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Infinite Systems of Linear Equations for Real Analytic Functions PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 132, Number 12, Pages 3607{3614 S 0002-9939(04)07435-0 Article electronically published on July 20, 2004 INFINITE SYSTEMS OF LINEAR EQUATIONS FOR REAL ANALYTIC FUNCTIONS P. DOMANSKI´ AND D. VOGT (Communicated by N. Tomczak-Jaegermann) Abstract. We study the problem when an infinite system of linear functional equations µn(f)=bn for n 2 N d has a real analytic solution f on ! ⊆ R for every right-hand side (bn)n2N ⊆ C and give a complete characterization of such sequences of analytic functionals d (µn). We also show that every open set ! ⊆ R has a complex neighbourhood Ω ⊆ Cd such that the positive answer is equivalent to the positive answer for the analogous question with solutions holomorphic on Ω. Introduction One of the first classical examples of an infinite system of linear equations is related to the so-called moment problem solved by Hausdorff. It is the question of finding a Borel measure µ on [0; 1]R such that a given sequence of reals (bn)n2N is 1 n the sequence of moments of µ (i.e., t dµ(t)=bn for n 2 N). More generally, we 0 0 can look for a functional f 2 X such that for the given sequence of scalars (bn) and of vectors (xn) belonging to a locally convex (Banach) space X the following holds: (1) f(xn)=bn for n 2 N: First solutions for spaces X = C[0; 1] or Lp[0; 1] are due to F. Riesz (1909) and the functional analytic approach to the problem is contained in the famous book of Banach [1, Ch. IV x7, x8]; see also [23, pp. 106-107]. Later on, Eidelheit, a colleague of Banach, characterized in all Fr´echet spaces what are now called Eidelheit sequences [9] (see [17, Th. 26.27], [12, II.38.6]). Let X be a locally convex space. We call a sequence (µn)n 2 N of continuous linear functionals on X an Eidelheit sequence on X if for every sequence of scalars (bn)n 2 N there is f 2 X satisfying (2) µn(f)=bn for n 2 N: The notion of Eidelheit sequences has been extensively studied in Fr´echet spaces later on; see [9], [18], [20], [21] and [22], compare also [17, x26]. Received by the editors January 28, 2003 and, in revised form, May 22, 2003 and July 9, 2003. 2000 Mathematics Subject Classification. Primary 46E10; Secondary 46A13, 26E05, 46F15. Key words and phrases. Space of real analytic functions, analytic functionals, interpolation of real analytic functions, Eidelheit sequence. c 2004 American Mathematical Society 3607 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 3608 P. DOMANSKI´ AND D. VOGT We will prove a completely analogous result for the highly non-metrizable space of real analytic functions A (!) on an arbitrary open set ! ⊆ Rd (Theorem 2.2). We show that for every open set ! ⊆ Rd there is a domain of holomorphy Ω ⊆ Cd, Ω \ Rd = !, such that every Eidelheit sequence on A (!) is automatically Eidelheit on the space H(Ω) of holomorphic functions on Ω (Theorem 2.2). The main tool in the proof is given by the so-called distinguished sets, which might be of independent interest (see Lemma 1.1 and Lemma 1.3). Let (f1;n)n 2 N,(f2;n)n 2 N be two Eidelheit sequences on A (!1)andA (!2), respectively. We consider the question if there is a continuous linear map (= oper- ator) T : A (!1) ! A (!2) such that f1;n = f2;n ◦ T for every n 2 N (compare an analogous problem on Fr´echet spaces due to Mityagin [18] and its solution in [21]). It turns out that for every (f1;n)n2N there is (f2;n)n 2 N without such factoriza- tion (Proposition 3.1). On the other hand, we show a positive result for sequences related to the interpolation problem for analytic functions (Theorem 3.2). Let us recall that the space A (!), ! ⊆ Rd an open subset, of real analytic func- −→ C tions f : ! is equipped with the topology of the projective limit projN H(KN ), where (KN )isanexhaustionof! by a sequence of compact sets [ K1 ⊂⊂ K2 ⊂⊂ :::⊂⊂ KN ⊂⊂ :::⊂ !; KN = !; N2N and H(K) denotes the space of germs of analytic functions over K with its natural LB-space topology. It is known (see [16, Prop. 1.7, 1.2]) that this topology is equal to the inductive limit topology indH(U), where U runs over all open neighbor- hoods of ! in Cd and H(U) denotes the Fr´echet space of holomorphic functions on U with the compact open topology. Thus A (!) is a complete, separable, ultra- bornological, reflexive, webbed nuclear space with the approximation property (but non-metrizable and without a basis [7]) with the dual being a complete LF-space A 0 0 A (!)