Asymptotic behavior of best approximations of classes of Poisson integrals of functions from \(H_\omega\)
Let \(C\) be the space of \(2\pi\)-periodic continuous functions \(\phi\) with the norm \(\|\phi\|_c= \max_t|\phi(t)|\), \(L^\infty\) be the space of \(2\pi\)-periodic functions \(\phi\) which are Lebesgue measurable and essentially bounded with the norm \(\|\phi\|_\infty= \text{ess\,sup}_k |\phi(t)|\), and \(L_1\) be the space of \(2\pi\)-periodic functions \(\phi\) which are Lebesgue summable on the period \((0,2\pi)\) with the norm \(\|\phi\|_1= \int^\pi_{-\pi}|\phi(t)|\,dt\). Let \(B_p= \{\phi:\| f\|_p\leq 1\}\), \(p= 1,\,\infty\) and \(H_\omega= \{\phi\in C:\omega(\phi, t)\leq \omega(t)\}\), where \(\omega(\phi;t)\) is the modulus of continuity of the function \(\phi\) and \(\omega(t)\) is a fixed modulus of continuity. Define \(C^q_{\beta,p}\) and \(C^q_\beta H_\omega\) to be sets of all continuous \(2\pi\)-periodic functions \(f\) representable by convolutions \[ f(x)= A_0+{1\over\pi} \int^\pi_{-\pi} \phi(x- t)P^q_\beta(t)\,dt \] with the Poisson kernel \[ P^q_\beta(t)= \sum^\infty_{k=1} q^k\cos\Biggl(kt- {\beta\pi\over 2}\Biggr), \] \(q\in (0,1)\), \(\beta\in\mathbb{R}\), where \(A_0\) is an arbitrary constant, and \(\phi\in B_p\) or \(\phi\in H_\omega\), respectively. In this paper, the authors find asymptotic formulae for the least upper bounds of approximation in the metric of the space \(C\) by using a linear method \(U^*_n\) of classes of Poisson integrals of continuous \(2\pi\)-periodic functions in the case where their moduli of continuity do not exceed fixed convex majorants [The method \(U^*_n\) is defined as: For each function \(f\in C^q_\beta H\omega\), we associate with it a trigonometric polynomial \(U^*_{n-1}(f; x)= U^*_{n-1}(q; \beta; f; x)\) of the form \[ U^*_{n-1} (f; x)= A_0+ \sum^{n-1}_{k=1} (\lambda^{(n)}_k(a_k\cos kx+ b_k\sin kx)+ v^{(n)}_k(a_k\sin k=- b_k\cos kx)), \] where \(a_k= a_k(\phi)\), \(b_k= b_k(\phi)\), \(k= 1,2,\dots\), are the Fourier coefficients of the function \(\phi\), and the number \(\lambda^{(n)}_k= \lambda^{(n)}_k(q;\beta)\), and \(v^{(n)}_k= v^{(n)}_k(q;\beta)\), \(k= 1,\dots, n-1\), \(n\in\mathbb{N}\), are respectively defined by the equalities \[ \lambda^{(n)}_k= (q^k- q^{2n-k})\cos{\beta\pi\over 2},\;v^{(n)}_k= (q_k- q^{2n-k})\sin{\beta\pi\over 2},\;k= 1,\dots, n-1. \] On the basis of these results, the authors obtain asymptotic formulae for the best uniform approximations by trigonometric polynomials and, under some restrictions, asymptotic formulae for Kolmogorov widths \[ d_N(C^q_\beta H\omega)_c= \underset{F_n}{}{\text{inf}} \sup_{f\in C^q_\beta} M\omega \underset{u\in F_N}{}{\text{inf}}\| f-u\|_c, \] when the outer infimum is taken over all subspaces \(F_N\subset C\) of dimension \(\leq N\), of these classes in the uniform metric.
scientific article; zbMATH DE number 5965562
scientific article