΐfir, we have, using Re — iz = Im z, Ao = (P - 2) I / r" 2 sgn / I m ( | / | sgn# ^-almost everywhere on X, we have \\kn — \f\ sgn g|| p —>0 and since &(P) is closed |/1 sgn# e &{P) which proves (i) for p > 2. Suppose 1 < j> < 2; as we have already stated P* is a contractive projection on Lq, and q > 2. By Lemma 2.2, / t = I / Γ ^ s g n / and are in ^ ( P * ) . By our proof above |/i|sgnflr1 = ), and, by Lemma 2.2 again, \f\sgnge&(P). This completes the proof of (i). For (ii) we have by (i), that | /1 sgn Pk e &(P) (keLP). By (i) again, JfPk

= \Pk\ s g n ( I / I s g n Pk) e

Thus JfP — PJfP. Further, since P* is a contractive projection on Lq, and I/I'- 1 sgn feέ&(P*) we have J . P * - P*JgP* with

In addition Jg — J*, since J"^ and Jf are each multiplication by the same characteristic function. We conclude JfP = PJfP = (P*JfP*)* = (P*JgP*)* = (JgP*)* - PJf , which is (ii). (iii) The proof is like the proof of Lemma 2.1(ii). Suppose 0 ^ h ^ | / | . By (i), | / | sgn P(hsgnf) e &(P), so by Lemma 2.2, I /1*- 1 sgn P(h sgn /) e Hence, I P(h sgn /) I I / \*-*dμ - \P{h sgn /) -1 / Γ 1 sgn P(hsgnf)dμ = \^h sgn / I / Γ 1 sgn P{hsgnf)dμ

Also

THE RANGE OF A CONTRACTIVE PROJECTION ON AN L^-SPACE

27

Hence,

= j I P(Λ sgn /) + P((| /1 - h) sgn f)\\f

\^dμ

ι

ι

sgn /) | | / Γ d μ + j | P((|/1 - h) sgn /1 | / \'~ dμ

- h)\fΓ'dμ We have equality at each stage and hence, (/^-almost everywhere), /I = | P ( | / | s g n / ) | - \P(hsgnf)\

+ \f - P(h sgn / ) | .

This proves (iii) for 0 ^ h ^ | / | . The extension to 0 ^ fee LP is the same as in the proof of Lemma 2.1(ii) and (iii) so we are done. 3* Contractive projections and conditional expectations* In this section we describe the contractive projections on LP(X, Σ, μ) (1 g p < oo, p φ 2) in terms of conditional expectation. We first need the necessary σ-subring. LEMMA 3.1. Suppose 1 :S p < CXD, p ^ 2, απd ieί P be a contractive projection on LP(X, Σ, μ). Define ΣQ to be the set of supports of all functions whose equivalence classes are in &(P)\ then (i) PJJ = JJ (f,ge&(P)); (ii) Σo is a σ-subring of Σ.

Proof, (i) By Lemma 2.3(ii), (i) is valid if p Φ 1. We give a proof that uses only the identity JgPJg = PJg valid for 1 ^ p < ©o, ι p =£ 2 (Lemma 2.1(i) or 2.3(ii) weakened). Since f-Jgfeg and L1 Jgf - PJgfeg , we have

\\P(f -

= \\f- JJW

+ \\J,f -

Thus P/,/ - /,/ which is (i). (ii) By (i), S(f) - S(YL there exists f e &{P) such that 11 hef . COROLLARY

Proof. By Lemma 1.1 there is a sequence (/Λ) in &(P) such that h = \imn^χS{fn)h. Choose / e ^ T ( P ) such that S(f) = \J S(fn), 11 then hef . Observe now that if / e Lp the measure \f\pμ restricted to any σ-subring, Σo, of Σ, is finite. By the Radon-Nikodym theorem we may define the conditional expectation operator, &f = ^{ΣQj \f\v), for the measure \f\pμ relative to ΣQ. equation

ί h\f\'dμ

ί?f is uniquely determined by the

μ

JA

(AeΣ0)

v

for he Lλ{Xy Σ, \f\ dμ), and the condition that g*/fe is Jo-measurable. LEMMA 3.3. Suppose 1 ^ p < co, p φ 2; let P be a contractive projection on LV{X, Σf μ) and let Σo be the σ-subring of Σ, consisting of supports of functions in &{P). If Mf = f~ιJf&{P) = {f^J/Q- 9 € ^ ( P ) } then Mf - L9(S(f), Σo \ S(f), \f\pμ) where Σo \ S(f) = {AeΣ0:Aa S(f)} and we make the obvious identification of functions on S(f) and functions on X which vanish off S(f). In addition the map h—*f~Ίι is an isometric isomorphism between Jf&(P) and Lp(S(f), ΣQ\S(f), \f\pμ).

Proof. Observe that 1/1*7* is finite on S(f), and that the isometry claim is obviously true. If AeΣQ\S(f) then A = S(g) for some ge &(P). By Lemmas 2.1 and 3.1 (if p = 1) or 2.3 (if p > 1) we have Jgf = PJgf so that χA = f~xJgf e Mf. Let h be a simple function with respect to Σ0\S(f). Then heMf and hfe&{P). In addition )S(f)

\ h \ p \f\pdμ=

\\hf\'dμ.

We conclude that MfiiLP(S(f),ΣQ\S(f),\f\*μ). Conversely, let heMf, then h e Lp(S(f), Σ\S(f), \f\pμ) and it is enough to show that h is 2Vπieasurable. Let g = (Re /?)", then gfe Lp(Xf Σ, μ). By Lemma 2.1(ii) or 2.3(iii) P(9f) = P(\ Of I sgn /) = I P(| gf \ sgn /) | sgn / so ΓιP(gf)

= \f\-1\P(\gf\ sgn /) | e Mf. It follows that

THE RANGE OF A CONTRACTIVE PROJECTION ON AN LP-SPACE

29

+

Reh = /-\P((Re h) f) - /-'PtfRe h)~f) e Mf . Since each of these functions is nonnegative it is sufficient to consider 0 0 and put k = h V aχsιf). Arguing as above, we have fιP{kf) ^ λ and f~ιP{kf) ^ αχ S ( / ) so that f~ιP{kf) ^ k ^ 0. Since P is contractive we have p

HOT ^ lliW)ll = IIW) -kf + kf\γ This gives P(kf) = kf, so that & 6 Λf/. This shows, incidently, that Mf is a lattice. For our purpose, however, we have {ί G S(f): h(t) >a} = {te S(f): (k - aχS[f))(t)

Φ 0}

- S(kf - α/) e I'o . Thus Mf consists of immeasurable functions and we are done. THEOREM 3.4. Suppose

l ^ p < oo, p φ 2 α^id that

tractive projection on LP(X, Σ, μ). If f e &(P) and hef

P is a con1L

then

Ph = f&(ΣQ,\f\*)(hf-1). Proof. Since f~λPh we have only to show

e Mf we know f~λPh

( f-1Ph\f\pdμ=

\ hf-^\f\*dμ

is Jo-measurable.

Thus

(AeΣQ).

Choose g e &(P) such that A = S(g). By Lemma 3.1(i), u = Jgf e P). Suppose p = 1 and 0 ^ ke L,. By Lemma 2.1(ii) and (iii), ί ksgnf'f-ί\f\dμ=

\

kdμ = \\Juk\\ = \\P(ksgnu)\\

= ( tιP(Jaksgnf)

\f\dμ.

JA

Putting v = f — u = / — Jgf G &(P), we have, by Lemma 2.1(1), P(k sgn /) - JuP(Juk sgn /) + J,P(Jwfc sgn /) . Hence ( ΓiP(J9ksgnf).\f\dμ= JA

We conclude that

\ f'Pik }A

sgn

f)-\f\dμ.

30

S. J. BERNAU AND H. ELTON LACEY

\hf-1

\f\dμ=

JA

\ Γ JA

11

for all hef and all AeΣ0 so we are finished for p = 1. If p > 1 we have PJg - JgP by Lemma 2.3(ii) and \f\p~ι sgn/e by Lemma 2.2. Hence,

\ PJMfl'-'agnfdμ ix

= ( f-ιPh

\f\'dμ

(

Thus 1

-)

(h e

as claimed. Our theorem has useful consequences. THEOREM 3.5. Suppose l^p ^ ( X , I , ^) be a linear isometry with range M. Suppose a, be LP{Ω, Ξ, λ) and \a\ Λ |δ| = 0, we claim that I Ta\ A | Tb\ = 0. This is essentially proved by Lamperti [6], Since | α | Λ |6| - 0, \\a + δH' + \\a - b\\p = 2\\a\\p + 2||δ|Γ Since T p

p

p

is a n i s o m e t r y , \ \ T a + Tb\\ + \\Ta - Tb\\ = 2\\Ta\\

p

+ 2\\Tb\\ .

Since

p Φ 2, the equality condition for Clarkson's inequality [6, Corollary 2.1] shows that | Tα| Λ | Tb\ = 0. Take a maximal subset of Ξ consisting of sets of nonzero finite

THE RANGE OF A CONTRACTIVE PROJECTION ON AN Lp-SPACE

33

λ-measure which intersect pairwise in sets of λ-measure zero and let 5$Γ be the corresponding set of characteristic functions. Let a e SΓ and suppose BeΞ and Bc:S(a). Write b = χB, then T(a - b), Tb are disjoint in M so we have Tb — \ Tb | sgn Ta. This extends to non11 negative simple functions δ in α and then to all nonnegative be 11 a . Define U: LP(X, Σ, μ)~*Lp{X, Σ, μ) to be the direct sum of the unitary multiplications sgn Ta{a e ^%Γ). It is easy to see that U is an isometry of M such that UT is positive and hence UM = UT LP(Ω, Ξ, λ) is a closed vector sublattice of LP(X, Σ, μ) (compare the proof in Lemma 3.3 where we showed that functions in Mf were immeasurable). Assume (iii) and let Σo be the set of supports of functions (whose equivalence classes are) in M. Then ΣQ is a σ-subring of Σ. (If (fn) is a sequence in M, S(fn) = S(Ufn) = S(\Ufn\) so

Ίif,ge M, Jg = JUg; Jg\Uf\ = \im\Uf\ Λn\Ug\e UMand S(f) ~ S(g) = S(U"\\Uf\ - Jg\Uf\)).) Let f,geUM and suppose / is real, g ^ 0 and / e g11, then {ί 6 X: (f/g)(t) > a} = S((f - ag)+) e Σo. Thus f/g is 2Ό-measurable. This extends to all fe UMf] g11 and hence Jgf/g is 2Ό-measurable if /, g e UM and g ^ 0. This now extends to all f,ge UM and, since U~ιJgfjU~ιg — Jgf/gf we have f/g, Immeasurable 1L for /, geM and feg . It follows that M is the set of all elements in LP(X, Σ, μ) which can be written in the form hf with h, Jo-measurable and f e M. (If h = χs{g) with ge M, hf — JJ — 1 U^JUgUfe U~ (UM) = M.) Let J be the band projection on ML1, let he LP(X, Σ, μ), choose feM such that Jh — Jfh, (such an / exists by the arguments used in Corollary 3.2) and define Ph =

f&(Σf>\f\')(hf-1).

