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- Title
Haar's Condition and the Joint Polynomiality of Separately Polynomial Functions.
- Authors
Voloshyn, H.; Kosovan, M.; Maslyuchenko, V.
- Abstract
For systems of functions F = { f ∈ K : n ∈ ℕ} and G = { g ∈ K : 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 ( f ⊗ g )( x, y) = f ( x) g ( 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 h = h( x, ·) are G-polynomials and all y -sections h = h( ·, y) are F-polynomials, is an F ⊗ G-polynomial? A similar problem is investigated for functions of n variables.
- Subjects
MATHEMATICAL functions; POLYNOMIALS; HAAR system (Mathematics); APPROXIMATION theory; MATHEMATICAL variables
- Publication
Ukrainian Mathematical Journal, 2017, Vol 69, Issue 1, p19
- ISSN
0041-5995
- Publication type
Article
- DOI
10.1007/s11253-017-1345-3