Comments on the absolute convergence of Fourier series (Q5937626)

Sufficient conditions are given for the convergence of the series NEWLINE\[NEWLINE\sum\limits_{k=1}^\infty k^\delta(\varphi(|a_{n_k}|)+\varphi(|b_{n_k}|)),NEWLINE\]NEWLINE where \(a_{n_k},\) \(b_{n_k}\) are the \(n_k\)th Fourier coefficients of the Fourier series NEWLINE\[NEWLINE\sum\limits_{k=1}^\infty(a_{n_k}\cos n_kx+b_{n_k}\sin n_kx),\qquad \delta\geq 0,\varphi(u),u\geq 0,NEWLINE\]NEWLINE is an increasing concave function, and \(\{n_k\}\) is a certain increasing sequence of natural numbers. The case \(\delta=0\) reduces to an earlier, of 1999, result by N. Ogata, which, in turn, generalizes preceding results by O. Szász, S. B. Stechkin and J. R. Patadia and V. M. Shah. A related theorem due to A. A. Konyushkov is generalized as well.