Then Phe M and this definition is independent of the choice of / in M such that hef11. To see this suppose geM and hegλL. Then h is zero outside S(f) n S(g)eΣ0 and so is &(ΣOf I/ΓXΛ/""1), ^-almost everywhere. Let B = S(f) n S(g), then ft = χBf e M and 1

1 / \*dμ = ( ft/"11 / Γ # - ί hfT11Λ \pdμ (A e ΣQ), v

1

so that fί?(Σ0, \f\ ){hf- ) = f^(Σ0, sume S(f) = S(g). Now

ι

| Λ |')(λ/r ).

Thus we may as-

eLtX, Σo, \g\*μ) ,

34

S. J. BERNAU AND H. ELTON LACEY

so we have, for A e Σo,

p

ι

g\'^(Σ,, \f\ )(hf- )\fγdμ . Because g'1/ and f~ιg are Immeasurable and the integrals are finite, the second integral is \g-1f\Γ1g\PhΓ1\f\pdμ - \hg-ι\g\'dμ. Thus and our definition of Ph is unambiguous. If hlf h2 e Lp we can take / e M such that Jh1 = Jfh1 and Jh2 = Jfh2. Thus P is linear. Since f-iph = &(Σ09 ifΠihf-1) we see P 2 = P. Finally, if p > 1, write p u= &(Σ0, \f\ )(hr% we have \\Ph\\l= ^uΓ'sgnΰ.&ίΣo,

v

ι

v

\f\ ){hΓ )\f\ dμ.

Since ^ is J0-measurable, this is ^ul^sgnΰ

hf-'lfl'dμ

= [\Ph\v~ι$gnfΰ hdμ £\\\PhΓ%\\h\\, = \\Ph\\y\\h\\9.

(We used Holder's inequality and q for the conjugate index to p.) We conclude that \\Ph\\P g \\h\\P. Since Ph = h(h e M) we have shown that M is the range of the contractive projection P. REMARK 4.2. The results (iii) implies (i) (with the same proof) and (i) is equivalent to (ii) are valid if p = 2; in fact (i) and (ii) are equivalent for any Hubert space. If we assume the projection P, is positive as well as contractive the proof in Lemma 3.3 that Mf is a lattice shows ,^{P) is a sublattice of L2 and Theorem 4.1 is valid for L2 with the projection and the isometry both required to be positive and in (iii) M required to be a closed vector sublattice. We use this remark in our next result. COROLLARY 4.3. If M is a closed vector sublattice of Lp (1 ^ p < co) then M is the range of a positive contractive projection.

THE RANGE OF A CONTRACTIVE PROJECTION ON AN L^-SPACE

35

Proof. Clearly M satisfies condition (iii) with U = I. In the definition of Ph we may always choose a positive f e M such that hefλl. Positivity of P follows from positivity of conditional expectation. REMARK 4.4. In the introduction we referred to Rao's paper [8] and claimed that its treatment of contractive projections contained errors. In particular, his Theorem II. 2.7 asserts that if M is the range of a contractive projection P on a Banach function space LP(Σ) there is, under suitable conditions, a unitary multiplication U such that UPU~ι is a positive contractive projection. The conditions are all satisfied if M is the subspace of Ϊ2(3) = C3 spanned by (1, 1, 1) and (1, 2, —3). Rao's theorem now claims the existence of a unitary multiplication, say by u — (λ1? λ2, λ3), such that uM is a vector sublattice of C3. This is impossible, as we show. First, uM contains the elements (0, λ2, — 4λ3), (λlf 0, 5λ3), and (4λlf 5λ2, 0). If Re λ2λ3 = 0 we have λaλg"1 = λ2λ3 = ±i and uM contains Im(0, λ2λ3, -4) = (0, ± 1 , 0); so that (0, 1, 0)euM, and uM = Cz. If all Reλp^ Φ 0 (i Φ j), then uM contains Re(0, 1, — 4λ3λ2) and Re(l, 0, δλgλj; hence, taking a multiple of their infimum, (0, 0, 1) e uM and again uM = C3. Exactly the same counterexample vitiates the proof of Rao's Theorem II. 2.8 see p. 177 lines -15 to - 1 1 . The error in both cases seems to be the reduction of the general case of LP(Σ) to the Lx situation. Vital to this reduction, but invalid, is the assertion that if LP(Σ) c Lι(Σ, G) and || || lfff ^ p( ) then a contraction on LP(Σ) for the p-norm can be extended to the closure of LP(Σ) in U{Σ, G) with the 1, G-norm and that the extension is contractive for the 1, G-norm.

5* The theorem of Lindenstrauss, Pelczynski, and Zippiru We begin by recalling some definitions. If E, F are isomorphic Banach spaces, d(E, F) = inf {||L|| U-Z/"1!): L is a linear isomorphism between E and F}. A Banach space E is an £fPtX space (for 1 1. As outlined in the introduction we discuss some details of the proof for the complex case. Observe first that (3) is a trivial consequence of (1). Simply identify E with LP(X, Σ, μ) and take for % the subspaces spanned by finite sets of (pih power)-integrable characteristic functions.

The proof that (3) implies (2). This result is certainly part of the folklore. It can be obtained quite efficiently as follows. LEMMA 5.2. Let xl9 *"9xn be n linearly independent elements of a normed space E then there exists ε > 0 such that if yt e E, and \\Xi — Vi\\ < ε(i — 1, 2, , n) then {yu , yn} is a linearly independent subset of E.

Proof. (This is standard but our proof may be novel.) Let K denote the scalar field and S the unit sphere in Kn, S = {XaKn: \\X\\ = 1}. The map g: S x E* — E defined by g((\, , λ j , (y19 -, yn)) \Vι + + ^nVn is continuous. By linear independence, the compact set S x (xlf •••,#„) does not meet the closed set ^ ( O ) . Hence there are open neighborhoods Ut of xif i = 1, , n, such that (S x Uι x ••• x ί / J ί l ^ ( O ) = 0. If yt e Ui(ί = 1, , u) it follows that {y19 • - , Vn) is linearly independent. LEMMA 5.3. every λ > 1.

Let E be a Zv-space, then E is an JzfPj-space for

Proof. Let F be a finite dimensional subspace of E. Let {xlf • , xn) be a basis for F, such that H^H = l(i = 1, , n). Let xf, • , x* G E* be such that x}(xj) - δij9 and let M = Σ?=i IK* H Choose ε > 0 such that Me < 1 and | | ^ — yt\\ < ε for i = 1, , n implies that {yl9 •••,!/»} is linearly independent. By the Zp-hypothesis there is a finite dimensional subspace H of E and points yί9 --,yn in H, such that H is isometrically isomorphic to ^(dim H), and \\%i — yΛl < ε(ΐ = 1, , w). Then {^/i, •••,!/»} is a linearly independent subset of

THE RANGE OF A CONTRACTIVE PROJECTION ON AN LP-SPACE

37

H. If Σ aiVi e ή i=i

i=i

then

Σ 3=1

=Σ 3=1

Since itfε < 1 we conclude that at = 0 for each ΐ. Thus we can extend 2/1, , 2/» to a basis, say ylf , # Λ , α?n+1, •••,#*, of H with the property that {xn+u -, xp) c fl?=i ^ ( « * ) . Let G be the subspace of E spanned by xu , xn, xn+1, , av Then F c G . If y = Σ ? = 1 aiVi + JJ=n+1 a,x, e Hdefine Ty = Σ?=i«A + Σf=n+i ^ A e G. We have

111/-

Σ This gives (1 - Me)\\ Ty\\ ^ \\y\\ ^ (1 + ikZε)|| Ty\\(ye H); so that T ι is an isomorphism between F and if such that || Γ|| || T~ \\ S (1 + Me)l(l - ikfε). If λ > 1 we can choose ε such that (1 + Me)/(I - Me) < λ. Thus E is an ^^-space for all λ > 1. The proof that (2) implies (1). Here the plan is first to embed E, isometrically, in an Lp-space, and then to use the theory of contractive projections of Lp-spaces. This is carried out in detail for the real reparable case in [7] and for the real nonseparable case in [5] The generalizations to cover the complex case are mostly obvious. For 1 < p < 00 our Theorem 4.1 is used. For p = 1, it follows as in the real case that E* is a ^ space whence by the complex version of Grothendieck's theorem [9] E is an Lλ{μ) space. There is an aspect of the construction which needs a little elaboration. At one stage of the proof we have a complex vector space, say V, consisting of complex valued functions on a set U. V is a vector sublattice of the space of all complex functions on U. There

38

S. J. BERNAU A N D H. ELTON LACEY

is a seminorm π on V such that π(f) ^ π(g) whenever | / | ^ \g\f and + #)*> = 7r(/y + π(g)p whenever | / | Λ |flf| = 0. We then need to π(f embed the quotient V/N, where N = {f e V:π(f) = 0}, isometrically in a concrete, complex, Lp-space. For this, let VB and NR denote the spaces of real-valued functions in V and N respectively. The quotient VB/NR with the norm induced by π is then linearly and lattice isomorphic, and isometric, to a vector sublattice of real LP(X, Σ, μ) just as in [7]. Let hL denote the composition of the quotient map UR —•> VBjNR and the isometric isomorphism into real LP(X, Σ, μ). Then h,1 is a linear and lattice homomorphism and ||/&i/|| = π(/)(/ £ VR). We construct the required embedding of V/N into complex LP(X, Σ, μ) by defining h(f + N) - ^(Re /) + i^ίlrn /) . Then h is clearly well defined. need the next lemma.

To verify that h is an isometry we

LEMMA

5.4. The map h constructed above satisfies h\f\ — \hf\,

Proof.

