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 $f={\sum}_{k=1}^n{\uplambda}_k{f}_k$, 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 _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.