Asymptotic equalities for best approximations for classes of infinitely differentiable functions defined by the modulus of continuity (Q325559)

This paper is about trigonometric polynomial approximation of \(2\pi\)-periodic functions \(f\in L_1\) given by the integral transform \(f(z)=\frac{a_0(f)}{2}+ \frac{1}{\pi}\int_{-\i}^\pi \varphi(x-t)\Psi_\beta(t)dt\) whose kernel \(\Psi_\beta(t)\sim \sum_{k=1}^\infty \psi(k)\cos(kt-\beta\pi/2)\), \(\beta\in\mathbb{R}\), has Fourier coefficients \(\psi(k)>0\) that go to zero faster than any power, like in the generalized Poisson kernel \(P_\beta^{\alpha,r}\). The function \(\varphi\) belongs to \(C\) or \(L_p\) (\(1\leq p<\infty\)) and its modulus of continuity is bounded by a (convex) function \(\omega(t)\). The paper gives (under additional technical conditions) asymptotic error estimates (as \(n\to\infty\)) for an \(n\)-th order trigonometric approximation of such functions \(f\) either in \(C\) or in \(L_p\). It is shown that in \(C\) and \(L_1\) the estimates are sharp. For the special case of generalized Poisson kernels, such results were given in [\textit{A. S. Serdyuk} and \textit{I. V. Sokolenko}, Ukr. Mat. Zh. 62, No. 7, 979--996 (2010); translation in Ukr. Math. J. 62, No. 7, 1139--1157 (2010; Zbl 1224.41027); J. Approx. Theory 163, No. 11, 1692--1706 (2011; Zbl 1232.41034)].