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

### Abstract

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 Ψ

_{β}⊂

*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 Ψ

_{β}⊂

*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.

Published

25.09.2013

How to Cite

*Ukrains’kyi Matematychnyi Zhurnal*, Vol. 65, no. 9, Sept. 2013, pp. 1186–1197, http://umj.imath.kiev.ua/index.php/umj/article/view/2500.

Issue

Section

Research articles