Generalized Nörlund summability of Fourier series and its conjugate series (Q1813276)

Let \(\sum^\infty_{n=0} a_n\) be a given infinite series with the sequence of partial sums \(\{S_n\}\). For any sequence \(\{S_ n\}\) we write \(t_n^{p,q}=(1/(p * q)_ n)\sum^n_{k=0}p_{n-k}q_k S_k\), where for two sequences \(\{p_n\}\) and \(\{q_n\}\) the convolution is defined by \((p*q)_ n=\sum^n_{k=0}p_{n-k}q_k\). If \((p * q)_n\neq 0\) for all \(n\), then \(t_n^{p,q}\) is the generalized Nörlund transform of the sequence \(\{S_ n\}\). If \(t_n^{p,q}\to S\) as \(n\to\infty\), then the sequence \(\{S_n\}\) is said to be summable to \(S\) by generalized Nörlund method \((N,p,q)\). The \((N,p,q)\) method reduces to the Nörlund method \((N,p_n)\) if \(q_ n=1\) for all \(n\) and reduces to the Riesz method \((\overline N,q_n)\) if \(p_n=1\) for all \(n\). The author proves following result: Theorem 1: Let the regular generalized Nörlund method \((N,p,q)\) be defined as follows. Let \(\{p_n\}\) be a non-negative, monotonic non- increasing sequence of real constants and \(\{q_n\}\) be a non-negative monotonic non-decreasing sequence of real constants. If, for any \(\delta\), \(0<\delta<\pi\), \[ \int^\delta_{1/n}\frac{|\phi(t)|}{t}P\left(\frac1t\right)\,dt=O(R_nq^{-1}_n),\quad\text{as } n\to\infty, \] then the Fourier series at \(t=x\) is summable \((N,p,q)\) to \(f(x)\). A similar result is established for the conjugate series (Theorem 2). For \(q_ n=1\), Theorem 1 reduces to the theorem of \textit{K. S. K. Iyengar} [Proc. Indian Acad. Sci., Sect. A 18, 113--120 (1943; Zbl 0060.18501)]. In case \((N,p_n)\) implies \((N,p,q)\), Theorem 1 gives nothing new. Theorem 2 is the generalization of results proved by \textit{A. N. Singh} and \textit{S. P. Khare} [Proc. Natl. Acad. Sci. India, Sect. A 58, No. 4, 509--515 (1988; Zbl 0706.42005)]. In this paper the author has generalized his own results.