Vector Valued Polynomials, Exponential Polynomials and Vector Valued Harmonic Analysis

Abstract

AbstractLet 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 $$f:G \rightarrow E$$ 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).