AbstractâA sufficient condition for the stability of completeness of systems of ... Key words: stability of completeness on a compact convex set, Banach space of ...
Mathematical Notes, vol. 72, no. 4, 2002, pp. 542–550. Translated from Matematicheskie Zametki, vol. 72, no. 4, 2002, pp. 587–596. c Original Russian Text Copyright 2002 by B. N. Khabibullin.
Stability of Completeness for Systems of Exponentials on Compact Convex Sets in C B. N. Khabibullin Received November 28, 2000; in final form, November 20, 2001
Abstract—A sufficient condition for the stability of completeness of systems of exponentials {exp λn z} with exponent shifts λn in spaces of functions continuous on and holomorphic inside a compact convex set K ⊂ C with nonempty interior is obtained. This condition is a generalization of an important special case of the corresponding result for the closed interval due to Redheffer and Alexander. Key words: stability of completeness on a compact convex set, Banach space of continuous
functions, system of complex exponentials, determining measure, holomorphic function.
INTRODUCTION All functions and measure are assumed to be complex-valued, unless otherwise specified or implied by the context. Let K be a compact subset of the complex plane C . As usual, C(K) denotes the Banach space of all continuous functions on K with the natural norm f K = sup{|f (z)| : z ∈ K},
f ∈ C(K).
(0.1)
By A(K) we denote the subspace of C(K) consisting of functions holomorphic in the interior Int K of the compact set K , provided Int K = ∅ , with the same norm (0.1). In particular, if Int K = ∅ , we have A(K) = C(K) . Let Λ = {λn } ⊂ C , where n ∈ N , be a sequence of complex numbers (points) having no limit points in C . To the sequence Λ ⊂ C we assign the exponential system Exp Λ = z k−1 eλz : λ ∈ Λ, 1 ≤ k ≤ Λ(λ), k ∈ N ,
(0.2)
where Λ(λ) is the number of occurrences of the point λ in the sequence Λ . Following, e.g., [1, 2], we say1 that the system Exp Λ is complete on the compact set K if the closure of its linear hull in the space A(K) coincides with A(K) . The following theorem was proved in [3, Theorem 3] in a weakened form; its final form is given in the survey [4, Theorem 14]. 1 Some authors use the dual definition of the notion of completeness of a system of functions in a topological vector space E or, in other words, with respect to E ; namely, E is strongly dual to the space containing the system, and the system is annihilated only by the zero functional from E (see, in particular, [3, Theorem 3; 4, p. 5]). This definition sometimes leads to inaccuracies [5, p. 212] or ambiguities [6, p. 213] in the statement of the initial version of the Alexander–Redheffer theorem, which is our starting point.
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c 2002 Plenum Publishing Corporation
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Alexander–Redheffer Theorem. If two sequences of complex numbers Λ = {λn } , Γ = {γn } , where n = 1, 2, . . . , satisfy the relation ∞
|λn − γn | < +∞, 1 + | Re λn | + | Re γn | n=1
(0.3)
then Exp Λ and Exp Γ are complete or incomplete on an arbitrary closed interval K ⊂ R simultaneously. The Alexander–Redheffer theorem is beautiful if only because condition (0.3) does not imply any regularity in the distribution of either of the individual sequences Λ or Γ . This result readily implies the following assertion. Corollary. If the sequences Λ and Γ are related by ∞
|λn − γn | < +∞,
(0.4)
n=1
then the systems Exp Λ and Exp Γ are complete or incomplete on each line segment K in C simultaneously. Many other results on the stability of the completeness of systems of exponentials for the spaces C[−a, a] and Lp (−a, a) , obtained before 1977, were cited in Redheffer’s survey [4]. A series of delicate results in this direction is due to Sedletskii (see, e.g., [6–11]). We are aware of only few works on the completeness of exponential systems on compact sets with nonempty interiors proper (see, e.g., [12–14]). At the same time, the study of the completeness of such systems on compact subsets of C with nonempty interior is very topical and even crucial to a certain degree, the more so since the existence of “good” bases or representing systems for the entire space A(K) , not to mention those of the form (0.2), is uncertain. Thus it is known that there exist no absolutely representing systems of exponentials for A(K) (see [15]); A(K) , as well as C[a, b] , has no unconditional bases. Moreover, very probably, A(K) even has no bases of the form (0.2). In [14, Theorem 2], a weakened version of the generalizations of the Alexander– Redheffer theorem to compact convex sets K with nonempty interior is proved; it states that, under certain “nearness” condition on the sequences Λ and Γ , which naturally develops relation (0.3) with taking into account the geometry of the compact convex set K , the systems Exp Λ and Exp Γ are complete or incomplete on K simultaneously, possibly, with a “gap” in two exponentials. But this does not solve the problem of completely carrying over the Alexander–Redheffer theorem, as well as its corollary stated above, to arbitrary compact convex sets K ⊂ C with nonempty interior. The main result of this paper is the following completeness stability theorem, which fully generalizes the corollary to arbitrary convex compact sets in C . Completeness Stability Theorem. If the sequences Λ and Γ are related by (0.4), then the systems Exp Λ and Exp Γ can be complete on a compact convex set in C only simultaneously. A special case of this theorem was announced in [16]; in final form, the theorem was announced in [17]. 1. ON THE DUAL TO A(K) From now on, we assume without explicit mention that the compact set K is convex. By A∗ (K) and C ∗ (∂K) we denote the spaces strongly dual to A(K) and to the space C(∂K) of functions continuous on ∂K , respectively. Each function f ∈ A(K) is uniquely determined by its values on the boundary, and the set of the restrictions of functions from A(K) to ∂K is a closed subspace in C(∂K) ; therefore, by the MATHEMATICAL NOTES
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Hahn–Banach theorem, each continuous linear functional L ∈ A∗ (K) with norm L∗ in A∗ (K) can be assigned a continuous linear functional l ∈ C ∗ (∂K) with the same norm. By the Riesz theorem, each functional l ∈ C ∗ (∂K) can be uniquely identified with a finite Borel measure on ∂K ; we denote this measure by the same symbol l and write l ∈ C ∗ (∂K) for the measure l . Such a correspondence L → l is far from being unique. It obeys the rule L(f ) =
f (ζ) dl(ζ), ∂K
f ∈ A(K).