F o r a n y r e a l θ \f\ ^ R e (e f) s o

iθ

iθ

iθ

h\f\ = hlfl ^ h^Ree*'/) - Reh(e f) - Ree hf . Hence h\f\ ^ |fe/|. For the converse, let ft) be a complex wth root of unity and observe that for any complex z max {Re ωrz: r = 1, 2,

, n) ^ cos (τr/w) 121 .

Hence, cos (7r/w)λ I /1 ^ &(sup {(Re ωrf): r = 1, = sup {Re ωr/^/: r = 1,

, n})

Letting n—> co we have ft|/| = |fe/| as required. This completes our discussion of the proof of Theorem 5.1. We add a comment. It seems that a more elementary proof that a space which is an ^ >rspace for all λ > 1, is an Lp(μ) space, should be possible. Certainly the result should not depend on the entire theory of contractive projections for such spaces. Indeed if p = 2 the ^f2j condition already implies the parallelogram law and this makes the space a Hubert space. For p Φ 2 we can see that the Clarkson inequalities are valid and these with enough finite dimensional lpsubspaces might give a more elementary proof.

THE RANGE OF A CONTRACTIVE PROJECTION ON AN LP-SPACE

39

6* Appendix* We prove two technical results used in [1], [10]. The first is also an extension of that in [1]. LEMMA 6.1. [1]. Suppose 0 < p < co and let M be a closed subspace of LP(X, Σ, μ). If (fn) is a sequence in M, then there exists f e M such that S(f) = U~=i S(fn). In particular if μ is finite or M is separable there exists f e M such that Jf = JM .L i that is, f is a function in M of maximum support.

Proof. If f,ge Lp and a is a scalar, the zero sets {t e I : (/ -f ag)(t) = 0} have disjoint intersection with S(f) U S(g) for differing values of a. Since S(f) U S(g) is σ-finite, μ(S(f) U S(g) ~ S(f + ag)) = 0 except, perhaps for countably many values of a. Assume, as we may, that \\fn\v = 1 for all n. We define, inductively, two sequences {an), (en) of positive real numbers such that, if we write gn = aj, + -. - + ajnf An = {te X: \gn(t)\ £ εn}, and Bn = {teX:\an+1fn+1(t)\^en/2}, then ( i ) an+ί < 2 - " and εn+1 < eJ2; (ii) μ(S(gn) U S(fn+1) - S(gn+1)) = 0; (iii)

\

\fi\pdμ)

Thus | / ( 0 | - Hm,^ \gk(t)\ ^ |

Vol. 53, No. 1

1 11 21 43 59 73 85 95 109 117 131 137 153 163 171 189 195 203 207 211 217 229 239 253 259 267 273 281 301 307

March, 1974

PACIFIC JOURNAL OF MATHEMATICS EDITORS RICHARD ARENS (Managing Editor)

University of California Los Angeles, California 90024

R. A.

J. DUGUNDJI Department of Mathematics University of Southern California Los Angeles, California 90007 D. GlLBARG AND J. MlLGRAM Stanford University Stanford, California 94305

BEAUMONT

University of Washington Seattle, Washington 98105

ASSOCIATE EDITORS E. F. BECKENBACH

B. H. NEUMANN

F. WOLF

K. YOSHIDA

SUPPORTING INSTITUTIONS UNIVERSITY OF BRITISH COLUMBIA CALIFORNIA INSTITUTE OF TECHNOLOGY UNIVERSITY OF CALIFORNIA MONTANA STATE UNIVERSITY UNIVERSITY OF NEVADA NEW MEXICO STATE UNIVERSITY OREGON STATE UNIVERSITY UNIVERSITY OF OREGON OSAKA UNIVERSITY

UNIVERSITY OF SOUTHERN CALIFORNIA STANFORD UNIVERSITY UNIVERSITY OF TOKYO UNIVERSITY OF UTAH WASHINGTON STATE UNIVERSITY UNIVERSITY OF WASHINGTON * * * AMERICAN MATHEMATICAL SOCIETY NAVAL WEAPONS CENTER

Printed in Japan by Intarnational Academic Printing Co., Ltd., Tokyo, Japan

PACIFIC JOURNAL OF MATHEMATICS Vol. 53, No. 1, 1974

STRONGLY UNIQUE BEST APPROXIMATES TO A FUNCTION ON A SET, AND A FINITE SUBSET THEREOF M. W.

BARTELT

Let X be a compact Hausdorff space and let C(X) denote the space of continuous real valued functions defined on X, normed by the supremum norm ||/|| = m3Lxχeχ\f(x) |. Let M be a finite dimensional subspace of C(X). This note examines the problem of whether every best (unique best, strongly unique best) approximate to / on X is also a best (respectively: unique best, strongly unique best) approximate to / on some finite subset of X. Appropriate converse results are also considered.

The Kolmogorov criterion for best approximates shows that π e M is a best approximate to / on X if and only if it is a best approximate to / on a finite subset of E π = { x e X: \f(x) - π(x)\ = \\f - π\\} .

Example 1 shows that the corresponding result does not hold for unique best approximates. It can easily be shown that when π is a strongly unique best approximate to / in C[a, b] from a Haar subspace then there is a finite subset A of [α, 6] such that π is a strongly unique best approximate to / on A. In Theorem 2 the latter result is extended to an arbitrary finite dimensional subspace M of C(X) and in Theorem 3 a converse is proven in this general setting. The second algorithm of Remez [11] is an important method for the computation of the best approximate to a function / in C[a9 b] from a finite dimensional Haar subspace. This algorithm depends on the fact that a best approximate to / on [a, b] is a best approximate to / on some finite subset of [α, 6]. (One can think of the algorithm as a search for this subset.) In fact, the proof of the convergence of the algorithm given by E. W. Cheney [3] indicates that the algorithm depends more precisely on the facts that the best approximate π to / on [α, b] is strongly unique and that π is also a strongly unique best approximate to / on some finite subset of [α, b]. It would also be natural to consider In Lp[a, b] for 1 :g p < co the relationship between strongly unique best approximates on [a, b] and on finite subsets of [α, 6]. However, D. E. Wulbert ([15], [16]) has shown that strong unicity does not occur (nontrivially) in any smooth space and Lp[a, b] for 1 ^ p < co is smooth. In the last section a different proof of Wulbert's result is given because the

2

M. W. BARTELT

method of the proof enables one to study strong unicity in IΛ It should be observed (see Example 3) that even though there are no finite dimensional subspaces of &[a, 6] containing a unique best approximate to every / in U[a, 6], a given / in &[a, b] may have a strongly unique best approximate. The result mentioned above on the relationship between the best approximates to / on X and the best approximates to / on a finite subset of X can be found in [8], [13], and [18]. The results of this note hold with obvious modifications for the complex case. 2* DEFINITIONS. An element π in M is a best approximate to / in C(X) if | | / - m\\ ^ \\f - π\\ for all m in M; π is a unique best approximate if the inequality is strict for all m in M, m Φ π; and π is a strongly unique best approximate to / if there exists a real number r > 0 such that \\f - m\\ ^ | | / — π\\ + r\\π — m\\ for all m in M. Let M have dimension n. The subspace M is called a Haar (Chebyshev) subspace if no nonzero function in M has more than n — 1 zeros in X. If X is the finite interval [α, 6], then Mis called a weak Chebyshev subspace if no nonzero function in M has more than n — 1 sign changes on [α, b]. (For properties of Haar and weak Chebyshev systems, see e.g. [4], [5], [6], and [17].) In particular it is known that if If is a Haar subspace of C[a, b] then π is a best approximate to / on a closed set X in [a, b] (where X contains at least n + 1 points) if and only if there exists an equioscillation set for / — π, i.e., a subset A of X containing n + 1 points xί < x2 < • < xn+ί such that f(xi+ι) - π(xi+1) = - [f(xτ) - π(xt)], i = 1, 2, ,n and 1/(0?,) - π(x 0 such that THEOREM.

There exists a real

11/ - m|| ^ | | / - π\\ + r\\π - m\\VmeM if and only if max [f(x) — π(x)]m(x) > 0 Vm e M,

m ^ 0.

In proofs we assume without loss of generality that the best approximate to / is 0.

STRONGLY UNIQUE BEST APPROXIMATES

3

3* Results* The relationship between a strongly unique best approximate to a given / on [a, b] and on a finite subset A of [a, b] is especially simple when M is a Haar subspace. Recall that when M is a Haar subspace of C[a, b] every / in C{X), where X is a compact subset of [α, 6], has a strongly unique best approximate from M [9]. Hence by the strong Kolmogorov criterion we have the following, result: THEOREM 1. Let π be a best approximate from the Haar subspace M of C[a, b] to a given f in C[α, 6]. Then for every equioscillation set A S Eπ,

max [f(x) — π(x)]m(x) > 0 Vm e M, m =£ 0 . xe A

If we only assume that π is a strongly unique best approximate from a weak Chebyshev subspace, then the conclusion of the previous theorem does not hold. For example, in C[0, Aπ] let f(x) = sin x and let M be the linear span of ίSπ/2 - α 0 g x g 3ττ/2 flr(α) - I 0 3ττ/2 ^ α? ^ 5τr/2 (5τr/2 - a; 5ττ/2 ^ a; ^ 4ττ . Then 0 is strongly unique to / since md»xxeEQf(x)m(x) > 0, VnieM, m Φ 0, but m&xxeA f(x)(- g(x)) = 0 where A = {5ττ/2, 7ττ/2} is an equioscillation set for / — 0. However, we now show that when π is a strongly unique best approximate from an arbitrary subspace M in C(X), it follows that there does exist some finite subset A of Eπ such that π is a strongly unique best approximate to f on A. THEOREM 2. Let π be a strongly unique best approximate from a subspace M of C(X) to an element f in C(X). Then there exists a finite subset A of Eπ with ^ 2n points such that

max [f(x) - π(x)]m(x) > 0 Vm e M, m/A =£ 0 . xeA

Proof

Let M be the span of {glf

, gn}.

Let Eo = {(f(x)gί(x)9

--, f(x)9n(oή):xeE0}. Then it follows ([2], Theorem 6) that 0 is in the interior of the convex hull of Eo. Hence (see e.g. Theorem 3.13 in [14]) 0 is in the interior of the convex hull of A, where A is a finite subset of EQ consisting of g 2n points. It follows ([2], Theorem 6) that 0 is a strongly unique best approximate to / on A. By the strong Kolmogorov criterion max, e i f(x)m(x) > 0 for all m in M with m/A =£ 0.

