Solution of the Kolmogorov-Nikol'skij problem for the Poisson integrals of continuous functions (Q2759297)

The celebrated Kolmogorov-Nikol'skij type asymptotic relation is obtained for the classes each element of which is the convolution of the Poisson kernel and a continuous function with modulus of continuity not exceeding a given majorant. The proof is lengthy and sophisticated as all proofs of such results. The author starts with a detailed historical survey and finishes with the 37 items long list of references. Under these circumstances, it is quite strange that nothing is mentioned about very important and recent results on the second term of asymptotics due to \textit{V. O. Leont'ev} [``The second term in Kolmogorov's asymptotic formula for approximation of Fourier series by partial sums'', Dokl. Akad. Nauk Ukr. SSR, Ser. A 1990, No. 5, 17-21 (Russian)(1990; Zbl 0712.42004); and ``Asymptotics of the approximation of differentiable functions by a Fourier series'', Russ. Acad. Sci., Dokl., Math. 46, 210-213 (1993); translation from Dokl. Akad Nauk, Ross. Akad. Nauk 326, No. 1, 31-34 (1992; Zbl 0789.42003)]. See also \textit{S. A. Telyakovskij} [``On approximation of differentiable functions of high smoothness by Fourier sums'', Proc. Steklov Inst. Math. 198, 183-201 (1994); translation from Tr. Mat. Inst. Steklova 198, 193-211 (1992; Zbl 0821.42002)].