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Let \(X=L_{p}(\mathbb {T})\) , \(1\leq p \leq \infty \) , or \(C(\mathbb {T})\) . The Kolmogorov–Nikolsky problem deals with establishing of asymptotic equalities for exact upper bounds for \({\sup \limits _{f\in \mathfrak {N}}\| f-S_{n}(f) \|_{X}}\) , where \(f\in \mathfrak {N} \subset X\) , and \(S_{n}\) is the n-th partial sum of the Fourier series of f. After giving a short survey over some classical results, we discuss in a more detailed way the above problem in \(C(\mathbb {T})\) for classes generated by convolutions of the functions, which belong to the unit balls of the spaces \(L_{p}\) , \(1\leq p\leq \infty \) , with generalized Poisson kernels \(P_{\alpha ,r,\beta }(t)=\sum \limits _{k=1}^{\infty }e^{-\alpha k^{r}}\cos \big (kt-\frac {\beta \pi }{2}\big ), \ \alpha >0, r>0, \ \beta \in \mathbb {R}.\)

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Approximation by Fourier Sums on the Classes of Generalized Poisson Integrals

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  • 发表日期：

    2026年

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  • Anatoly S. Serdyuk,
  • Tetiana A. Stepanyuk

摘要

Let \(X=L_{p}(\mathbb {T})\) , \(1\leq p \leq \infty \) , or \(C(\mathbb {T})\) . The Kolmogorov–Nikolsky problem deals with establishing of asymptotic equalities for exact upper bounds for \({\sup \limits _{f\in \mathfrak {N}}\| f-S_{n}(f) \|_{X}}\) , where \(f\in \mathfrak {N} \subset X\) , and \(S_{n}\) is the n-th partial sum of the Fourier series of f. After giving a short survey over some classical results, we discuss in a more detailed way the above problem in \(C(\mathbb {T})\) for classes generated by convolutions of the functions, which belong to the unit balls of the spaces \(L_{p}\) , \(1\leq p\leq \infty \) , with generalized Poisson kernels \(P_{\alpha ,r,\beta }(t)=\sum \limits _{k=1}^{\infty }e^{-\alpha k^{r}}\cos \big (kt-\frac {\beta \pi }{2}\big ), \ \alpha >0, r>0, \ \beta \in \mathbb {R}.\)