4

M. W. BARTELT

It is not known in general whether it is possible to find a finite set A satisfying the conditions of the previous theorem such that if m is in M and m/A = 0, then m = 0. However, if Eπ is finite then by setting A — Eπ one can add to the conclusion of Theorem 2 that m/Eπ == 0 implies m = 0. This follows from the strong KoImogorov criterion. Also if Eπ is not finite but it is known that any nonzero function in M has at most N — 1 zeros for some integer N (for example N = n when M is a Haar set), then one can just add to the set A of the previous theorem enough points of Eπ so that A has N or more points. It would be of interest to determine whether the 2n of the theorem is in general best possible. If π is a unique best approximate to / on X> then it does not follow that π is a unique best approximate to f on Ez. This can be seen in the next example which will also be used later. EXAMPLE

1.

Let M be the subspace of C[0, 3π ] spanned by

g,(x) = 1 and 'π — x

0

0

π ^ x g5τr/2

βπ/2 - x

5τr/2

^ 3π .

Let f(x) = sin x. Then M is a weak Chebyshev system, but it is not a Haar set on [0, 3τr], Because f(x) has a horizontal tangent at x = 5ττ/2, the function —g2(%) is not as good an approximate to f(x) as 0 is. Clearly then, 0 is a unique best approximate to / on [0, 3ττ], Now EQ = {τr/2, 3τr/2, 5ττ/2}. Since M has dimension 2, 2?0 is an equioscillation set for / — 0 on [0, 3ττ]. Now 0 is not a unique best approximate on EQ = A since g2(x) is also a best approximate. Also observe that 0 is not a strongly unique best approximate to / on [0, 3ττ] since maxxe^0 /(«)[-flr2(»)] = 0. In fact even more holds. Let a? --π/2

0

x -- π

0 ^ a? ^ π/2

τr/2 ^ α; ^ π 7Γ ^ a; ^ 3τr/2

2(7ττ/4 - α?)

3ττ/2 ^ a? ^ 7π/4

,χ -- Ίπ/A

7τr/4 ^ a; ^ 3τr .

Then let M be the subspace of C[0, 3π] spanned by &(&) and gz(x), and let /(x) = sin x. Then by consideration of the values of any me M at points τr/2, 3ττ/2, and 5π/2, it is easy to verify that zero is a unique best approximate to / on [0, 3τr] and Eo = {π/2, 3π/2, 5ττ/2}.

STRONGLY UNIQUE BEST APPROXIMATES

5

Moreover on each subset A of E09 there is a function g e M such that g/A =£ 0 and g is a best approximate to / on A. Thus zero is not a unique best approximate to / on any finite subset A of Eo. The next proposition summarizes the results for an arbitrary subspace M of C(X). For the result on best approximates see [8], [13], and [18]. PROPOSITION. If π is a best (strongly unique best) approximate to f on X, then there exists a finite subset A of X with less than or equal to n + 1 (resp. 2ri) points such that π is a best (strongly unique best) approximate on A.

The Kolmogorov and strong Kolmogorov criteria and Example 1 also yield the relationship between the best approximate to / on X and on all of Eπ. As expected, π is a best (strongly unique best) approximate to / on X if and only if it has the same property on Eπ. This does not hold for a unique best approximate. REMARK.

4* Converse results* of the next theorem.

The Kolmogorov criterion shows part (i)

THEOREM 3. (i) If π is a best approximate to f on a finite subset of Eπ, then π is a best approximate to f on X. (ii) If π is a unique (strongly unique) best approximate to f on a finite subset A of Eπ, then π is a unique (strongly unique) best approximate to f on X, except possibly for those m in M with ml A = 0. In fact more than this holds. The following result says that if π is a unique best approximate to / on a finite subset A of X, then π is also a strongly unique best approximate to / on A. THEOREM 4. Let π be a unique best approximate to f on a finite subset A of X. Assume f(x) — π(x) Ξ£ 0 on A. Then

max [f(x) — π(x)]m(x) > 0

Vm Ξ£ 0 on

A.

xeΛ

Proof. (We show that if m a x ί 6 i f(x)q(x) ^ 0 for some qeM, then there exists a real number λ > 0 such that — Xq is a best approximate to / on A.) Let A' = {xeA: f(x)q(x) < 0}. Let λ > 0 be such that both the following hold: ( 1 ) λmax. β ^|?(a?)|2

^ Σ l/(i)l + r{\π(ί) - m(l)| + |τr(2) where one can choose r = 1 to be the strong unicity constant. 1 The space L contains a finite dimensional subspace M which contains a strongly unique best approximate to every element fe 1 L - M if and only if (Γ, Σ, μ) contains an atom ([1], [10]). To obtain further information about strong unicity in L\ let f e L\ \\f\\ = 1 and f £ M. Assume without loss of generality that 0 is a best approximate to / and let £?Q - {L e *: Lf = 1 = ||L||}. For a

8

M. W. BARTELT

given L e ^f0, there exists by the Riesz Representation Theorem a function he L°° such that 1

and

Lg =[ hgdμVgeL Thus for a given L e ^ (1)

1 = \hfdμ

we have £ \\h\ \f\dμ

£

L = 1.

The condition for equality in Holders inequality implies that \h\ \f\ = || AIU I/I = | / | a . e . Also (1) shows that hf = \h\\f\ a.e. Thus jδ^ can be identified with {heL00:

\f\(\h\ - 1) = 0 a.e.

and

(Λ/)(l-sgnΛsgn/) - 0 a.e.} .

This characterization of ^ can be used to study strong unicity in L 1 . For example if μ{x: f(x) = 0} = 0, then \h\ = 1 a.e., sgn A sgn / = 1 a.e. and therefore h is uniquely determined a.e. Since J*fo contains a unique element it follows as before that 0 is not a strongly unique best approximate to / . We have shown the following: THEOREM 6. Let f in L\T, Σ, μ) have a strongly unique best approximate π from a subspace M. Then μ{x: f(x) — π(x) = 0} > 0.

It should be pointed out that it is possible for an element f e L1 to have a strongly unique best approximate from a subspace M even when (T, Σ, μ) does not have an atom. It is not known whether a result like Theorem 2 exists for Lι[a, 6], 3. Let

EXAMPLE

Z/[-2, 2].

Let M be the constant functions, a subspace of 'x + 1 0

- 2 g . t g -1

ί» — 1

1 ^ x^ 2.

Then one can verify that

11/ — cjf α =

(\c+lY 4|c|

Thus 0 is a best approximate to / and also

11/ - c l l ^ 11/11, + 1/2Hell,.

STRONGLY UNIQUE BEST APPROXIMATES

9

REFERENCES 1. D. A. Ault, F. R. Deutsch, P. D. Morris, and J. E. Olson, Interpolating subspaces in approximation theory, J. Approx. Theory, 3 (1970), 164-182. 2. M. W. Bartelt and H. W. McLaughlin, Characterizations of strong unicity in approximation theory, J. Approx. Theory, 9 (1973), 255-266. 3. E. W. Cheney, Introduction to Approximation Theory, McGraw Hill, New York, 1966. 4. A. Haar, Die Minkowschische Geometrie und die Annaherung an stetige Funktionen, Math. Annalen, 7 8 (1918), 294-311. 5. R. C. Jones and L A. Karlovitz, Equioscillation under nonuniqueness in the approximation of continuous functions, J. Approx. Theory, 3 (1970), 138-145. 6. S. Karlin and W. J. Studden, Tschebycheff Systems: With Application in Analysis and Statistics, John Wiley and Sons, Inc., New York, 1966. 7. A. N. Kolmogorov, A remark on the polynomials of P. L. Cebysev deviating the least from a given function, Uspehi Mat. Nauk., 3 (1948), 216-221 (Russian). 8. G. G. Lorentz, Approximation of Functions, Holt, Rinehart, and Winston, New York, 1966. 9. D. J. Newman and H. S. Shapiro, Some theorems on Cebysev approximation, Duke Math. J., 30 (1963), 673-681. 10. R. R. Phelps, Uniqueness of Hahn-Banach extensions and unique best approximation, Trans. Amer. Math. Soc, 95 (1960), 238-255. 11. E. Ya. Remez, General computational methods of Tchebycheff approximation, Kiev (Russian), (1957), AECT No. 4491. 12. T. J. Rivlin and E. W. Cheney, A comparison of uniform approximations on an interval and a finite subset thereof, J. SIAM Number Anal., 3 No. 2 (1966), 311-320. 13. T. J. Rivlin and H. S. Shapiro, A unified approach to certain problems of approximation and minimization, J. SIAM, (9) 670-699. 14. C. P. Valentine, Convex Sets, McGraw Hill, New York, 1964. 15. D. E. Wulbert, Uniqueness and differential characterization of approximation from manifolds of functions, Bull. Amer. Math. Soc, 7 7 (1971), 88-91. 16. _ , Uniqueness and differential characterization of approximation from manifolds of functions, Amer. J. Math., 9 3 (1971), 350-366. 17. J. W. Young, General theory of approximation by functions involving a given number of arbitrary parameters, Trans. Amer. Math. Soc, 8 (1907), 331-344. 18. S. I. Zuhovickii, On approximation of real functions in the sense of P. L. Chebysev, Uspehi Mat. Nauk., 11 (1956), 125-159 (Russian), AMS Translation, Series 2, 19 221-252. Received July 13, 1973 and in revised form September 21, 1973. This research was partially supported by N. S. F. Grant GU-2605. RENSSELAER POLYTECHNIC INSTITUTE

PACIFIC JOURNAL OF MATHEMATICS Vol. 53, No. 1, 1974

THEOREMS OF KOROVKIN TYPE FOR ^-SPACES S. J. BERNAU

Suppose (X, Σ, μ) is a measure space, 1 < p < oo, p Φ2, and that (Tn) is a net of linear contractions on (real or

complex) LP(X, Σ, μ).

Let M = {xeLp:

Tnx->x]

(M is the

convergence set for (Tn)). It is obvious that M is a closed siabspace of Lp; indeed this would be true for an arbitrary normed space. In this paper we shall show that M is the range of a contractive projection on Lp and hence is itself isometrically isomorphic to an Lp-space. If S c LP(X, Σ, μ) we can define the shadow, x for every net of linear contractions (Tn) such that Tny -> y for all yeS. We shall also give a complete description of ^(S) (for p Φ 1, 2, oo).

