On the degrees of approximation of functions belonging to \(L^{p} (\widetilde \omega)_{\beta }\) class by matrix means of conjugate Fourier series (Q2911487)

Let \(L^{p}\left( 1\leq p<\infty \right) \) be the class of all \(2\pi\)-periodic real-valued functions whose \(p\)-th power is integrable in the Lebesgue sense over \(\left[ -\pi ,\pi \right] \). Denote by \(S_{k}f\left( x\right) \) and \(\widetilde{S}_{k}f\left( x\right) \) the partial sums of the trigonometric Fourier series and the conjugate Fourier series of the function \(f\), respectively.NEWLINENEWLINESet NEWLINE\[NEWLINE \widetilde{f}\left( x,\varepsilon \right) :=\frac{-1}{\pi } \int\limits_{\varepsilon }^{\pi }\left( f\left( x+t\right) -f\left( x-t\right) \right) \cot \left( t/2\right) dt. NEWLINE\]NEWLINENEWLINENEWLINELet the \(A\)-transformation of \(\widetilde{S}_{k}f\left( x\right) \) be given by NEWLINE\[NEWLINE \widetilde{T}_{n,A}f\left( x\right) :=\sum\limits_{k=0}^{n}a_{nk}\widetilde{S }_{k}f\left( x\right) . NEWLINE\]NEWLINE In this paper the deviation \(\widetilde{T}_{n,A}f-\widetilde{f}\) is estimated at a concrete point as well as in the norm of \(L^{p}\). The obtained results generalize the results of \textit{S. Lal} and \textit{H. K. Nigam} [Int. J. Math. Math. Sci. 27, No. 9, 555--563 (2001; Zbl 1040.42010)], \textit{S. Lal} [Tamkang J. Math. 31, No. 4, 279--288 (2000; Zbl 1032.42005)], and \textit{W. Lenski} and \textit{B. Szal} [Acta Comment. Univ. Tartu. Math. 13, 11--24 (2009; Zbl 1193.42032)].