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arXiv · 2303.05568

Approximation of generalized Poisson integrals by interpolation trigonometric polynomials

Anatoly Serdyuk·Tetiana Stepaniuk
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Abstract

In this paper we establish asymptotically best possible interpolation Lebesgue-type inequalities for $2\pi$-periodic functions $f$, which are representable as generalized Poisson integrals of the functions $\varphi$ from the space $L_p$, $1\leq p\leq \infty$. In these inequalities the deviation of the interpolation Lagrange polynomials $|f(x)- \tilde{S}_{n-1}(f;x)|$ for every $x\in\mathbb{R}$ is expressed via the best approximations $E_{n}(\varphi)_{L_{p}}$ of the functions $\varphi$ be trigonometric polynomials in $L_{p}$-metrics. We also find asymptotic equalities for the exact upper bounds of points approximations by interpolation trigonometric polynomials on the classes $C^{\alpha,r}_{\beta,p}$ of generalized Poisson integrals of the functions, which belong to the unit balls of the spaces $L_p$, $1\leq p\leq\infty$.

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Anatoly Serdyuk, Tetiana Stepaniuk. 2023-03-09. Approximation of generalized Poisson integrals by interpolation trigonometric polynomials. https://arxiv.org/abs/2303.05568

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