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On certain generalizations of the Levi-Civita and Wilson functional equations. (English) Zbl 1382.43006

The authors investigate the functional equation \[ \sum_{i=1}^m f_i(b_ix+c_iy)=\sum_{k=1}^n u_k(y)v_k(x) (1) \] for all \(x, y \in \mathbb R^d\), where \(f_i, u_k, v_k\), \(1\leq i\leq m\), \(1\leq k\leq n\), are functions defined on \(\mathbb R^d\), and \(b_i, c_i \in GL(d,\mathbb R)\). This functional equation is a generalization of the Wilson equation, the equation of iterated differences and the equation characterizing the harmonic polynomials.
Moreover, they study the more general functional equation \[ \sum_{i=1}^m f_i(b_ix+c_iy)=\sum_{k=1}^n u_k(y)v_k(x)+\sum_{s=1}^N P_s(x)w_s(y)\exp \langle x,\phi_s(y)\rangle (2) \] where the \(P_s\) are polynomials and the functions \(w_s, \phi_s\) are arbitrary.
The following theorem is proved:
Theorem. If the functions \(v_k\) are continuous and the matrices \(b_i, c_i\) and \(b_i^{-1}c_i-b_j^{-1}c_j\) (for \(i\neq j\)) are invertible, then all solutions \(f_i\in C(\mathbb R^d)\) of equation (1) and of equation (2) are exponential polynomials.
In the last section of the paper, equation (1) is studied in the distributional setting.

MSC:

43A45 Spectral synthesis on groups, semigroups, etc.
39A70 Difference operators
39B52 Functional equations for functions with more general domains and/or ranges
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