SOLUTIONS AND STABILITY OF A GENERALIZATION OF WILSON'S EQUATION

doi:10.1016/S0252-9602(16)30040-6

Abstract

Abstract:

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.

Key words: d'Alembert's functional equation, locally compact group, involution, character, complex measure, Wilson's functional equation, Hyers-Ulam stability

CLC Number:

39B82

Bouikhalene BELAID, Elqorachi ELHOUCIEN. SOLUTIONS AND STABILITY OF A GENERALIZATION OF WILSON'S EQUATION[J].Acta mathematica scientia,Series B, 2016, 36(3): 791-801.

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