On the generalized absolute convergence of Fourier series (Q5937628)

In this paper further generalizations of the results given by the author in [Hokkaido Math. J. 30, No.~1, 221-230 (2001; Zbl 1002.42003, preceding review)] are obtained. Sufficient conditions are established for the convergence of the series NEWLINE\[NEWLINE\sum_{n=1}^\infty\omega_n\varphi(\sqrt{a_n^2+b_n^2}),NEWLINE\]NEWLINE where \(a_n,\) \(b_n\) are the Fourier coefficients of a function in \(L^p,\) \(1<p\leq 2,\) \(\varphi(u),\) \(u\geq 0,\) is an increasing concave function, and \(\{\omega_n\}\) is a sequence of positive numbers satisfying some special conditions. The problem of the convergence of the series in question for an arbitrary sequence \(\{\omega_n\}\) still remains open.