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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2409.10629 |
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| _version_ | 1866914950568476672 |
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| author | Serdyuk, Anatoly Stepaniuk, Tetiana |
| author_facet | Serdyuk, Anatoly Stepaniuk, Tetiana |
| contents | We present a survey of results related to the solution of Kolmogorov--Nikolsky problem for Fourier sums on the classes of generalized Poisson integrals $C^{α,r}_{β,p}$, which consists in finding of asymptotic equalities for exact upper boundaries o f uniform norms of deviations of partial Fourier sums on the classes of $2π$--periodic functions $C^{α,r}_{β,p}$, which are defined as convolutions of the functions, which belong to the unit balls pf the spaces $L_{p}$, $1\leq p\leq \infty$, with generalized Poisson kernels $$ P_{α,r,β}(t)=\sum\limits_{k=1}^{\infty}e^{-αk^{r}}\cos \big(kt-\frac{βπ}{2}\big), \ α>0, r>0, \ β\in \mathbb{R}.$$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_10629 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Approximation by Fourier sums on the classes of generalized Poisson integrals Serdyuk, Anatoly Stepaniuk, Tetiana Classical Analysis and ODEs We present a survey of results related to the solution of Kolmogorov--Nikolsky problem for Fourier sums on the classes of generalized Poisson integrals $C^{α,r}_{β,p}$, which consists in finding of asymptotic equalities for exact upper boundaries o f uniform norms of deviations of partial Fourier sums on the classes of $2π$--periodic functions $C^{α,r}_{β,p}$, which are defined as convolutions of the functions, which belong to the unit balls pf the spaces $L_{p}$, $1\leq p\leq \infty$, with generalized Poisson kernels $$ P_{α,r,β}(t)=\sum\limits_{k=1}^{\infty}e^{-αk^{r}}\cos \big(kt-\frac{βπ}{2}\big), \ α>0, r>0, \ β\in \mathbb{R}.$$ |
| title | Approximation by Fourier sums on the classes of generalized Poisson integrals |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2409.10629 |