@article{Hrabova_Serdyuk_2013, title={Order Estimates for the Best Approximations and Approximations by Fourier Sums of the Classes of (ψ, β)-Differential Functions}, volume={65}, url={https://umj.imath.kiev.ua/index.php/umj/article/view/2500}, abstractNote={We establish exact-order estimates for the best uniform approximations by trigonometric polynomials on the classes <em class="a-plus-plus">C</em> <sup class="a-plus-plus">ψ</sup> <sub class="a-plus-plus">β, <em class="a-plus-plus">p</em> </sub> of 2π-periodic continuous functions f defined by the convolutions of functions that belong to the unit balls in the spaces <em class="a-plus-plus">L</em> <sub class="a-plus-plus"> <em class="a-plus-plus">p</em> </sub>, 1 ≤ <em class="a-plus-plus">p</em> < ∞, with generating fixed kernels Ψ<sub class="a-plus-plus">β</sub> ⊂ <em class="a-plus-plus">L</em> <sub class="a-plus-plus"> <em class="a-plus-plus">p</em>′</sub>, <span class="a-plus-plus inline-equation id-i-eq1"> <span class="a-plus-plus equation-source format-t-e-x">\( \frac{1}{p}+\frac{1}{p^{\prime }=1 \)</span> </span>, whose Fourier coefficients decrease to zero approximately as power functions. Exactorder estimates are also established in the <em class="a-plus-plus">L</em> <sub class="a-plus-plus"> <em class="a-plus-plus">p</em> </sub> -metric, 1 < <em class="a-plus-plus">p</em> ≤ ∞, for the classes <em class="a-plus-plus">L</em> <sup class="a-plus-plus">ψ</sup> <sub class="a-plus-plus">β,1</sub> of 2π -periodic functions f equivalent in terms of the Lebesgue measure to the convolutions of kernels Ψ<sub class="a-plus-plus">β</sub> ⊂ <em class="a-plus-plus">L</em> <sub class="a-plus-plus"> <em class="a-plus-plus">p</em> </sub> with functions from the unit ball in the space <em class="a-plus-plus">L</em> <sub class="a-plus-plus">1</sub&gt;. It is shown that, in the investigated cases, the orders of the best approximations are realized by Fourier sums.}, number={9}, journal={Ukrains’kyi Matematychnyi Zhurnal}, author={Hrabova, U. Z. and Serdyuk, A. S.}, year={2013}, month={Sep.}, pages={1186–1197} }