Partitions of Unity and Approximation C. De Boor; R. Devore Proceedings of the American Mathematical Society, Vol. 93, No. 4. (Apr., 1985), pp. 705-709. Stable URL: http://links.jstor.org/sici?sici=0002-9939%28198504%2993%3A4%3C705%3APOUAA%3E2.0.CO%3B2-O Proceedings of the American Mathematical Society is currently published by American Mathematical Society.
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PROCEEDINGS O F THE AMERICAN MATHEMATICAL SOCIETY Volume 93. Number 4. Apnl 1985
PARTITIONS O F UNITY AND APPROXIMATION
ABSTRACT. We show that for certain translation invariant spaces S , a necessary and sufficient condition for the eventual denseness of the corresponding scaled spaces S, is that S contain a stable and locally supported partition of unity. These results have been motivated by recent work on approximation by multivariate piecewise polynomials on regular meshes.
1. Introduction. We are interested in the existence of partitions of unity in certain translation invariant spaces. We are motivated by recent investigations (listed in the references) of approximation by multivariate piecewise polynomial functions on regular partitions-a partition 7 of R"' is called regular if j + r = r for all j E Zm. These investigations study the degree of approximation from the spaces .rr,P(h~), h > 0, which consist of pp (i.e., piecewise polynomial) functions in CP(Rm)of total degree k on the dilated partition h7. It has been possible to determine the approximation order from these spaces only for very special partitions and certain values of p and k. In particular, there are no general results which characterize when U, , .rr,P( h r ) is dense in the space of continuous functions with compact support. The main technique for proving denseness or giving lower estimates for the degree of approximation has been to construct a partition of unity in m,P(r) consisting of certain multivariate B-splines known as box splines and to use this partition of unity to construct suitable approximation schemes called quasi-interpolants. It has not been clear to us whether thls represents the only viable approach to the construction of suitable approximants. For example, in [BDH] we have asked the following: contains a Is U, ,.rr,P(h T ) dense in C,(Rs) if and only if .rr,P(~) (I.') good partition of unity? By a good partition of unity we mean a (countable) collection 9 of functions with the properties that (i) sup,, a diam supp @ < oo ;
,
(I .2)
(ii) E,=@@
=
1;
(iii) E6,al@l < w . Received by the editors May 4. 1984.
1980 Murhemurrcs Subject Clu~~ificutrot~. Primary
41A15. 41A25. Ker. words undphruses. Degree of approximation, partition of unity. piecewise polynomial. 'Both authors were sponsored by the United States Army under Contract No. DAAG29-80-C-0041. 'This material is based upon work supported in part by the National Science Foundation under Grant NO. DMS-8320562. 01985 American Mathematical Society 0002-9939/85 $1.00 + $.25 per page
706
C. D E BOOR AND R. DEVORE
Actually, it is easy to see that the existence of a good partition of unity is sufficient for denseness. Namely, if x, E supp then Lf:= C@ , f(x,)+ g'ives an operator L whose dilate LA:= o,,,La,, with (ohf ) ( x ) := f(hx), provides an approximation L, f in .rr,P(h~) whch converges to f as h + 0. It turns out, but is much more difficult to prove, that the existence of a good partition of unity is necessary for denseness. Here we shall present a proof of this fact for the univariate case but for much more general spaces S than just pp ones, in an attempt to extract the essential features of pp function spaces on regular partitions. The ideas of the proof seem to carry over to the multivariate case, but the constructions involved are much more involved. For this reason, we feel that it is beneficial to present here the basic ideas in their simplest setting and report elsewhere on their multivariate analogues.
+,
2. The main results. We let S be a space of functions on R with the following properties: (i) S is translation invariant, i.e., f (2.1)
E
S
-
f (.
+ 1) E S ;
(ii) p := dim S I [,, < co ; (iii) dim S = w ; (iv) S is closed under uniform convergence on compact sets.
We note for later reference that (i) and (iii) imply (v) dim S I
[0, m
)
= @2
These are the essential properties of pp spaces over regular meshes. However, it is easy to give examples of such spaces S which are not pp. For example, if @ is a finite collection of compactly supported functions, then the closure under uniform convergence on compact sets of the span of the translates := $(. - j ) , j E Z, + E @, has these properties. The results of this section will show that any space S satisfying (2.1) can be obtained this way modulo some finite-dimensional space. Associated with S we have the dilated spaces S, := ol,,S.
