On approximation by bivariate incomplete polynomials

Abstract

In this paper we prove that given certain convex domains Δ on the plane, ε>0, andf∈C(Δ) such thatf=0 on θ²Δ={(θ² x,θ² y):(x,y)∉Δ} (0<θ<1), a polynomialp(x, y) of the form

$$p(x,y) = \sum\limits_{\theta n \leqslant k + l \leqslant n} {a_{kl} x^k y^l }$$

exists such that ∥f−p∥ _C(Δ) ≤ε. The admissible convex domains include triangles and parallelograms with a vertex at the origin and sections of unit disk.