We establish exact-order estimates for the best uniform approximations by trigonometric polynomials on the classes C
 ψ
 β, p
 of 2π-periodic continuous functions f defined by the convolutions of functions that belong to the unit balls in the spaces L
 p
 , 1 ≤ p < ∞, with generating fixed kernels Ψβ ⊂ L
 p′, $ \frac{1}{p}+\frac{1}{{p^{\prime}}}=1 $
 , whose Fourier coefficients decrease to zero approximately as power functions. Exactorder estimates are also established in the L
 p
 -metric, 1 < p ≤ ∞, for the classes L
 ψ
 β,1 of 2π -periodic functions f equivalent in terms of the Lebesgue measure to the convolutions of kernels Ψβ ⊂ L
 p
 with functions from the unit ball in the space L
 1. It is shown that, in the investigated cases, the orders of the best approximations are realized by Fourier sums.

Order Estimates for the Best Approximations and Approximations by Fourier Sums of the Classes of (ψ, β)-Differential Functions

We generalize some known results on the best, best linear, and best one-sided approximations by trigonometric polynomials from the classes of 2π-periodic functions represented in the form of convolutions to the case of classes of set-valued functions.

Approximation of Some Classes of Set-Valued Periodic Functions by Generalized Trigonometric Polynomials

We study some problems of approximation of functions by the linear methods of summation of their Fourier–Laplace series.

Approximation of Functions on the Sphere by Linear Methods

Generalizations of some known results on the best, best linear and best one-sided approxima- tions by trigonometric polynomials of the classes of 2\pi - periodic functions presented in the form of convolutions to the case of set-valued functions are obtained

Approximation of some classes of set-valued periodic functions by generalized trigonometric polynomials

We expand upon the known results on sharp linear Fourier methods of approximation where the approximation is the best in terms of both rate and constant among all polynomial procedures of approximation. So far these results have been studied due to their mathematical beauty rather than their practical importance. In this paper we show that they are the core mathematics underlying best statistical methods of solving noisy ill-posed problems. In particular, we suggest a procedure for recovery of noisy blurred signals based on samples of small sizes where a traditional statistics concludes that the complexity of such a setting makes the problem not worthy of a further study. Thus, we present a problem where a combination of the classical approximation theory and statistics leads to interesting practical results.

V. F. Babenko

Papers

Best linear approximation of some classes of differentiable periodic functions