On Hyers-Ulam stability of Wilson's functional equation
The paper investigates the stability problem for spherical functions in the Hyers-Ulam sense. Let \((G,+)\) be a topological abelian group and let \(K\) be a compact subgroup of automorphisms of G with the normalized Haar measure \(\mu\). Further, let the map \[ k\mapsto ky\in G,\qquad k\in K , \] where \(ky\) stands for the action of \(k\in K\) on \(y\in G\), is continuous for each fixed \(y\in G\). A continuous function \(f:G\to\mathbb C\) is called \(K\)-spherical if there exists a nonzero continuous function \(g:G\to\mathbb C\) such that the generalized Wilson's functional equation \[ \int_Kg(x+ky) d\mu (k)=g(x)f(y) \tag{1} \] holds for all \(x,y\in G\). The author proves the following result on the Hyers-Ulam stability of equation (1): Let \(f,g:G\to\mathbb C\) be continuous functions. Assume that there exists \(c\geq 0\) such that \[ \left |\int_Kg(x+ky) d\mu (k)-g(x)f(y)\right |\leq c \] for all \(x,y\in G\). Then either \(f,g\) are bounded or \(g\) is unbounded, \(f\) satisfies the integral equation \[ \int_Kf(x+ky) d\mu (k)=f(x)f(y) \] for all \(x,y\in G\) or \(f\) is unbounded, \(f,g\) satisfy (1) (if \(g\neq 0\) then \(f\) satisfies (2)). This result is applied to the study of the Hyers-Ulam stability of equation (2) and, more generally, the signed functional equation \[ \int_Kg(x+ky)\overline{\chi (k)} d\mu (k)=f(x)g(y), \] where \(\chi :K\to T\) is a continuous homomorphism and \(T=\{z\in\mathbb C:|z|=1\}\).
generalized Wilson's functional equation
scientific article; zbMATH DE number 1556596