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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| Published in: | Aequationes mathematicae Vol. 94; no. 4; pp. 689 - 701 |
|---|---|
| Main Author: | |
| Format: | Journal Article |
| Language: | English |
| Published: |
Cham
Springer International Publishing
01.08.2020
Springer Nature B.V |
| Subjects: | |
| ISSN: | 0001-9054, 1420-8903 |
| Online Access: | Get full text |
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| Summary: | 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
. |
|---|---|
| Bibliography: | 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 |