β =indH(KN )β.Thespace (!) is the projective limit of a sequence of LB-spaces. Spaces of this type are called PLB-spaces. For more information on the space of real analytic functions, see [2], [4], [5], [7], [8] and [16]. For smooth functions on ! ⊆ Rd we use the standard multi-index notation. So d if α =(α1,...,αd) 2 N ,then @jαj @jαj @jαj f = f; δ (f):=f(z),δα(f):=(−1)jαj f(z); α α1 αd z z α @x @x1 :::@xd @x where jαj = α1 + :::+ αd. The elements µ 2 H(Cd)0 are called analytic functionals. A compact subset K ⊂ Cd is called a carrier for µ if for every neighborhood U ⊂⊂ Cd of K there is a continuity estimate jµ(f)|≤C sup jf(z)j: z2U We may identify A (Rd)0 with the analytic functionals that are carried by some K ⊂ Rd. In this case there is a smallest real carrier which is called the support of µ and denoted by supp µ (see [19, p. 44 ff.]). For open ! ⊂ Rd we may identify A (!)0 with the analytic functionals µ with supp µ ⊂ !. The other notation is standard. We refer for functional analysis to [17] and for complex analysis to [13]. For analytic functionals see [19] or [15]. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use INFINITE SYSTEMS OF LINEAR EQUATIONS 3609 1. Distinguished open sets We will need some auxiliary notions. We call a nonnegative C1-function : Rd −→ R distinguished whenever it vanishes at infinity and sup kH (x)k < 1; x2Rd where H denotes the Hessian of and k·kis the norm in the space of quadratic forms on the d-dimensional euclidean space. For every distinguished function we define Ω := fz = x + iy 2 Cd : jyj2 < (x)g and call it a distinguished open set with basis ! := Ω \ Rd. These sets and the construction below are variations of the proof of the Cartan-Grauert Theorem as given in [3, Prop. 1] and [10, Prop. 6 and Prop. 7]. Lemma 1.1. (a) Every distinguished open set Ω is a domain of holomorphy and f j j2 − }⊂⊂ for every α>0, Ωα := z : y < (x) α Ω . Moreover, polynomials are dense in H(Ω ) equipped with the compact-open topology. (b) For every pair of open sets U ⊂ W ⊂ Cd, U distinguished, there is a distin- guished set Ω with the basis W \ Rd such that U ⊆ Ω ⊆ W .IfU ⊂⊂ W ,then we can choose such that U ⊂⊂ Ω . (c) Every open set ! ⊂ Rd has a basis of open neighbourhoods in Cd consisting of distinguished sets. (d) Every compact set in Cd is contained in some bounded distinguished set. Parts (a) and (c) imply a corollary going back to Cartan (see [11, Cor. II. 3.15]): Corollary 1.2. Every open set ! in Rd has a basis of neighbourhoods in Cd con- sisting of domains of holomorphy. Proof of Lemma 1.1. (a): Let f(z):=jyj2 − (x)forz = x + iy 2 Cd.Since kH (x)k < 1, the Levi form of f is positive definite. Thus, f is plurisubharmonic Cd f g ⊂⊂ on . Clearly Ω = z : f(z) < 0 .Since vanishes at infinity, Ωα Ω . By [13, Cor. 5.4.3], every function holomorphic on a neighbourhood of Ωα can be approximated uniformly on this set by entire functions. (b): Let U =Ω 0 and 2 s(x):=inffjwj : x + i(y + w) 62 W for some y; jyj ≤ 0(x)g: Observe that (i) since 0 is continuous and W is open, s is lower semi-continuous; (ii) V := fx 2 W \ Rd : s(x) > 0g is open and contains (W n U) \ Rd; (iii) if U ⊂⊂ W ,thenV = W \ Rd. We choose a covering of V by sets f 2 Rd j − j }⊂⊂ \ Rd Urj (xj)= x : x xj <rj W : We set ( 1 X1 e jxj2−1 for jxj < 1, 1 '(x)= (x)= " ' (x − x ) : j |≥ 1 j r j 0forx 1; j=1 j License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 3610 P. DOMANSKI´ AND D. VOGT By (i) and a suitable choice of ("j), we get that the series is convergent and 2 (iv) 1(x) <s (x); k k − k k (v) supx2Rd H 1 (x) < 1 supx2Rd H 0 (x) . By (v), := 0 + 1 is a distinguished function. It is immediate that U ⊆ Ω and, by (iv), we get Ω ⊆ W . By (ii), the basis of is equal to W \ Rd. Finally, ⊂⊂ j ⊂ ⊂⊂ if U W , then , by (iii), 1 U\Rd >δ>0.
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