Approximation of conjugate functions by general linear operators of their Fourier series at the Lebesgue points

TL;DR

This paper investigates how well general linear operators approximate conjugate functions of Fourier series at Lebesgue points, providing pointwise and norm estimates that extend previous theorems.

Contribution

It generalizes Mittal's theorem by establishing new pointwise and norm approximation estimates for conjugate functions using general linear operators.

Findings

Pointwise deviation estimates in terms of moduli of continuity.

Norm approximation results with corollaries.

Extension of Mittal's theorem to broader operators.

Abstract

The pointwise estimates of the deviations $\widetilde{T}_{n,A,B}^{\text{}%}f\left(\cdot \right) -\widetilde{f}(\cdot)$ and $\widetilde{T}_{n,A,B}^{% \text{}}f\left(\cdot \right) -\widetilde{f}(\cdot,\varepsilon)$ in terms of moduli of continuity $\widetilde{\bar{w}}_{\cdot}f$ and $\widetilde{w}%_{\cdot}f$ are proved. Analogical results on norm approximation with remarks and corollary are also given. These results generalized a theorem of Mittal.