Approximation of integrable functions by general linear matrix operators of their Fourier series

Vishnu Narayan Mishra, Włodzimierz Łenski, Bogdan Szal

Type: Article

Publication Date: 2022-01-01

Citations: 1

DOI: https://doi.org/10.1515/dema-2022-0009

Abstract

Abstract The pointwise estimates of the deviation <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>T</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> <m:mo>,</m:mo> <m:mi>A</m:mi> </m:mrow> </m:msub> <m:mi>f</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mo>⋅</m:mo> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>−</m:mo> <m:mi>f</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mo>⋅</m:mo> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> {T}_{n,A}f(\cdot )-f\left(\cdot ) in terms of pointwise moduli of continuity based on the points of differentiability of indefinite integral of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>f</m:mi> </m:math> f , with application of the r th differences of the entries of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>A</m:mi> </m:math> A , are proved. The similar results in case of the Lebesgue points are considered, too. Analogical results on norm approximation with remarks and corollaries are also given.

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