SOME PROBLEMS IN THE THEORY OF ALMOST PERIODIC FUNCTIONS

The set of almost periodic functions f(x) ∼ Σ a(λ) eiλx on the additive group of real numbers R is considered as an algebra A under convolution multiplication. The isometric automorphisms of A are determined. A characterization is given of all multipliers q(λ) that transfer every almost periodic Fou...

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Bibliographic Details
Published inMathematica scandinavica Vol. 3; no. 1; pp. 49 - 67
Main Author HELGASON, SIGURÐUR
Format Journal Article
LanguageEnglish
Published KØBENHAVN 01.12.1955
Subjects
Algebra
Automorphisms
Fourier series
Mathematical functions
Mathematical induction
Mathematical permutation
Mathematical theorems
Periodic functions
Topology
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Summary:The set of almost periodic functions f(x) ∼ Σ a(λ) eiλx on the additive group of real numbers R is considered as an algebra A under convolution multiplication. The isometric automorphisms of A are determined. A characterization is given of all multipliers q(λ) that transfer every almost periodic Fourier series Σ a(λ) eiλx into an almost periodic Fourier series Σ q(λ) a(λ) eiλx. In the special case when q(λ) is the characteristic function of a proper subset S of R the following is proved: (i) S has mean-density 0. (ii) If fS (x) ∼ Σ λ ∈ S a(λ) eiλx is positive whenever f(x) ∼ Σ a(λ) eiλx is positive, then S is a subgroup of R and all subgroups have this property. The results hold for every abelian group G, with the exception of (i) which however holds if the character group G* is infinitely divisible.
ISSN:0025-5521
1903-1807
DOI:10.7146/math.scand.a-10425