On mean periodic functions (Q1818254)

Based on fundamental results on spectral synthesis the author introduced a Fourier type transformation for mean periodic functions on the real line. In this paper there is a continuation of some ideas presented in his earlier papers. Some properties of the Fourier-like transformation are presented. Applying this transformation the author introduces some kind of conditional expectation for mean periodic functions and exhibits its basic properties. Moreover, it is shown that the Fourier-like transformation can be used to solve convolution-type functional equations. The functional equation \(f(x+y)+ g(xy)= 2f(x)g(y)\), where \(f,g: \mathbb{R}\to \mathbb{C}\) are unknown continuous functions illustrates how the Fourier transform of mean periodic functions can be used to solve this equation.