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 K_n (θ) coincide with positive cosine polynomials generated by Fejér, our proof differs from others known in the literature.

Keywords:

• Univalent polynomial
• Positive summability kernel
• Jackson's theorem

AMS Subject Classifications: :

• Primary 30C10
• 42A10

Acknowledgement

This Research was supported by the Brazilian Science Foundation FAPESP under Grant 03/10469-4.