Haar’s condition and joint polynomiality of separate polynomial functions

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

Abstract

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.
Published
25.01.2017
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.
Section
Research articles