Approximation of generalized Poisson integrals by interpolating trigonometric polynomials
UDC 517.5 We establish asymptotically unimprovable interpolation analogs of Lebesgue-type inequalities for $2\pi$-periodic functions $f$ that can be represented in the form of generalized Poisson integrals of  functions $\varphi$ from the space $L_p,$ $1\leq p\leq \infty.$ ...
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| Date: | 2023 |
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| Main Authors: | , , , |
| Format: | Article |
| Language: | Ukrainian |
| Published: |
Institute of Mathematics, NAS of Ukraine
2023
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/7523 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
| Download file: |  |
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Ukrains’kyi Matematychnyi Zhurnal| Summary: | UDC 517.5
We establish asymptotically unimprovable interpolation analogs of Lebesgue-type inequalities for $2\pi$-periodic functions $f$ that can be represented in the form of generalized Poisson integrals of  functions $\varphi$ from the space $L_p,$ $1\leq p\leq \infty.$  In these inequalities, the moduli of deviations of the interpolation Lagrange polynomials $|f(x)- \tilde{S}_{n-1}(f;x)|$ for every $x\in\mathbb{R}$ are expressed via the best approximations $E_{n}(\varphi)_{L_{p}}$ of the functions $\varphi$ by trigonometric polynomials in the $L_{p}$-metrics. We also deduce asymptotic equalities for the exact upper bounds of pointwise approximations  of the  generalized Poisson integrals of functions that belong to the unit balls in the spaces $L_p,$ $1\leq p\leq\infty,$ by interpolating trigonometric polynomials on the classes $C^{\alpha,r}_{\beta,p}.$ |
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| DOI: | 10.37863/umzh.v75i7.7523 |