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Solution of the generalized cube and octahedron functional equations. (English) Zbl 0664.39007

The authors consider equations \[ \sum^{1}_{j_ 1,...,j_ n=0}f(x_ 1+(-1)^{j_ 1}y,...,x_ n+(-1)^{j_ n}y)=2^ nf(x_ 1,...,x_ n) \] and \[ \sum^{1}_{i=0}[f(x_ 1+(-1)^ iy,x_ 2,...,x_ n)+f(x_ 1,x_ 2+(-1)^ iy,x_ 3,...,x_ n)+ \]
\[ f(x_ 1,...,x_{n-1},x_ n+(-1)^ iy)]=2nf(x_ 1,...,x_ n) \] with \(n=4\) for f: \({\mathbb{R}}^ 4\to {\mathbb{R}}\). It was proved by L. Sweet [Aequationes Math. 22, 29-38 (1981; Zbl 0475.39016)] that in this case \((n=4)\) these equations are equivalent for f: \(G^ n\to V\) under the assumption that G is an abelian group divisible by 2 and V is a \({\mathbb{Q}}\)-vector space. In the paper under review this equivalence is proved for continuous f: \({\mathbb{R}}^ 4\to {\mathbb{R}}\) using distributions and the form of continuous solutions is obtained (a polynomial of the seventh degree in each of the four variables). A recurrence relation relating the general solutions of the n-dimensional octahedron equation (the second one) and its (n-1)-dimensional version is also formulated.
Reviewer: K.Baron

MSC:

39B99 Functional equations and inequalities

Citations:

Zbl 0475.39016
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