The univalent polynomial of Suffridge as a summability kernel

Abstract

A positive summability trigonometric kernel is generated through a sequence of univalent polynomials constructed by Suffridge. We prove that the convolution {K n *f } approximates every continuous 2π-periodic function f with the rate ω( f,1 / n), where ω( f,δ) denotes the modulus of continuity, and this provides a new proof of the classical Jackson's theorem. Despite that it turns out that Kn (θ) coincide with positive cosine polynomials generated by Fejér, our proof differs from others known in the literature.