One-Sided Integral Approximations of the Generalized Poisson Kernel by Trigonometric Polynomials

Abstract

We consider the generalized Poisson kernel Π q,α = cos(απ/2)P + sin(απ/2)Q with q ∈ (−1, 1) and α ∈ ℝ, which is a linear combination of the Poisson kernel $$P(t) = 1/2 + \sum\nolimits_{k = 1}^\infty {{q^k}} \cos kt$$ and the conjugate Poisson kernel $$Q(t) = \sum\nolimits_{k = 1}^\infty {{q^k}} \sin kt$$ . The values of the best integral approximation to the kernel Π q,α from below and from above by trigonometric polynomials of degree not exceeding a given number are found. The corresponding polynomials of the best one-sided approximation are obtained.