The Levi-Civita equation in function classes

Let G be an Abelian topological semigroup with unit. By a classical result (called Theorem  A ), if V is a finite dimensional translation invariant linear space of complex valued continuous functions defined on G , then every element of V is an exponential polynomial. More precisely, every element o...

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Veröffentlicht in:Aequationes mathematicae Jg. 94; H. 4; S. 689 - 701
1. Verfasser: Laczkovich, Miklós
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Cham Springer International Publishing 01.08.2020
Springer Nature B.V
Schlagworte:
Analysis
Combinatorics
Continuity (mathematics)
Functions (mathematics)
Mathematical analysis
Mathematics
Mathematics and Statistics
Polynomials
Theorems
Topology
Primary 39B52
Exponential polynomials
Translational invariant subspaces of function classes
Secondary 43A45
ISSN:0001-9054, 1420-8903
Online-Zugang:Volltext
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Zusammenfassung:Let G be an Abelian topological semigroup with unit. By a classical result (called Theorem  A ), if V is a finite dimensional translation invariant linear space of complex valued continuous functions defined on G , then every element of V is an exponential polynomial. More precisely, every element of V is of the form ∑ i = 1 n p i · m i , where m 1 , … , m n are exponentials belonging to V , and p 1 , … , p n are polynomials of continuous additive functions. We generalize this statement by replacing the set of continuous functions by any algebra A of complex valued functions such that whenever an exponential m belongs to A , then m - 1 ∈ A . As special cases we find that Theorem  A remains valid even if the topology on G is not compatible with the operation on G , or if the set of continuous functions is replaced by the set of measurable functions with respect to an arbitrary σ -algebra. We give two proofs of the result. The first is based on Theorem  A . The second proof is independent, and seems to be more elementary than the existing proofs of Theorem  A .
Bibliographie:ObjectType-Article-1
SourceType-Scholarly Journals-1
ObjectType-Feature-2
content type line 14
ISSN:0001-9054
1420-8903
DOI:10.1007/s00010-019-00686-1