Approximation of function of class \(\text{Lip}(\alpha,p)\) by \([F,d_ n]\) mean (Q1355423)

Let \(f\in L[-\pi,\pi]\) and be periodic with period \(2\pi\) outside this range. Let \((d_n)\), \(n\in\mathbb{N}= \{1,2,\dots\}\) be a sequence of positive numbers. If \(x\) is a real number then the numbers \(P_{nk}\) are defined by the relations: \[ P_{00}= 1,\quad \sum^\infty_{k=0} P_{nk}x^k= \prod^n_{j=1} {x+ d_j\over 1+ d_j}. \] Let \(\sigma(f_i,x)= \sum^\infty_{k=0} P_{nk}S_k(f; x)\) denote the \([F,d_n]\) mean of the Fourier series of \(f\in L[-\pi,\pi]\) at \(x\), where \(S_k(f;x)\) is the \(k\)th partial sum of the Fourier series. The aim of this paper is to obtain results which are \([F,d_n]\) analogues of the results obtained by \textit{R. N. Mohapatra}, \textit{A. S. B. Holland} and \textit{B. N. Sahney} [J. Approximation Theory 45, 363-374 (1985; Zbl 0603.42009)].