Haar’s condition and joint polynomiality of separate polynomial functions

  • H. A. Voloshyn
  • V. M. Kosovan
  • V. K. Maslyuchenko


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 $f = \sum^n_{k=1}\lambda_k f_k$, a $G$-polynomial $h = \sum^n_{k,j=1} \lambda_{k,j} f_k \otimes g_j$, 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 the following question: under what assumptions every function $h : X \times Y \rightarrow K$, such that all $x$-sections $h^x = h(x, \cdot )$ are $G$-polynomials and all $y$-sections $h_y = h(\cdot , y)$ are $F$ -polynomials, is an $F \otimes G$-polynomialy. A similar problem is investigated for functions of $n$ variables.
How to Cite
VoloshynH. A., KosovanV. M., and MaslyuchenkoV. K. “Haar’s Condition and Joint Polynomiality of separate Polynomial Functions”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 69, no. 1, Jan. 2017, pp. 17-27, http://umj.imath.kiev.ua/index.php/umj/article/view/1673.
Research articles