Our results are new for finite p not equal to 1 or 2. In the case p — 2 the assertions about Mare trivial and S^(S) is the closed subspace spanned by S. The case p = 1 was first considered by Wulbert [9] for Lebesgue measure on [0, 1], He showed that if S — {1, x, x2} then S^{S) = LJO, 1]. (Actually he considered sequences of contractions and required only Tnl —> 1 and Tnf weakly convergent to / for / = x and / = x2.) Wulbert's results were inspired by and generalized the classical theorem of Korovkin [7] which contains the result that if S = {1, x, of) then the shadow of S in C[0, 1] is C[0, 1]. In [8] Lorentz considered separable Lλ spaces on finite measure spaces. He showed that for sequences of contractions such that Tnl —* 1 the convergence set is a closed sublattice of Lx. A corollary of this, which he noted, is that for LJO, 1], S^(S) = L, if S = {1, x}. This last result and some further discussion of LX{X, Σ, μ), with μ(X) = 1 is also contained in [1], The methods we use are suggested by the methods used in [3] in discussing contractive projections. I am very grateful to Professors Lorentz and Berens for discussions of this material and for supplying me with preprints of [1], [2], [8]. My first introduction to this circle of ideas was a colloquium lecture by Professor Lorentz in which some of the results from [2] and [8] were presented. 2» The convergence set* We shall fix notation as in the first paragraph of the introduction. It does not seem to matter whether our measure space is taken over a cr-ring, σ-algebra or δ-ring. For definiteness we shall assume that Σ is a cr-ring and measurability is as defined by Halmos [5]. We shall let q be the conjugate index to p, defined by 1/p + 1/q = 1. Since p Φ 1, 2, o, the same is true 11

12

S. J. BERNAU

for q and Lq(X, Σ, μ) is the topological dual of LP(X, Σ, μ) with the usual identifications. We shall consider the complex case; i.e., Lp (and Lq) are (equivalence classes of) complex valued functions. The real case is a little easier, but the methods are the same. If T is a bounded linear operator on Lp, the conjugate operator T* is defined on Lq by the identity x (T*y)dμ = j (Γa?) ydμ (x eLP,ye

Lq) .

DEFINITION. The conjugate convergence set M* for the net of contractions (Tn) is defined by M* = {yeLq: T*y~>y}.

2.1. (Compare [3, Lemma 2.2].) Let xeLp, if and only if | x \v~ι sgn x e M*. LEMMA

then

xeM

Proof. Suppose xe M and write u — | x I2'"1 sgn x. Then \\u\\q ~ \\x\\pq and (T*u) is a bounded net in Lq. Let w be a weak-* limit point of this net. We have I x-(w — T*u)dμ = \ x-wdμ — \ (Tnx)-udμ = \x

> \ x wdμ — \x udμ

(w — u)dμ .

Taking a subnet such that T*u —> w (weak — *), we conclude that

Since the T* are contractions, || T * ^ | | g ^ \\u\\q = ||^||p / ? and hence II w||? ^ ll^l!?/9 Holders' inequality now gives 11 χ 11 v _ [ x . wdμ < 11 α; 11 ll^li ^ l l ^ l l llα l ^ ^ ^ l l α ll27. This gives equality throughout so [6, § 13.5] we have 1

ιv = j x l^" sgn x = u . Thus u is the unique weak-* limit point of the net (T*u). Since every subnet of (T*u) has a convergent subnet (by weak-* compactness of the unit ball in Lq), we see that T*u is weak-* convergent to u. Hence || u \\q S lim inf || T^u \\q ^ lim sup || T%u \\q £ \\ u \\q, because the T* are contractions; and we also have \\u\\ = lim || T*u ||. Because Lq is uniformly convex [4; 6, § 15.17] it follows that T%u—>u

THEOREMS OF KOROVKIN TYPE FOR LP-SPACES p

13

ι

in the norm of Lq, which gives | x \ ~ sgn x e M* as required. The same argument applied to Lq shows that if u = I x \p~ι sgn x e M* ,

x = \ u \q'x sgn ΰ e M** = M,

so we are done. We now apply differentiation arguments like those in [3, Lemma 2.3]. Recall that if z, w are complex, X is real and h(X) — \z + Xw\ then, if z + Xw Φ 0, h is differentiable at λ with h'(X) = Re [w sgn (z + λw)] . LEMMA 2.2. I x I sgn yeM.

(Compare [3, Lemma 2.3(i)].) // x,yeM,

then

Proof. Assume first that p > 2 and define, for λ e iϋ, and 0 < λ < 1, Zx = λ-χ[| a? + Xy \p-' sgn (a; + Xy) - \ x I*"1 sgn ^] - λ-χ[(| x + Xy \v~2 - I a? \p~2)(x + λ»)] + | a? | p - 2 ^ . Now, make a fixed choice of functions from the equivalence classes determined by x, y and observe that, except for the null-set where x or y is infinite, our differentiation result quoted above shows that as λ—>0, tλ —• (p - 2) I a; p" 3 Re [T/ sgn ^] x + | a; | p " 2 ^ at all points where x Φ 0. Also, since #> > 2, ^-->0(λ—>0) at points where x = 0. Let ^ denote this (almost everywhere) pointwise limit of ^ as λ->0. At points where 2 | Xy \ < |α?|, the mean value theorem gives a θ, 0 < 0 < 1 such that s* = (p - 2) I a? + θXy \p~3 Re fo sgn (x + 0λi/)](a? + λy) + | a? l Since | x |/2 < | a? + 0λi/1 < 2 | a? |, w e h a v e \zx\ ^ (p - 2)2l2?~31 |a?| p - 8 |2/| 2 | a ? | + p

^(2 (p-2)

\x\*-ι\v\

2

+

l)\xr \y\eLq.

At points where | x | ^ 2 | Xy |, we have

21 1

31 1

1

= (3 - + 2 " ) I y p" 1 λ Γ ^(S*-1 +

2

2p-1)\y\p-1eLq.

14

S. J. BERNAU

Thus the pointwise convergence of zλ to p

Zl

3

p

2

= (p — 2) I x \ ~~ x Re (y sgn x) + | x \ ~ y

is dominated by an element of Lq. The dominated convergence theorem then shows that \\zλ — z11| —•> 0. Now, by Lemma 2.1, ^ e M* and ilf* is closed. Hence zιeM*. Apply this result to a; and — ί?/ to see that p

3

p

2

z2 — (p — 2) |a?| " ^Re ( — ίy sgnx) + i \x\ ~ yeM* . Thus ^ — iz2 e M* and zλ — iz2 — (p — 2) j x \p~3x [Re y sgn x — ίlmy 2

p

sgnx]

2

+ 2 I # p~ i/ — p j α; \ ~ y . Use Lemma 2.1 again to conclude t h a t 11 x \v~2y \q~ι sgn (I x \p~2y) - | x l 1 "^-^ 12/ Γ 1 sgn y e M . L e t kn = \x\ι-{q-1)n\y\{q-1)n$gny. Observe t h a t for each n, kn+1 = |α;| 1 ~ ( 9 " 1 ) [ kn I 9 " 1 sgn kn. The argument we have just provided shows inductively that kne M for all n. Since 0 < q — 1 < 1, kn—*\x\ sgn 2/, //-almost everywhere, and clearly I kn I ^ max (| x [, 12/1) e L P . By dominated convergence again, \\kn — | x \ sgny \\p —>0 and hence, [ x I sgn ^/ G M as required. If 1 < p < 2, then by Lemma 2.1, x1 = {x^sgnx and yι = p 1 |^/| ~ sgn^ are in Λί*. Since g > 2, our proof above shows that x11 sgny^ = \χ Y'1 sgn y e M*. Apply Lemma 2.1 again to get I x I sgn y = \\χ p" 1 sgn y\q~ι sgn (|a; j^^sgn y) e M. For our next result we need some terminology from [3]. A

subspace N of LP(X, Σ, μ) is a vector suhlattice if for each xe N, (Re x)+ G N; this means that N is closed under taking real (or imaginary) parts and that the set of real functions in N is a real subspace and a sublattice of Lp. For a nonempty subset K of L^, the polar K- = {xe Lp: \x\ Λ\y\ = 0(^/ e K)}. A δα^d in L^ is a subset JSΓ such 11 that K = K (a band is necessarily a solid vector sublattice). If K is a band in Lp there is a natural direct sum decomposition Lv = K@KL and the associated projections are positive and contractive. We write Jκ for the band projection on K. If K — y11 (the only case we need) we write Jy for the band projection and note that Jy is multiplication by the characteristic function of the set on which y is nonzero.

THEOREMS OF KOROVKIN TYPE FOR LP-SPACES

2.3. If yeM and Jy is the associated JyMdM.

LEMMA

then

hand

15

projection,

Proof. Let x e M, then Jyx = 11 x | sgn y | sgn x and this element is in M by two applications of Lemma 2.2. LEMMA

+

2.4. If x,ye M, then (Re (x sgn y)) sgn yeM.

Proof. Suppose λ e R, λ Φ 0, then by Lemma 2.2, vλ = λ -1 (| ?/ + Xx I — I y I) sgn yeM. Since ^ = 0 at points where y — 0 we see that as λ —» 0, vλ converges pointwise (/^-almost everywhere) to (Re (x sgn y)) sgn y. Since I vλ I ^ I λ I"1 j I y + Xx I - 12/11 ^ I λ I' 1 1 y + λ α - y | = | x | , dominated convergence shows t h a t [| vλ — (Re (x sgn ^7)) sgn y \\p —> 0; so t h a t (Re (# sgn y)) sgn yeM. Another application of Lemma 2.2 gives I Re (x sgn y) \ sgn yeM and our results follows. THEOREM 2.5. The convergence set M is the range of a contractive projection on Lp.

Proof. For yeM, Lemma 2.4 shows that the map Uy: LP—>L P9 defined by Uyx — x sgn y, is norm decreasing, linear, and maps M onto a closed vector sublattice of Lp. Such a map was called a unitary multiplication operator in [3]. Choose, by Zorn's lemma, a maximal subset Y of M such that | y1 \ Λ | y21 = 0 if 2/i, 2/2 ^ Y a n ( ^ 2/i ^ 2/2 (a maximal pairwise disjoint subset of M). If f e Lp the set (ίGZ:/(ί) ^ 0} is cr-finite so the set {ye Y: Uyf ψ 0} is countable. Thus we can define the direct sum U of the unitary multiplications Uy(yeY) by Uf = Σver Uyf, and the defining sum has at most countably many nonzero terms and is convergent in Lp norm. Clearly UM is a closed vector sublattice of Lp. We show that U is isometric on M. Suppose x eM.xΦ 0. Let yl9 - - -, yn— - be an enumeration of the countable set of y e Y such that \y \ A \ x \ Φ 0. Let y0 = Σu^~n WVnW^Vn' Then yoeM and, by Lemma 2.3, JVoxeM. Hence x — Jyox e M and | x — JyQx \ A \ y \ = 0(2/ e Γ ) . By maximality of Y x — JyQx. Hence

« = Σ ^ and II .τ | p - Σ I I Λ * IP = Σ I! ^ IΓ = II K8II' yeY

yeY

yeY

It now follows by Theorem 4.1 of [3] that M is the range of a contractive projection on Lp and, which is an equivalent condition, that M is isometrically isomorphic to some LP(XO, Σo, μ0).

