On local property of \(|\overline N, p_n,\delta|_k\) summability of factored Fourier series (Q1827124)

The author precisely has shown that \(\sum^\infty_{n=1} a_n X_n\lambda_n\) is \(|\overline N,p_n,\delta|_k\) (\(k\geq 1\), \(\delta\geq 0\)) summable if [see \textit{H. Bor}, J. Math. Anal. Appl. 179, No. 2, 464--469 (1993; Zbl 0797.42005)], along with other conditions, \(s_n= O(1)\) and \[ \sum^\infty_{n=1}(P_n/p_n)^{\delta k} X_n|\Delta\lambda_n|< \infty\tag{1} \] hold, \(\Delta\lambda_n= \lambda_n- \lambda_{n+1}\), \(X_n= P_n/np_n\), \(p_n> 0\), \(P_n= p_0+ p_1+\cdots+ p_n\to\infty\) with \(n\to\infty\). Earlier, this result was obtained by H. Bor (loc. cit.) under the condition: \[ \sum^\infty_{n=1} (P_n/p_n)^{\delta k}(1+ X^k_n)|\Delta\lambda_n|< \infty,\tag{2} \] for (1), which is weaker than (2). An aspect of local property of a factored Fourier series was deduced from it by using the fact that the convergence of the Fourier series at a point is a local property and convergence ensures the boundedness. Misprint. Read ``\(\delta k+ k-1\)'' for ``\(k-1\)'' in line 10, p. 342.