(1.1)
The norm of the functional (measure) l ∈ C ∗ (∂K) in C ∗ (∂K) is the total variation Var[l] of the measure l . Conversely, every continuous linear functional l on C(∂K) considered as a measure l on ∂K , uniquely determines a continuous linear functional L ∈ A∗ (K) according to (1.1). In what follows, we call each measure l on ∂K related to a functional L ∈ A∗ (K) by (1.1) a determining measure of the functional L and say that the measure l determines the functional L . Thus, according to (1.1), every measure l determining a functional L ∈ A∗ (K) satisfies the estimate (1.2) L∗ ≤ Var[l] ; in addition, for any L ∈ A∗ (K) , there exists a measure l ∈ C ∗ (∂K) for which (1.2) becomes an equality. For our purposes, it is convenient to introduce the notation [0, S] for the interval [0, S] with identified points 0 and S , i.e., for the circle of circumference S with marked initial point 0 , which is endowed with the usual metric, that is, the length of the shortest arc between points. Consider the natural parametrization by arc length (in the counterclockwise direction) of the boundary of the compact convex set K , i.e., ζ : [0, S] → ∂K ,
where S is the length of ∂K ,
ζ(0) = ζ(S) = ζ0 .
(1.3)
Obviously, it satisfies the Lipschitz condition |ζ(s) − ζ(t)| ≤ |s − t| ;
(1.4)
hence it is an absolutely continuous function of bounded variation on [0, S] . In addition, the parametrization (1.3)is a homeomorphism between [0, S] and ∂K as a continuous bijection of the compact space [0, S] onto ∂K . Under the parametrization (1.3), every functional L from A∗ (K) with determining measure l can be specified by a right-continuous function of bounded variation l on [0, S] , where l(s) = l ζ (0, s] ,
0 < s ≤ S,
l(0) = 0.
(1.5)
According to (1.1), we then have L(f ) =
S
f (ζ(s)) dl(s), 0
f ∈ A(K),
(1.6)
where the integral is understood in the sense of Stieltjes. Conversely, under the parametrization (1.3), every right-continuous function l of bounded variation on [0, S] such that l(0) = 0 uniquely determines a Borel measure l according to (1.5), and this measure uniquely determines a functional L ∈ A∗ (K) according to (1.6) or (1.1). The total variation Var[l] of the function l on [0, S] coincides with the total variation Var[l] of the MATHEMATICAL NOTES
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measure l , because the parametrization (1.3) is a homeomorphism; thus the “amount” of continuous functions on [0, S] and on ∂K is the same, and the norms of the corresponding continuous linear functionals are defined as the total variations of the corresponding measures (functions). In what follows, if l is a right-continuous function of bounded variation on [0, S] such that l(0) = 0 and it is related to a functional L ∈ A∗ (K) by (1.6) under the parametrization considered, then we call l a determining function of the functional L and say that it determines the functional L . According to the remarks on determining measures l ∈ C ∗ (∂K) made above, each function l determining a functional L ∈ A∗ (K) satisfies the estimate L∗ ≤ Var[l] ;
(1.7)
in addition, for any L ∈ A∗ (K) and any parametrization (1.3), there exists a right-continuous function l of bounded variation on [0, S] , l(0) = 0 for which (1.7) becomes an equality. The characteristic function (Laplace transform) of a functional L ∈ A∗ (K) with determining measure l and determining function l satisfying (1.1) and (1.6) is uniquely defined as the entire function of exponential type S zζ L(z) = L(e ) = exp(zζ) dl(ζ) = exp(zζ(s)) dl(s), z ∈ C. (1.8) 0
∂K
zζ
Since the system of exponentials {e : z ∈ C} , where ζ ∈ K , is complete in A(K) , if L is the zero function, then the functional L is also zero. 2. THE LEVINSON–ALEXANDER–REDHEFFER CONSTRUCTION FOR A COMPACT CONVEX SET Under the assumptions and in the notation of the preceding sections (see (1.3)–(1.8)), the following version of the Levinson identity is valid (Levinson applied this version to the case of a closed interval K ∈ R [18]). Proposition 1. If L is the characteristic function of the functional L ∈ A∗ (K) with determining function l and L vanishes at a point λ , then the functional Lλ ∈ A∗ (K) with characteristic function Lλ = L(z)/(z − λ) has a determining function lλ (under the same parametrization (1.3) as for l ) with distribution density s S λ(ζ(t)−ζ(s)) λ(ζ(t)−ζ(s)) e dl(t) dζ(s) = e dl(t) dζ(s). (2.1) dlλ (s) = − 0
s
Proof. Let us rewrite the characteristic function L in the form S S (z−λ)ζ(s) λζ(s) (z−λ)ζ(s) e e dl(s) = e d L(z) = 0
0
s
eλζ(t) dl(t) ;
(2.2)
0
applying integration by parts2 [19, Chap. VIII] and taking into account that L(λ) = 0 , we obtain S s S (z−λ)ζ(s) λζ(t) λζ(t) e dl(t) − e dl(t) de(z−λ)ζ(s) L(z) = e 0
(z−λ)ζ(S)
=e
L(λ) −
= −(z − λ)
S
0
S s
λζ(t)
e 0
zζ(s)
0
s
e 0
0
dl(t) (z − λ)e(z−λ)ζ(s) dζ(s)
λ(ζ(t)−ζ(s))
e
dl(t) dζ(s).
