Decomposable functions of several variables

Summary

The paper deals with the functions which admit the decomposition

$$H(x,y,z) = \sum\limits_{i = 1}^m {f_i (x)\varphi _i (y,z)}$$

or

$$H(x,y,z) = \sum\limits_{i = 1}^m {\sum\limits_{j = 1}^n {\sum\limits_{k = 1}^p {\alpha _{ijk} f_i (x)g_j (y)h_k (z)} } }$$

and so it solves the problemP286 of these Aequationes, proposed also byH. Gauchman andL. A. Rubel in [3]. Necessary and sufficient conditions onH to have such a decomposition are formulated both for differentiable functions in terms of partial derivatives and for functions without any regularity assumptions. Many of these results can be extended to the case of more than three variables.