Haar’s condition and joint polynomiality of separate polynomial functions - Voloshyn, H. A.; Kosovan, V. M.; Maslyuchenko, V. K. (Ukrains’kyi Matematychnyi Zhurnal, 2017)
For systems of functions $F = \{ f_n \in K^X : n \in N\}$ and $G = \{ g_n \in K^Y : n \in N\}$ we consider an $F$ -polynomial$f = \sum^n_{k=1}λ_k f_k$, a $G$-polynomial $h = \sum^n_{k,j=1}λ_{k,j} f_k \otimes g_j$, and an $F \otimes G$-polynomial $(f_k\otimes g_j)(x, y) = = f_k(x)g_j(y)$, where$(f_k\otimes g_j)(x, y) = f_k(x)g_j(y)$. By using the well-known Haar’s condition from the approximation theory we study thefollowing question: under what assumptions every function $h : X × Y → K$, such that all $x$-sections $h^x = h(x, · )$are $G$-polynomials and all $y$-sections $h_y = h(· , y)$ are $F$ -polynomials, is an $F \otimes G$-polynomialy. A similar problem isinvestigated for functions of $n$ variables.