0
2 In
the book of Natanson, the majority of assertions are proved for functions and measures with values on [−∞, +∞] , but they are readily carried over to functions and measures with values in C ∪ {∞} by separately considering their imaginary and real parts. MATHEMATICAL NOTES
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Dividing the obtained equality by z − λ and choosing lλ (s) as in (2.1), we obtain (2.1), because each functional is uniquely determined by its characteristic function. According to (2.1), the function lλ has bounded variation on [0, S] ; therefore, the functional Lλ with determining function lλ belongs to A∗ (K) . The independence of ζ0 is seen from the construction. The last relation in (2.1) follows from the identity
s
λζ(t)
e
L(λ) =
S
dl(t) +
0
s
eλζ(t) dl(t) ≡ 0.
(2.3)
Proposition 2. Under the conditions and in the notation of Proposition 1, the functional G ∈ A∗ (K) with characteristic function G(z) =
z−γ L(z) z−λ
(2.4)
has a determining function g related to l by dg(s) = dl(s) + (λ − γ) dlλ (s).
(2.5)
Proof. It suffices to apply the identity G(z) = L(z) + (λ − γ)
L(z) z−λ
and recall that each functional is uniquely determined by its characteristic function. The inclusion G ∈ A∗ (K) follows from the form of its determining function (2.5). The following assertion extends an estimate obtained in [3] and [4, Sec. 6] for closed intervals over compact convex sets. Proposition 3. If the characteristic function L of a nonzero functional L ∈ A∗ (K) satisfies the condition L(λ) = 0 , then the function G defined by (2.4) is the characteristic function of some nonzero functional G ∈ A∗ (K) , and G − L∗ ≤ S · |λ − γ| · L∗ ,
(2.6)
where S is the perimeter of the compact convex set K . Proof. We can assume without loss of generality that the point λ lies on the positive semiaxis (0, +∞) (otherwise, we translate the entire plane C and rotate it through a suitable angle). As mentioned in Sec. 1, we can select a determining measure l ∈ A∗ (K) of the functional L so that (2.7) L∗ = Var[l]. Now, let us describe the parametrization (1.3) (see Fig. 1). Let l+ and l− be two support straight lines of the compact set K which are orthogonal to the positive semiaxis and l− lies on the left of l+ . As the initial point ζ0 = ζ(0) in the parametrization (1.3), we take one of the support points of the support line l− . Next, pick a support point ζ+ = ζ(s+ ) of the support line l+ . Under this parametrization, we can construct a determining function l for the measure l satisfying (2.7) so that (2.8) L∗ = Var[l]. MATHEMATICAL NOTES
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Fig. 1. The parametrization (the direction is counterclockwise)
By Propositions 2 and 1, G is the characteristic function of some functional G ∈ A∗ (K) , for which we can select a determining function g (under the same parametrization) so that S s λ(ζ(t)−ζ(s)) λ(ζ(t)−ζ(s)) e dl(t) dζ(s) = (λ − γ) e dl(t) dζ(s), d(g − l)(s) = (λ − γ) − 0
s
whence
Var[g − l] ≤ |λ − γ| sup min
s
s
λ(ζ(t)−ζ(s))
e 0
dl(t)
,
S
λ(ζ(t)−ζ(s))
e
s
dl(t)
· Var[ζ] ;
the variation Var[ζ] of the parametrization (1.3) is the perimeter of S . Continuing this estimate, we obtain (2.9) Var[g − l] ≤ |λ − γ| sup min{B0 (s), BS (s)} · Var[l] · S , s
where
B0 (s) = sup |eλ(ζ(t)−ζ(s)) |,
BS (s) = sup |eλ(ζ(t)−ζ(s)) |,
0
