Solutions and stability of a generalization of wilson's equation

In this paper we study the solutions and stability of the generalized Wilson's functional equation ∫Gf(xty)dμ(t)+∫Gf(xtσ(y))dμ(t)=2f(x)g(y), x,y∈G, where G is a locally compact group, σ is a continuous involution of G and μ is an idempotent complex measure with compact support and which is σ-in...

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Bibliographic Details
Published in:Acta mathematica scientia 2016-05, Vol.36 (3), p.791-801
Main Authors: BELAID, Bouikhalene, ELHOUCIEN, Elqorachi
Format: Article
Language:English
Subjects:
39B32
39B52
39B82
character
complex measure
d'Alembert's functional equation
Hyers-Ulam stability
involution
locally compact group
Mathematical analysis
Stability
Theorems
Wilson's functional equation
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Summary:In this paper we study the solutions and stability of the generalized Wilson's functional equation ∫Gf(xty)dμ(t)+∫Gf(xtσ(y))dμ(t)=2f(x)g(y), x,y∈G, where G is a locally compact group, σ is a continuous involution of G and μ is an idempotent complex measure with compact support and which is σ-invariant. We show that ∫Gg(xty)dμ(t)+∫Gg(xtσ(y))dμ(t)=2g(x)g(y)  if f≠0 and ∫Gf(t.)dμ(t)≠0,  where [∫Gf(t.)dμ(t)](x)=∫Gf(tx)dμ(t). We also study some stability theorems of that equation and we establish the stability on noncommutative groups of the classical Wilson's functional equation f(xy)+χ(y)f(xσ(y))=2f(x)g(y) x,y∈G where χ is a unitary character of G.
ISSN:0252-9602
1572-9087
DOI:10.1016/S0252-9602(16)30040-6