+,
THEOREM 1. If U , S, is dense in CO(R),then S contains a good partition of unity. In preparation for the proof, we now construct certain compactly supported functions in S. Let S- denote the subspace of S consisting of those functions which vanish on (-a, 01. It will follow later that S-is nontrivial. For now, we prove LEMMA2. For each integer n > 0, there is an R, such that any [-R,,, 0] agrees on [0, n] with some function from S-.
+ E S vanishing on
PROOF.For r , j > 0, we let L,,, denote the space of functions which are restrictions of functions in S which vanish on [-r,O]. whenever r > r'. Hence (dim Lr,J)F=l is a decreasing sequence integers and therefore is eventually constant. This implies that
defined on [0, j] Then L,,, L,I,, of (nonnegative) we can find an
707
PARTITIONS O F UNITY AND APPROXIMATION
increasing sequence (R,) such that L,,, Now given n > 0 and @ E L,, we define n,:= n
+ i,
=
LJ:= L,,., for all r >, R,, j i
r,:= m a x { ,,,, ~ n,),
=
=
1,2,. ...
0,1,
Then, since r, 2 R,,, >, R,,, there is an f, E S which vanishes on [-r,, 01 and agrees with @ on [0, n]. Similarly, r, 2 R n z2 R,,, and so there is an f, E S which vanishes on [-r,, 01 and agrees with f, on [0, n l ] In this manner, we obtain a sequence such that f , vanishes on [-r,,0] and agrees with f, on [0, n,] for all i < j. Hence f, converges uniformly on compact sets to somef , necessarily in S - because of (2.l)(iv), with the desired properties. 13 Now let R := R, be the constant of Lemma 2. We improve Lemma 2 in the following way:
(4)
3. Iff E S vanishes on [-R, 01, then @ := f x ,[ LEMMA
,
is in S -
PROOF.F rom Lemma 2, we know an f, E S - for which f - f, vanishes on [-R, I]. Hence there is some f, with f l ( . 1) E S - , therefore also f, E S - , for which f - f , - f, vanishes on [-R, 21. In this manner, we construct a sequence (f,) in S . . . f , converges to @. Since S - is closed under uniform converfor which f, gence on compact sets (by (2.l)(iv)), @ E S-. 13 Next, we construct a basis of compactly supported functions for S - . Recall the definition p := dim SI I,,l1 and set M := pR.
+
+
+
LEMMA4. Iff E S - , then there is a function agrees with f on [O,l].
+ E S which is supported on [0, M ] and
PROOF. Set J : = [M, M + R ] and consider the functions f , := f ( . - j), j = 0,. . . ,M. Since dim SI, G M, these functions are linearly dependent on J . Hence there are numbers c,, not all zero, such that #,:= EcJf, vanishes on J. Let i be the smallest value of j for which c, # 0. Then #:= #,(. - i)/c, agrees with f on [O,1] and vanishes on a + [-R, 0] with a := M + R - i. Therefore, by Lemma 3, 4, := # ( . + a ) x ,[ is in S - and so @ := # - #,(. - a ) has the desired properties.
,
It follows from Lemma 4 that there is afinite collection B of functions supported on [0, MI whose restrictions to [O, 11 form a basis for (S-)lro,ll.We use the abbreviation b, := b(. - j ) , b E B, j E Z. The peeling-off argument of Lemma 3 shows that the functions b,, b E B, j = 0,1,2,. . ., form a basis for S-. But, in order to insure local linear independence of this basis, we need to choose B with some care. Precisely, we choose B so that length(B) := C,,,length(b) is minimal. Here, we mean by length(b) the smallest integer r for which supp b [i, i + r ] for some integer i. LEMMA5. For any interval J : = [r, r before, the collection
is linearly independent over J.
