Asymptotic estimates for the widths of classes of functions of high smothness

Keywords:
Bernstein width, Kolmogorov width, linear width, projection width, Fourier sum, Weyl-Nagy class, class of the generalized Poisson integrals, (ψ,ˉβ)-integral, asymptotic equalityAbstract
We find two-sided estimates for Kolmogorov, Bernstein, linear and projection widths of the classes of convolutions of 2π-periodic functions φ, such that ‖φ‖2≤1, with fixed generated kernels Ψˉβ, which have Fourier series of the form ∞∑k=1ψ(k)cos(kt−βkπ/2), where ψ(k)≥0, ∑ψ2(k)<∞,βk∈R. It is shown that for rapidly decreasing sequences ψ(k) (in particular, if limk→∞ψ(k+1)/ψ(k)=0) the obtained estimates are asymptotic equalities. We establish that asymptotic equalities for widths of this classes are realized by trigonometric Fourier sums.