On almost \((N,p,q)\) summability of Fourier series. (Q2785042)

The series \(\sum a_n\) or the sequence \(\{S_n \}\) of its \(n\)-th partial sums is said to be almost generalized Nörlund \((N,p,q)\) summable to \(S\) if NEWLINE\[NEWLINE\left(\sum _{k=o}^{n}p_kq_{n-k}\right)^{-1}\sum _{\nu =0}^{n}p_{n-\nu }q_{\nu }{1\over {\nu +1}}\sum _{k=m}^{\nu +m}S_kNEWLINE\]NEWLINEtends to \(S\), as \(n\to \infty \), uniformly with respect to \(m\;(\{ p_n\} \) and \(\{ q_n\} \) are sequences of non-zero constants ). The almost \((N,p,q)\) method reduces to the almost Nörlund method \((N,p_n)\), if \(q_n=1\) for all \(n\). A new theorem on almost \((N,p,q)\)-summability of Fourier series, which improves a theorem of \textit{B. N. Pandey} [J. Indian Math. Soc., New Ser. 47, 87--94 (1983; Zbl 0604.42004)] on \((N,p_n)\) summability of Fourier series is proved.