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

For systems of functions F = { f n ∈ K X : n ∈ ℕ} and G = { g n ∈ K Y : n ∈ ℕ} , we consider an F -polynomial f = ∑ k = 1 n λ k f k , a G -polynomial g = ∑ k = 1 n λ k g k , and an F ⊗ G -polynomial h = ∑ k , j = 1 n λ k , j f k ⊗ g j , where ( f k ⊗ g j )( x, y ) = f k ( x ) g j ( y ) . By using th...

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Bibliographic Details
Published in:Ukrainian mathematical journal 2017-06, Vol.69 (1), p.19-31
Main Authors: Voloshyn, H. A., Kosovan, M. V., Maslyuchenko, V. K.
Format: Article
Language:English
Subjects:
Algebra
Analysis
Applications of Mathematics
Functions (mathematics)
Geometry
Mathematical analysis
Mathematics
Mathematics and Statistics
Polynomials
Statistics
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Summary:For systems of functions F = { f n ∈ K X : n ∈ ℕ} and G = { g n ∈ K Y : n ∈ ℕ} , we consider an F -polynomial f = ∑ k = 1 n λ k f k , a G -polynomial g = ∑ k = 1 n λ k g k , and an F ⊗ G -polynomial h = ∑ k , j = 1 n λ k , j f k ⊗ g j , where ( f k ⊗ 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 problem: Under what assumptions every function h : X × Y → K, such that all x -sections h x = h ( x, · ) are G -polynomials and all y -sections h y = h ( ·, y ) are F -polynomials, is an F ⊗ G -polynomial? A similar problem is investigated for functions of n variables.
ISSN:0041-5995
1573-9376
DOI:10.1007/s11253-017-1345-3