16

S. J. BERNAU

3* The shadow of subset S. In a certain sense characterization of shadows is trivial. Call a subspace M of Lp an exchange suhspace if \x\ sgn y e M for any x, y e M. Clearly an intersection of (closed) exchange subspaces is again a (closed) exchange subspace. Hence for a subset S of Lp we can determine the closed exchange subspace of Lp generated by S as the intersection of all closed exchange subspaces of Lp which contains S. THEOREM 3.1. If SaLP then the shadoiv, S^(S), of S is the closed exchange suhspace of Lp generated by S.

Proof. Lemma 2.2 shows that 6^{S) is a closed exchange subspace of Lp which contains S. If M denotes the closed exchange subspace of Lp generated by S a careful check of the proofs of Lemmas 2.3 and 2.4 and Theorem 2.5 show that these are valid for any closed exchange subspace of Lp. Hence M is the range of a contractive projection, say P, on Lp. Define a sequence (Tn) of linear contractions oΐ Lp by Tn = P(n = 1, 2 •)• Then M is the convergence set for (Tn) and hence, y ( S ) c I . This proves our theorem. As a corollary of the proof of Theorem 3.1 we digress to state the following result. THEOREM 3.2. Suppose l ^ p < ^ , pφ2, a subspace M of LP(X9 Σ, μ) is the range of a contractive projection if and only if M is a closed exchange subspace of LP(X, Σ, μ).

Proof. If M is a closed exchange subspace of Lp then just as in Theorem 3.1, Theorem 2.5 is valid for M. (This is equally true for p = 1 and p = 2 as can easily be checked.) By [3, Theorem 4.1J it follows that M is the range of a contractive projection on Lp. The converse result is the statement of [3 ? Lemma 2.3(i)] if p Φ 1 and an easy consequence of [3, Lemma 3.3] if p = 1. Returning to shadows we note that Theorem 3.1 is difficult to apply in practice. The following alternative seems a little more useful. Let έ%? be the smallest sub σ-ring of Σ such that the functions JyX/y are ^-measurable for all x, y e S. (To be precise here, we consider all choices of functions x, y in equivalence classes in S. The ratios are zero, by definition, wherever the denominators are zero.) Choose, by Zorn's lemma, a subset 3ίΓ of ^ x S which is maximal with respect to the properties: (i) if (A, y) e SΓ, then Aa{te X:y(t)Φ0}; (ii) if (A, y) e SΓ, μA > 0; (iii) if (Au yj, (A2, y2) are distinct elements of J?7 then μ{Aι Π A2) = 0. Define a measure λ on & by

THEOREMS OF KOROVKIN TYPE FOR L P -SPACES

AΠB

17

\y\'dμ.

Also define a map V: LP(X, ^ λ)-*LP(X, Σ, μ) by Vf = Σu,y)^f yχA (/ G LP(X, &, λ)). We note that the sets in & all have σ-finite ^-measure; that the sum defining Vf has, therefore, only countably many nonzero terms and is convergent in the norm of LP(X, Σ, μ). Furthermore, V is an isometry of LP(X, &, λ). THEOREM

3.3.

The shadow £S(S) = VLP(X, &, λ).

Proof. Let U be a direct sum of unitary multiplications such that U is an isometry of &*(&) and US^(S) is a closed vector sublattice of LP(X, Σ, μ). For f e Lp write T(f) = {t e X: f(t) Φ 0}, (we allow the ambiguity of sets of measure zero here); and let &s — {T(f):fe^(S)}. I claim that ^ is a sub σ-ring of Σ. For this, observe that T(f) = T(Uf) so we may assume that Sf(S) is a closed vector sublattice of Lp. Now T(f) = T(\ /1) = Γ(| Re / I) U Γ(| Im /1) and I / I 6 (S) so T(g) ~ T(f) = T(\ g | - / , | ^ [) e &B.

This

proves our claim about α > 0} = T((Re (Jyx - ay)) ) e &s. Hence, every Jyx/y (x, yeS) is ^-measurable and &s 3 &. Suppose B e & then B = T(fB) for some fB e £f{β). If λ(J5) < oo, then VχB = Έj(A,y)e.^y 'XBΠA- Since J5 is ^-finite we can enumerate the countable set of pairs (A, y) in 3ίΓ such that μ(B Π A) ^ 0 as (An, yn) and choose fn e S^{S) such that T(f%) = AnΓ\B. Then = Σ=-i Σ ~ i ^/l/* Since the An are disjoint and Vn\vdμ

= XB

w

n

the series for VχB converges in LP(X, Σ9 μ). Since each yneS and each f%e£S(S)9 each JfnyneS^(S) and F χ * G ^ ( S ) . Extending linearly to simple functions in LP{X, ^ μ) and then taking closure we conclude that VLP(X, ^ μ)a S*(S). If x e S and {A, y) e ^Γ then χAxjy is ^-measurable and

18

S. J. BERNAU

Hence z = Σ u ^ e j r XAΦ e LP(X, &, λ) and x = Vze VLP(X, &, λ).

This shows that VLP(X, ^ λ) z> S. If/, g e LP(X, ^ λ), then | f\ sgn g e LP(X, sup {||χS(/)fe||p: / e l ) ,

We omit the remaining details. 1.2. This lemma can be strengthened, in case M is closed, to say that for each he Lp there exists f e M such that Jh = Jfh = %s(f)h. This depends essentially on the fact that the set of supports of functions whose equivalence classes are in M is closed under countable union. This is proved by Ando [1, Lemma 3] for finite μ, and we give a rather easier alternative proof in our appendix. REMARK

2* Preliminary results* In this section the cases p = 1, and 1 < p < co, p Φ 2, are treated separately. Our first lemma is based on an argument of Douglas [2, p. 452]. LEMMA 2.1. Let P be a contractive projection on Lt(X, Σ, μ) and suppose f e &(P); then ( i ) PJf = JfPJ/, (ii) P(h sgn /) - I P(h sgn /) | sgn / (0 ^ h e LL); (iii) ||P(fcsgn/)|| =

Proof. Suppose 0 2, let λ e U, 0 < | λ | < 1, and let ge&(P). By Lemma 2.2, p

gλ — λ Wf + λ^| * sgn (/ + λ#) — Since p > 2,

0* = λ-^d/ + λflr|M - |/Γ 2 )(T + λsrΓ2 - l/I^XZ + Xfif)] + l/l"-2^ . Recall, that for real λ and complex w, z, d/dX | w + Xz 11; = Re [z sgn {w + λ«)], provided w + λz ^ 0. It follows that as X —> 0, Λ

3

(P - 2) I /1"" Re (fir sgn / ) • / + ! / Γ*9

at all points of X where / Φ 0. If 2|λff| < I/I we have |/|/2 < 1/ + ^λ^| < 2 | / | if 0 < θ < 1; and, by the mean value theorem there exists θ, 0 < θ < 1 such that | Re(firsgn(/ 2 ^ (p - 2)2i'-'|/|'-|flr| 2 I/I + | / Γ | ί / | If 2|λflr| ^ I/I, | / + λflr| ^ 3|λflr| and lfif;|^λ- 1 [(3|λflr|r i = (3"-1 + 2 p - ι )|flΊ'~ 1 |λ| The penultimate line above shows that gλ —> 0(λ —• 0) if / = 0. This shows that gr^ converges to g0 = (p - 2 ) | / r 2 s g n / R e ( s r s g n / ) + \f\'~'g , pointwise almost everywhere on X and that the convergence is dominated by an element of Lq. Hence \\gλ — g0\\g —* 0 and gr0ε &(P*) because ^ ( P * ) is closed. By the same argument, applied to — ΐfir, we have, using Re — iz = Im z, Ao = (P - 2) I / r" 2 sgn / I m ( | / | sgn# ^-almost everywhere on X, we have \\kn — \f\ sgn g|| p —>0 and since &(P) is closed |/1 sgn# e &{P) which proves (i) for p > 2. Suppose 1 < j> < 2; as we have already stated P* is a contractive projection on Lq, and q > 2. By Lemma 2.2, / t = I / Γ ^ s g n / and are in ^ ( P * ) . By our proof above |/i|sgnflr1 = ), and, by Lemma 2.2 again, \f\sgnge&(P). This completes the proof of (i). For (ii) we have by (i), that | /1 sgn Pk e &(P) (keLP). By (i) again, JfPk

= \Pk\ s g n ( I / I s g n Pk) e

Thus JfP — PJfP. Further, since P* is a contractive projection on Lq, and I/I'- 1 sgn feέ&(P*) we have J . P * - P*JgP* with

In addition Jg — J*, since J"^ and Jf are each multiplication by the same characteristic function. We conclude JfP = PJfP = (P*JfP*)* = (P*JgP*)* = (JgP*)* - PJf , which is (ii). (iii) The proof is like the proof of Lemma 2.1(ii). Suppose 0 ^ h ^ | / | . By (i), | / | sgn P(hsgnf) e &(P), so by Lemma 2.2, I /1*- 1 sgn P(h sgn /) e Hence, I P(h sgn /) I I / \*-*dμ - \P{h sgn /) -1 / Γ 1 sgn P(hsgnf)dμ = \^h sgn / I / Γ 1 sgn P{hsgnf)dμ

Also

THE RANGE OF A CONTRACTIVE PROJECTION ON AN L^-SPACE

27

Hence,

= j I P(Λ sgn /) + P((| /1 - h) sgn f)\\f

\^dμ

ι

ι

sgn /) | | / Γ d μ + j | P((|/1 - h) sgn /1 | / \'~ dμ

- h)\fΓ'dμ We have equality at each stage and hence, (/^-almost everywhere), /I = | P ( | / | s g n / ) | - \P(hsgnf)\

+ \f - P(h sgn / ) | .

