On Fréchet's functional equation (Q478508)

The paper is devoted to the basic theorem on polynomials, originally proved by \textit{D. Ž. Đoković} [Ann. Pol. Math. 22, 189--198 (1969; Zbl 0187.39903)], which states that a function \(f: G\to\mathbb{C}\), defined on an abelian group \(G\), is a generalized polynomial of degree at most \(n\), i.e., satisfies Fréchet's equation \[ \Delta_{y_1,y_2,\dots,y_{n+1}}f=0 \] if and only if it satisfies \[ \Delta^{n+1}_{y}f=0. \] The usual difference operators are here defined using characteristic functions \(\delta\) and their convolutions: \[ \Delta_y=\delta_{-y}-\delta_{0};\quad \Delta_{y_1,y_2,\dots,y_{n+1}}=\prod_{j=1}^{n+1}\Delta_{y_j};\quad \Delta^{n+1}_{y}=\Delta_{y,y,\dots,y}. \] Using the spectral synthesis a new short proof of the above mentioned result is given.