Vector Valued Polynomials, Exponential Polynomials and Vector Valued Harmonic Analysis
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Vector Valued Polynomials, Exponential Polynomials and Vector Valued Harmonic Analysis M. Laczkovich Abstract. Let G be a topological Abelian semigroup with unit, and let E be a Banach space. We define, for functions mapping G into E, the classes of polynomials, generalized polynomials, local polynomials, exponential polynomials, and some other relevant classes. We establish their connections with each other and find their representations in terms of the corresponding complex valued classes. We also investigate spectral synthesis and analysis in the class C(G, E) of continuous functions f : G → E. It is known that if G is a compact Abelian group and E is a Banach space, then spectral synthesis holds in C(G, E). We give a self-contained proof of this fact, independent of the theory of almost periodic functions. On the other hand, we show that if G is an infinite and discrete Abelian group and E is a Banach space of infinite dimension, then even spectral analysis fails in C(G, E). We also prove that if G is discrete, has finite torsion free rank and if E is a Banach space of finite dimension, then spectral synthesis holds in C(G, E). Mathematics Subject Classification. Primary 39B52; Secondary 22A20. Keywords. Banach space valued polynomials, exponential polynomials, spectral synthesis.
1. Introduction Let G be a topological Abelian semigroup with unit. For complex valued functions defined on G, the classes of polynomials, generalized polynomials, local Supported by the Hungarian National Foundation for Scientific Research, Grant No. K124749. 0123456789().: V,-vol
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M. Laczkovich
Results Math
polynomials, exponential polynomials have been defined and their basic properties have been established. (See, e.g., [5–8,10–12,14–16] and the references therein.) Our first aim is to extend these notions to the vector valued case. Let E be a Banach space, and let C(G, E) denote the set of continuous functions f : G → E. A function f ∈ C(G, E) is a generalized polynomial, if there is an n ≥ 0 such that Δh1 . . . Δhn+1 f = 0 for every h1 , . . . , hn+1 ∈ G, where Δh is the difference operator. We say that f ∈ C(G, E) is a polynomial, if it is a generalized polynomial, and the linear span of its translates is of finite dimension; f is a w-polynomial, if u ◦ f is a polynomial for every u ∈ E ∗ , and f is a local polynomial, if it is a polynomial on every finitely generated subsemigroup. We show that each of the classes of polynomials, w-polynomials, generalized polynomials, local polynomials is contained in the next class (Theorem 8). We also prove that if G is an Abelian group and has a dense subgroup with finite torsion free rank, then these classes coincide (see Theorem 9). We introduce the classes of exponential polynomials and w-exponential polynomials as well, establish their representations and connection with polynomials and w-polynomials. We also investigate spectral synthesis and analysis in the class C(G, E). It is known that if G is a compact Abelian group and E is a Banach space, t
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