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On Stetkær type functional equations and Hiers-Ulam stability. (English) Zbl 1111.39021

Given a locally compact group \(G\), a compact subgroup \(K\) of the group of morphisms of \(G\), a character \(\chi:K\to\mathbb C\), and a \(K\)-invariant bounded measure on \(G\), the authors investigate the functional equation \[ \int_G\int_K f(xtk\cdot y)\overline{\chi}(k)dkd\mu(t)=g(x)h(y), \qquad x,y\in G, \tag{(1)} \] where \(f,g,h\in C_b(G)\) are unknown functions. This equation contains, as a particular case, the D’Alembert and Wilson functional equations.
The main results of the paper describe the connection of the solutions equation (1) to the solutions of Badora’s functional equation \[ \int_G\int_K f(xtk\cdot y)dkd\mu(t)=f(x)f(y), \qquad x,y\in G. \] The Hyers-Ulam stability problem of equation (1) is also studied, and a superstability-type result is obtained.

MSC:

39B52 Functional equations for functions with more general domains and/or ranges
39B82 Stability, separation, extension, and related topics for functional equations
22D05 General properties and structure of locally compact groups
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