Solutions and stability of variant of Wilson's functional equation (original) (raw)

Solutions and stability of a variant of Wilson's functional equation

Proyecciones (Antofagasta), 2018

In this paper we will investigate the solutions and stability of the generalized variant of Wilson's functional equation (E) : f (xy) + χ(y)f (σ(y)x) = 2f (x)g(y), x, y ∈ G, where G is a group, σ is an involutive morphism of G and χ is a character of G. (a) We solve (E) when σ is an involutive automorphism, and we obtain some properties about solutions of (E) when σ is an involutive anti-automorphism. (b) We obtain the Hyers Ulam stability of equation (E). As an application, we prove the superstability of the functional equation f (xy) + χ(y)f (σ(y)x) = 2f (x)f (y), x, y ∈ G.

On Wilson’s functional equations

Aequationes mathematicae, 2014

We find on a group G the solutions f, g : G → C of the functional equation f (xy) + f (y −1 x) = 2f (x)g(y), x, y ∈ G, in terms of characters, additive maps and matrix-elements of irreducible, 2-dimensional representations of G.

Wilson’s functional equation with an endomorphism

2017

In the present paper, we determine the complex-valued solutions ( f ;g) of the functional equation f (xy)+ f (j(y)x) = 2 f (x)g(y); in the setting of groups and monoids, where j is an endomorphism not necessarity involutive. We prove that their solutions can be expressed in terms of multiplicative and additive functions. Many consequences of these results are presented.

A variant of Wilson's functional equation on semigroups

Cornell University - arXiv, 2022

We determine the complex-valued solutions of the following functional equation f (xy) + µ(y)f (σ(y)x) = 2f (x)g(y), x, y ∈ S, where S is a semigroup and σ an automorphism, µ : S → C is a multiplicative function such that µ(xσ(x)) = 1 for all x ∈ S.

A note on Wilson’s functional equation

Aequationes mathematicae

Let S be a semigroup, and F a field of characteristic = 2. If the pair f, g : S → F is a solution of Wilson's µ-functional equation such that f = 0, then g satisfies d'Alembert's µ-functional equation.

Variants of Wilson's functional equation on semigroups

arXiv: General Mathematics, 2019

Given a semigroup SSS generated by its squares equipped with an involutive automorphism sigma\sigmasigma and a multiplicative function mu:StomathbbC\mu:S\to\mathbb{C}mu:StomathbbC such that mu(xsigma(x))=1\mu(x\sigma(x))=1mu(xsigma(x))=1 for all xinSx\in SxinS, we determine the complex-valued solutions of the following functional equations \begin{equation*}f(xy)+\mu(y)f(\sigma(y)x)=2f(x)g(y),\, x,y\in S\end{equation*} and \begin{equation*}f(xy)+\mu(y)f(\sigma(y)x)=2f(y)g(x),\, x,y\in S\end{equation*}

Wilson's functional equations on groups

Aequationes Mathematicae, 1994

We study properties of solutions f, g, h ~ C(G) of the functional equation f(xk. y)z(k) dk = g(x)h(y), x, y ~ G (1) and of the special case ixf(xk • y))f(k) dk = g(x)f(y), G, X~ y (2) where G is a locally compact group, K a compact subgroup of Aut(G) and X a character on K. We show that g and h are associated to certain K-spherical functions and use that to compute the complete set of solutions in special examples; in particular in the case of G = R n.

Solutions and stability of a generalization of wilson's equation

Acta Mathematica Scientia, 2016

In this paper we study the solutions and stability of the generalized Wilson's functional equation G f (xty)dµ(t)+ G f (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 G g(xty)dµ(t) + G g(xtσ(y))dµ(t) = 2g(x)g(y), x, y ∈ G if f = 0 and G f (t.)dµ(t) = 0. We also study some stability theorems of that equation and we establish the stability on noncommutaive 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.