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Sums of an entire function in certain weighted \(L^2\)-spaces. (English) Zbl 1038.30012

The author considers the entire solutions of the functional equation \(f(z+\sigma)-f(z)=g(z)\), where \(\sigma\) is a nonzero constant and \(g(z)\) is a given entire function, with growth conditions. By an a priori estimate method in certain weighted \(L^{2}\)-spaces, the author proves the existence of certain types of solutions of this equation.

MSC:

30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
30D15 Special classes of entire functions of one complex variable and growth estimates
39A70 Difference operators
39B32 Functional equations for complex functions
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