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Haar’s Condition and the Joint Polynomiality of Separately Polynomial Functions

Research paper by H. A. Voloshyn, M. V. Kosovan, V. K. Maslyuchenko

Indexed on: 31 Oct '17Published on: 29 Jul '17Published in: Ukrainian Mathematical Journal



Abstract

For systems of functions F = {fn ∈ KX : n ∈ ℕ} and G = {gn ∈ KY : n ∈ ℕ}, we consider an F-polynomial \( f={\sum}_{k=1}^n{\uplambda}_k{f}_k \), a G-polynomial \( g={\sum}_{k=1}^n{\uplambda}_k{g}_k \), and an F ⊗ G-polynomial \( h={\sum}_{k,j=1}^n{\uplambda}_{k,j}{f}_k\otimes {g}_j \), where (fk ⊗ gj)(x, y) = fk(x)gj(y). By using the well-known Haar’s condition from the approximation theory, we study the following problem: Under what assumptions every function h : X × Y → K, such that all x-sections hx = h(x, ·) are G-polynomials and all y -sections hy = h(·, y) are F-polynomials, is an F ⊗ G-polynomial? A similar problem is investigated for functions of n variables.