This proves (iii) for 0 ^ h ^ | / | . The extension to 0 ^ fee LP is the same as in the proof of Lemma 2.1(ii) and (iii) so we are done. 3* Contractive projections and conditional expectations* In this section we describe the contractive projections on LP(X, Σ, μ) (1 g p < oo, p φ 2) in terms of conditional expectation. We first need the necessary σ-subring. LEMMA 3.1. Suppose 1 :S p < CXD, p ^ 2, απd ieί P be a contractive projection on LP(X, Σ, μ). Define ΣQ to be the set of supports of all functions whose equivalence classes are in &(P)\ then (i) PJJ = JJ (f,ge&(P)); (ii) Σo is a σ-subring of Σ.

Proof, (i) By Lemma 2.3(ii), (i) is valid if p Φ 1. We give a proof that uses only the identity JgPJg = PJg valid for 1 ^ p < ©o, ι p =£ 2 (Lemma 2.1(i) or 2.3(ii) weakened). Since f-Jgfeg and L1 Jgf - PJgfeg , we have

\\P(f -

= \\f- JJW

+ \\J,f -

Thus P/,/ - /,/ which is (i). (ii) By (i), S(f) - S(YL there exists f e &{P) such that 11 hef . COROLLARY

Proof. By Lemma 1.1 there is a sequence (/Λ) in &(P) such that h = \imn^χS{fn)h. Choose / e ^ T ( P ) such that S(f) = \J S(fn), 11 then hef . Observe now that if / e Lp the measure \f\pμ restricted to any σ-subring, Σo, of Σ, is finite. By the Radon-Nikodym theorem we may define the conditional expectation operator, &f = ^{ΣQj \f\v), for the measure \f\pμ relative to ΣQ. equation

ί h\f\'dμ

ί?f is uniquely determined by the

μ

JA

(AeΣ0)

v

for he Lλ{Xy Σ, \f\ dμ), and the condition that g*/fe is Jo-measurable. LEMMA 3.3. Suppose 1 ^ p < co, p φ 2; let P be a contractive projection on LV{X, Σf μ) and let Σo be the σ-subring of Σ, consisting of supports of functions in &{P). If Mf = f~ιJf&{P) = {f^J/Q- 9 € ^ ( P ) } then Mf - L9(S(f), Σo \ S(f), \f\pμ) where Σo \ S(f) = {AeΣ0:Aa S(f)} and we make the obvious identification of functions on S(f) and functions on X which vanish off S(f). In addition the map h—*f~Ίι is an isometric isomorphism between Jf&(P) and Lp(S(f), ΣQ\S(f), \f\pμ).

Proof. Observe that 1/1*7* is finite on S(f), and that the isometry claim is obviously true. If AeΣQ\S(f) then A = S(g) for some ge &(P). By Lemmas 2.1 and 3.1 (if p = 1) or 2.3 (if p > 1) we have Jgf = PJgf so that χA = f~xJgf e Mf. Let h be a simple function with respect to Σ0\S(f). Then heMf and hfe&{P). In addition )S(f)

\ h \ p \f\pdμ=

\\hf\'dμ.

We conclude that MfiiLP(S(f),ΣQ\S(f),\f\*μ). Conversely, let heMf, then h e Lp(S(f), Σ\S(f), \f\pμ) and it is enough to show that h is 2Vπieasurable. Let g = (Re /?)", then gfe Lp(Xf Σ, μ). By Lemma 2.1(ii) or 2.3(iii) P(9f) = P(\ Of I sgn /) = I P(| gf \ sgn /) | sgn / so ΓιP(gf)

= \f\-1\P(\gf\ sgn /) | e Mf. It follows that

THE RANGE OF A CONTRACTIVE PROJECTION ON AN LP-SPACE

29

+

Reh = /-\P((Re h) f) - /-'PtfRe h)~f) e Mf . Since each of these functions is nonnegative it is sufficient to consider 0 0 and put k = h V aχsιf). Arguing as above, we have fιP{kf) ^ λ and f~ιP{kf) ^ αχ S ( / ) so that f~ιP{kf) ^ k ^ 0. Since P is contractive we have p

HOT ^ lliW)ll = IIW) -kf + kf\γ This gives P(kf) = kf, so that & 6 Λf/. This shows, incidently, that Mf is a lattice. For our purpose, however, we have {ί G S(f): h(t) >a} = {te S(f): (k - aχS[f))(t)

Φ 0}

- S(kf - α/) e I'o . Thus Mf consists of immeasurable functions and we are done. THEOREM 3.4. Suppose

l ^ p < oo, p φ 2 α^id that

tractive projection on LP(X, Σ, μ). If f e &(P) and hef

P is a con1L

then

Ph = f&(ΣQ,\f\*)(hf-1). Proof. Since f~λPh we have only to show

e Mf we know f~λPh

( f-1Ph\f\pdμ=

\ hf-^\f\*dμ

is Jo-measurable.

Thus

(AeΣQ).

Choose g e &(P) such that A = S(g). By Lemma 3.1(i), u = Jgf e P). Suppose p = 1 and 0 ^ ke L,. By Lemma 2.1(ii) and (iii), ί ksgnf'f-ί\f\dμ=

\

kdμ = \\Juk\\ = \\P(ksgnu)\\

= ( tιP(Jaksgnf)

\f\dμ.

JA

Putting v = f — u = / — Jgf G &(P), we have, by Lemma 2.1(1), P(k sgn /) - JuP(Juk sgn /) + J,P(Jwfc sgn /) . Hence ( ΓiP(J9ksgnf).\f\dμ= JA

We conclude that

\ f'Pik }A

sgn

f)-\f\dμ.

30

S. J. BERNAU AND H. ELTON LACEY

\hf-1

\f\dμ=

JA

\ Γ JA

11

for all hef and all AeΣ0 so we are finished for p = 1. If p > 1 we have PJg - JgP by Lemma 2.3(ii) and \f\p~ι sgn/e by Lemma 2.2. Hence,

\ PJMfl'-'agnfdμ ix

= ( f-ιPh

\f\'dμ

(

Thus 1

-)

(h e

as claimed. Our theorem has useful consequences. THEOREM 3.5. Suppose l^p ^ ( X , I , ^) be a linear isometry with range M. Suppose a, be LP{Ω, Ξ, λ) and \a\ Λ |δ| = 0, we claim that I Ta\ A | Tb\ = 0. This is essentially proved by Lamperti [6], Since | α | Λ |6| - 0, \\a + δH' + \\a - b\\p = 2\\a\\p + 2||δ|Γ Since T p

p

p

is a n i s o m e t r y , \ \ T a + Tb\\ + \\Ta - Tb\\ = 2\\Ta\\

p

+ 2\\Tb\\ .

Since

p Φ 2, the equality condition for Clarkson's inequality [6, Corollary 2.1] shows that | Tα| Λ | Tb\ = 0. Take a maximal subset of Ξ consisting of sets of nonzero finite

THE RANGE OF A CONTRACTIVE PROJECTION ON AN Lp-SPACE

33

λ-measure which intersect pairwise in sets of λ-measure zero and let 5$Γ be the corresponding set of characteristic functions. Let a e SΓ and suppose BeΞ and Bc:S(a). Write b = χB, then T(a - b), Tb are disjoint in M so we have Tb — \ Tb | sgn Ta. This extends to non11 negative simple functions δ in α and then to all nonnegative be 11 a . Define U: LP(X, Σ, μ)~*Lp{X, Σ, μ) to be the direct sum of the unitary multiplications sgn Ta{a e ^%Γ). It is easy to see that U is an isometry of M such that UT is positive and hence UM = UT LP(Ω, Ξ, λ) is a closed vector sublattice of LP(X, Σ, μ) (compare the proof in Lemma 3.3 where we showed that functions in Mf were immeasurable). Assume (iii) and let Σo be the set of supports of functions (whose equivalence classes are) in M. Then ΣQ is a σ-subring of Σ. (If (fn) is a sequence in M, S(fn) = S(Ufn) = S(\Ufn\) so

Ίif,ge M, Jg = JUg; Jg\Uf\ = \im\Uf\ Λn\Ug\e UMand S(f) ~ S(g) = S(U"\\Uf\ - Jg\Uf\)).) Let f,geUM and suppose / is real, g ^ 0 and / e g11, then {ί 6 X: (f/g)(t) > a} = S((f - ag)+) e Σo. Thus f/g is 2Ό-measurable. This extends to all fe UMf] g11 and hence Jgf/g is 2Ό-measurable if /, g e UM and g ^ 0. This now extends to all f,ge UM and, since U~ιJgfjU~ιg — Jgf/gf we have f/g, Immeasurable 1L for /, geM and feg . It follows that M is the set of all elements in LP(X, Σ, μ) which can be written in the form hf with h, Jo-measurable and f e M. (If h = χs{g) with ge M, hf — JJ — 1 U^JUgUfe U~ (UM) = M.) Let J be the band projection on ML1, let he LP(X, Σ, μ), choose feM such that Jh — Jfh, (such an / exists by the arguments used in Corollary 3.2) and define Ph =

f&(Σf>\f\')(hf-1).

Then Phe M and this definition is independent of the choice of / in M such that hef11. To see this suppose geM and hegλL. Then h is zero outside S(f) n S(g)eΣ0 and so is &(ΣOf I/ΓXΛ/""1), ^-almost everywhere. Let B = S(f) n S(g), then ft = χBf e M and 1

1 / \*dμ = ( ft/"11 / Γ # - ί hfT11Λ \pdμ (A e ΣQ), v

1

so that fί?(Σ0, \f\ ){hf- ) = f^(Σ0, sume S(f) = S(g). Now

ι

| Λ |')(λ/r ).

Thus we may as-

eLtX, Σo, \g\*μ) ,

34

S. J. BERNAU AND H. ELTON LACEY

so we have, for A e Σo,

p

ι

g\'^(Σ,, \f\ )(hf- )\fγdμ . Because g'1/ and f~ιg are Immeasurable and the integrals are finite, the second integral is \g-1f\Γ1g\PhΓ1\f\pdμ - \hg-ι\g\'dμ. Thus and our definition of Ph is unambiguous. If hlf h2 e Lp we can take / e M such that Jh1 = Jfh1 and Jh2 = Jfh2. Thus P is linear. Since f-iph = &(Σ09 ifΠihf-1) we see P 2 = P. Finally, if p > 1, write p u= &(Σ0, \f\ )(hr% we have \\Ph\\l= ^uΓ'sgnΰ.&ίΣo,

v

ι

v

\f\ ){hΓ )\f\ dμ.

Since ^ is J0-measurable, this is ^ul^sgnΰ

hf-'lfl'dμ

= [\Ph\v~ι$gnfΰ hdμ £\\\PhΓ%\\h\\, = \\Ph\\y\\h\\9.

