On the integrability of trigonometric series (Q5919847)

Let \(\phi\) be a Young function which satisfies the following conditions: (1) \(\phi(1)=1\), (2) \(\phi(st)\leq \phi(s)\phi(t)\), for all \(s,t\in[0,1]\), (3) \(\int^ t_ 0 s^{-2}\phi(s)ds\leq Ct^{-1}\phi(t)\), for \(0<t\leq 1\), where \(C\) is a positive constant, and (4) \(t^{-q}\phi(t)\) is monotonic decreasing for \(0<t\leq 1\) and some \(q>1\). Generalizing the results of \textit{G. A. Fomin} [Mat. Zametki 23, 213-222 (1978; Zbl 0379.42004)] the present authors have proved the following two results: Theorem 1. The sum of the series \({1\over 2} a_ 0+\sum^ \infty_{k=1} a_ k \cos kx\) is integrable and (5) is the Fourier series of its sum if (6) \(a_ k=o(1)\), as \(k\to \infty\), and \[ \sum^ \infty_{m=0} 2^ m\phi^{-1} (2^{-m})\phi^{-1}\left(\sum^ \infty_{k=2^ m+1}\phi(a_ k- a_{k+1})\right)<\infty.\tag{7} \] Theorem 2. Let (6) and (7) hold. Then the sum of (8) \(\sum^ \infty_{k=1} a_ k \sin kx\) is integrable if and only if (9) \(\sum^ \infty_{k=1} k^{-1}| a_ k|<\infty\). However, if (6), (7), and (9) hold then series (8) is the Fourier series of its sum.