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
Voloshyn, H. A., V. M. Kosovan, and V. K. Maslyuchenko. “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