+ R], with
r an arbitrary integer and R as
708
C. DE BOOR A N D R. DEVORE
PROOF.Suppose that we have f := C,,,J c,+ = 0 on J . Then, by Lemma 3, we can find some g E S- so that f, := f - g(. - ( r + R ) )vanishes on [ r , m). If not all the coefficients c, were zero, then we could choose some @ := b, with c, =: c, + 0 and so that j is as small as possible. Since B is linearly independent over [O. 11, ( f l ) 1 [,, + 0, and hence we must have J < r. Since f, vanishes outside [ j ,r ] , while supp b, intersects [ j, j + 11 as well as J, this would imply that
, +
while f 2 := f,(. + J ) E S- and, on [O,l],is in the span of B with the coefficient of b not zero. But then B':= ( B \ { b ) ) U {f,} would also supply a basis and length (B') < length(B), contradicting our choice of a basis of minimal length. Suppose now that f = C c,+ with the sum over all @ = b, with j E Z and b E B. Then the local linear independence just proved allows the conclusion that
for some constant independent of b and j. is finite dimensional, we can choose a finite-dimensional subspace T Since SI [ S which is carried faithfully onto S I [ b y the restriction map f ++ f 1 [ of For s E S, let s, be the unique element of T whch agrees with s on [-R,O]. Then ( s - s , ) ~ [ E~ S, - b~ y~Lemma 3, hence on [0,oo),
,,,,
for suitable coefficients c,,. Since S/[,,,, is infinite dimensional by (2.l)(v), this shows that B + 0 and that S- is also infinite dimensional. Further, since the linear has dimension (dim T)2, the maps space of all linear maps from T to SI[-R,Ol
must be linearly dependent whenever A c Z and IAl > (dim T)2. This together with Lemma 3 implies that, for any A c Z with more than (dim T ) 2 elements, there exist coefficients c,, a E A , so that
PROOFOF THEOREM 1. Let g be the continuous piecewise linear function with integer breakpoints which has the value 1 at x = 1 , 2 , 3 and vanishes at all other integers. Since g E Co(R), there is S, E S, such that S, converges to g uniformly on R. From now on, we consider only h E 1 / N . We write s,?:= a,S, and g,:= a,g. Then /Ig, - shll -+ 0 as h + 0. Since s, E S, we have .I.,= t, + s, on [0,oo), with t, E T and s, E S-. Now choose A c 8 N / h as above. Then the functions g,(. - a), a E A, are supported in [0,oo) and have disjoint supports. We define G,:=
C cog,(.-
a),
USA
with the c, as given by (2.4). Thenf, := C,,,c,s,( as h 0. -+
- a ) is in S-, and 11 f,
-
G,II
-+
0
PARTITIONS O F UNITY AND APPROXIMATION
709
Now, because of (2.4), G, is identically one on an interval of the form [ m , - l / h , mi, + l,i'h]. Hence, with FA:= f,(. + m,), we have IIFh - lllI-l,h,l,hl + 0 as h + 0. Sincef, E S - , we can write F, =: CJ,,Cb,,cb,(h)bJ. In view of (2.2), c , ( h ) a const
for all b , j , h .
We can take therefore a subsequence h' of h so that, for all b and j , c,,(hr) converges as h' + 0 to some c,,. It follows that CJ,,C,cb1bJ converges uniformly on compact sets to some function F E S . Since we also have that F, converges to F uniformly on compact sets, we must have F = 1. Thus (cb1bJ),,,,,,, is a partition of unity in S . It is good because of (2.5).
[BD] C , de Boor and R. DeVore, Approzin1ution b j smooth mulrr~~urrute splines. Trans. Amer. Math. Soc. 276 (1983). 775-785. [BDH] C. de Boor. R. DeVore and K. Hollig, Approsrn~utron order fronl smooth pp functions. Approximation Theory. IV (C. Chui. L. L. Schumaker and J. Ward, editors). Academic Press, New York. 1983. pp. 353-357. [BH,] C. de Boor and K. Hollig. Bir.uriute box splrnes utld smooth pp functions on u three-direcrron mesh, J . Comput. Appl. Math. 9 (1983). 13-28. [BH,] Approsimutron order from hruurrute C1-cubrcs:A counterexun~ple,Proc. Amer. Math. Soc. 87 (1983). 649-655. [DM] W. Dahmen and C. Micchelli. Otr the optrmul order upprosimutron rates for crrss-cross finrte ele~netlts paces. 1983. [J] R.-q. Jia. Approsimutiorl hj, smooth brc.uriure splrtles otl u three-direction mesh. Approximation Theory. IV (C. Chui, L. L. Schumaker and J. Ward. editors). Academic Press. New York. 1983, pp. 539-545.
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Approximation by Smooth Multivariate Splines C. de Boor; R. DeVore Transactions of the American Mathematical Society, Vol. 276, No. 2. (Apr., 1983), pp. 775-788. Stable URL: http://links.jstor.org/sici?sici=0002-9947%28198304%29276%3A2%3C775%3AABSMS%3E2.0.CO%3B2-1 BH2
Approximation Order from Bivariate C1-Cubics: A Counterexample C. De Boor; K. Höllig Proceedings of the American Mathematical Society, Vol. 87, No. 4. (Apr., 1983), pp. 649-655. Stable URL: http://links.jstor.org/sici?sici=0002-9939%28198304%2987%3A4%3C649%3AAOFBAC%3E2.0.CO%3B2-7
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