(We used Holder's inequality and q for the conjugate index to p.) We conclude that \\Ph\\P g \\h\\P. Since Ph = h(h e M) we have shown that M is the range of the contractive projection P. REMARK 4.2. The results (iii) implies (i) (with the same proof) and (i) is equivalent to (ii) are valid if p = 2; in fact (i) and (ii) are equivalent for any Hubert space. If we assume the projection P, is positive as well as contractive the proof in Lemma 3.3 that Mf is a lattice shows ,^{P) is a sublattice of L2 and Theorem 4.1 is valid for L2 with the projection and the isometry both required to be positive and in (iii) M required to be a closed vector sublattice. We use this remark in our next result. COROLLARY 4.3. If M is a closed vector sublattice of Lp (1 ^ p < co) then M is the range of a positive contractive projection.

THE RANGE OF A CONTRACTIVE PROJECTION ON AN L^-SPACE

35

Proof. Clearly M satisfies condition (iii) with U = I. In the definition of Ph we may always choose a positive f e M such that hefλl. Positivity of P follows from positivity of conditional expectation. REMARK 4.4. In the introduction we referred to Rao's paper [8] and claimed that its treatment of contractive projections contained errors. In particular, his Theorem II. 2.7 asserts that if M is the range of a contractive projection P on a Banach function space LP(Σ) there is, under suitable conditions, a unitary multiplication U such that UPU~ι is a positive contractive projection. The conditions are all satisfied if M is the subspace of Ϊ2(3) = C3 spanned by (1, 1, 1) and (1, 2, —3). Rao's theorem now claims the existence of a unitary multiplication, say by u — (λ1? λ2, λ3), such that uM is a vector sublattice of C3. This is impossible, as we show. First, uM contains the elements (0, λ2, — 4λ3), (λlf 0, 5λ3), and (4λlf 5λ2, 0). If Re λ2λ3 = 0 we have λaλg"1 = λ2λ3 = ±i and uM contains Im(0, λ2λ3, -4) = (0, ± 1 , 0); so that (0, 1, 0)euM, and uM = Cz. If all Reλp^ Φ 0 (i Φ j), then uM contains Re(0, 1, — 4λ3λ2) and Re(l, 0, δλgλj; hence, taking a multiple of their infimum, (0, 0, 1) e uM and again uM = C3. Exactly the same counterexample vitiates the proof of Rao's Theorem II. 2.8 see p. 177 lines -15 to - 1 1 . The error in both cases seems to be the reduction of the general case of LP(Σ) to the Lx situation. Vital to this reduction, but invalid, is the assertion that if LP(Σ) c Lι(Σ, G) and || || lfff ^ p( ) then a contraction on LP(Σ) for the p-norm can be extended to the closure of LP(Σ) in U{Σ, G) with the 1, G-norm and that the extension is contractive for the 1, G-norm.

5* The theorem of Lindenstrauss, Pelczynski, and Zippiru We begin by recalling some definitions. If E, F are isomorphic Banach spaces, d(E, F) = inf {||L|| U-Z/"1!): L is a linear isomorphism between E and F}. A Banach space E is an £fPtX space (for 1 1. As outlined in the introduction we discuss some details of the proof for the complex case. Observe first that (3) is a trivial consequence of (1). Simply identify E with LP(X, Σ, μ) and take for % the subspaces spanned by finite sets of (pih power)-integrable characteristic functions.

The proof that (3) implies (2). This result is certainly part of the folklore. It can be obtained quite efficiently as follows. LEMMA 5.2. Let xl9 *"9xn be n linearly independent elements of a normed space E then there exists ε > 0 such that if yt e E, and \\Xi — Vi\\ < ε(i — 1, 2, , n) then {yu , yn} is a linearly independent subset of E.

Proof. (This is standard but our proof may be novel.) Let K denote the scalar field and S the unit sphere in Kn, S = {XaKn: \\X\\ = 1}. The map g: S x E* — E defined by g((\, , λ j , (y19 -, yn)) \Vι + + ^nVn is continuous. By linear independence, the compact set S x (xlf •••,#„) does not meet the closed set ^ ( O ) . Hence there are open neighborhoods Ut of xif i = 1, , n, such that (S x Uι x ••• x ί / J ί l ^ ( O ) = 0. If yt e Ui(ί = 1, , u) it follows that {y19 • - , Vn) is linearly independent. LEMMA 5.3. every λ > 1.

Let E be a Zv-space, then E is an JzfPj-space for

Proof. Let F be a finite dimensional subspace of E. Let {xlf • , xn) be a basis for F, such that H^H = l(i = 1, , n). Let xf, • , x* G E* be such that x}(xj) - δij9 and let M = Σ?=i IK* H Choose ε > 0 such that Me < 1 and | | ^ — yt\\ < ε for i = 1, , n implies that {yl9 •••,!/»} is linearly independent. By the Zp-hypothesis there is a finite dimensional subspace H of E and points yί9 --,yn in H, such that H is isometrically isomorphic to ^(dim H), and \\%i — yΛl < ε(ΐ = 1, , w). Then {^/i, •••,!/»} is a linearly independent subset of

THE RANGE OF A CONTRACTIVE PROJECTION ON AN LP-SPACE

37

H. If Σ aiVi e ή i=i

i=i

then

Σ 3=1

=Σ 3=1

Since itfε < 1 we conclude that at = 0 for each ΐ. Thus we can extend 2/1, , 2/» to a basis, say ylf , # Λ , α?n+1, •••,#*, of H with the property that {xn+u -, xp) c fl?=i ^ ( « * ) . Let G be the subspace of E spanned by xu , xn, xn+1, , av Then F c G . If y = Σ ? = 1 aiVi + JJ=n+1 a,x, e Hdefine Ty = Σ?=i«A + Σf=n+i ^ A e G. We have

111/-

Σ This gives (1 - Me)\\ Ty\\ ^ \\y\\ ^ (1 + ikZε)|| Ty\\(ye H); so that T ι is an isomorphism between F and if such that || Γ|| || T~ \\ S (1 + Me)l(l - ikfε). If λ > 1 we can choose ε such that (1 + Me)/(I - Me) < λ. Thus E is an ^^-space for all λ > 1. The proof that (2) implies (1). Here the plan is first to embed E, isometrically, in an Lp-space, and then to use the theory of contractive projections of Lp-spaces. This is carried out in detail for the real reparable case in [7] and for the real nonseparable case in [5] The generalizations to cover the complex case are mostly obvious. For 1 < p < 00 our Theorem 4.1 is used. For p = 1, it follows as in the real case that E* is a ^ space whence by the complex version of Grothendieck's theorem [9] E is an Lλ{μ) space. There is an aspect of the construction which needs a little elaboration. At one stage of the proof we have a complex vector space, say V, consisting of complex valued functions on a set U. V is a vector sublattice of the space of all complex functions on U. There

38

S. J. BERNAU A N D H. ELTON LACEY

is a seminorm π on V such that π(f) ^ π(g) whenever | / | ^ \g\f and + #)*> = 7r(/y + π(g)p whenever | / | Λ |flf| = 0. We then need to π(f embed the quotient V/N, where N = {f e V:π(f) = 0}, isometrically in a concrete, complex, Lp-space. For this, let VB and NR denote the spaces of real-valued functions in V and N respectively. The quotient VB/NR with the norm induced by π is then linearly and lattice isomorphic, and isometric, to a vector sublattice of real LP(X, Σ, μ) just as in [7]. Let hL denote the composition of the quotient map UR —•> VBjNR and the isometric isomorphism into real LP(X, Σ, μ). Then h,1 is a linear and lattice homomorphism and ||/&i/|| = π(/)(/ £ VR). We construct the required embedding of V/N into complex LP(X, Σ, μ) by defining h(f + N) - ^(Re /) + i^ίlrn /) . Then h is clearly well defined. need the next lemma.

To verify that h is an isometry we

LEMMA

5.4. The map h constructed above satisfies h\f\ — \hf\,

Proof.

F o r a n y r e a l θ \f\ ^ R e (e f) s o

iθ

iθ

iθ

h\f\ = hlfl ^ h^Ree*'/) - Reh(e f) - Ree hf . Hence h\f\ ^ |fe/|. For the converse, let ft) be a complex wth root of unity and observe that for any complex z max {Re ωrz: r = 1, 2,

, n) ^ cos (τr/w) 121 .

Hence, cos (7r/w)λ I /1 ^ &(sup {(Re ωrf): r = 1, = sup {Re ωr/^/: r = 1,

, n})

Letting n—> co we have ft|/| = |fe/| as required. This completes our discussion of the proof of Theorem 5.1. We add a comment. It seems that a more elementary proof that a space which is an ^ >rspace for all λ > 1, is an Lp(μ) space, should be possible. Certainly the result should not depend on the entire theory of contractive projections for such spaces. Indeed if p = 2 the ^f2j condition already implies the parallelogram law and this makes the space a Hubert space. For p Φ 2 we can see that the Clarkson inequalities are valid and these with enough finite dimensional lpsubspaces might give a more elementary proof.

THE RANGE OF A CONTRACTIVE PROJECTION ON AN LP-SPACE

39

6* Appendix* We prove two technical results used in [1], [10]. The first is also an extension of that in [1]. LEMMA 6.1. [1]. Suppose 0 < p < co and let M be a closed subspace of LP(X, Σ, μ). If (fn) is a sequence in M, then there exists f e M such that S(f) = U~=i S(fn). In particular if μ is finite or M is separable there exists f e M such that Jf = JM .L i that is, f is a function in M of maximum support.

Proof. If f,ge Lp and a is a scalar, the zero sets {t e I : (/ -f ag)(t) = 0} have disjoint intersection with S(f) U S(g) for differing values of a. Since S(f) U S(g) is σ-finite, μ(S(f) U S(g) ~ S(f + ag)) = 0 except, perhaps for countably many values of a. Assume, as we may, that \\fn\v = 1 for all n. We define, inductively, two sequences {an), (en) of positive real numbers such that, if we write gn = aj, + -. - + ajnf An = {te X: \gn(t)\ £ εn}, and Bn = {teX:\an+1fn+1(t)\^en/2}, then ( i ) an+ί < 2 - " and εn+1 < eJ2; (ii) μ(S(gn) U S(fn+1) - S(gn+1)) = 0; (iii)

\

\fi\pdμ)

Thus | / ( 0 | - Hm,^ \gk(t)\ ^ |