Approximation by Fourier sums and best approximation in classes of analytic functions (Q2739812)

Let \(f(x)\) be a \(2\pi\)-periodic function, let \({\mathcal T}_{2n-1}\) be the set of all trigonometric polynomials \(f_{n-1}(\cdot)\) of the order \(\leq n-1\), and let \(E_n(f)_s=\inf_{t_{n-1}\in{\mathcal T}_{2n-1}} \|f(\cdot)-f_{n-1}(\cdot)\|_s\) be the best approximation of the function \(f(\cdot)\) by trigonometric polynomials \(f_{n-1}(\cdot)\) from the set \({\mathcal T}_{2n-1}\) in the space \(L_p\). Let, for \(f\in L\), \(S_n(f;x)\) be a partial sum of the Fourier series and let \(\rho_n(f;x)=f(x)-S_{n-1}(f;x)\). The authors give the asymptotic estimates for the values \(\|\rho_n(f;x)\|_s\), \(E_n(f)_s\), \(f\in L^{\overline{\psi}}{\mathcal N}\), where \({\mathcal N}\) is a given subset of \(L_p, 0\leq p,s\leq\infty\), and the values NEWLINE\[NEWLINE{\mathcal E}_n(L^{\overline{\psi}}{\mathcal N})_s=\sup_{ f\in L^{\overline{\psi}}{\mathcal N}}\|f-S_{n-1}(f)\|_s NEWLINE\]NEWLINE and NEWLINE\[NEWLINEE_n(L^{\overline{\psi}}{\mathcal N})_s=\sup_{ f\in L^{\overline{\psi}}{\mathcal N}}E_n(f)_s= \sup_{f\in L^{\overline{\psi}}{\mathcal N}} \inf_{t_{n-1}\in{\mathcal T}_{2n-1}}\|f(\cdot)-f_{n-1}(\cdot)\|_s NEWLINE\]NEWLINE For more details see \textit{A. I. Stepanets} [Ukr. Math. J. 49, No. 8, 1201-1251 (1997; Zbl 0938.42004)].