Haar’s condition and joint polynomiality of separate polynomial functions - Voloshyn, H. A.; Kosovan, V. M.; Maslyuchenko, V. K. (Ukrains’kyi Matematychnyi Zhurnal, 2017)
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 = ^n_k=1λ_k f_k, a G-polynomial h = ^n_k,j=1 λ_k,j f_k g_j, and an F G-polynomial (f_kg_j)(x, y) = = f_k(x)g_j(y), where (f_kg_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 × 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-polynomialy. A similar problem is investigated for functions of n variables.