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

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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.

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Correspondence to A. G. Babenko.

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Original Russian Text © A.G. Babenko, T.Z. Naum, 2016, published in Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2016, Vol. 22, No. 4, pp. 53–63.

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Babenko, A.G., Naum, T.Z. One-Sided Integral Approximations of the Generalized Poisson Kernel by Trigonometric Polynomials. Proc. Steklov Inst. Math. 300 (Suppl 1), 38–48 (2018). https://doi.org/10.1134/S0081543818020050

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