Integrability of the majorants of Fourier sums and the divergence of Fourier series of functions with restrictions on the integral modulus of continuity.
The author proves three essential theorems. The first one asserts that there exists a function from the class \(H_1^{\omega^*},\) where \[ \omega^*(t):=\{\log\log(t^{-1})/\log(t^{-1})\}^{1/2}, \] whose trigonometric Fourier series is divergent almost everywhere. The Theorems 2 and 3 state that if \[ \sum^\infty_{k=1}{\omega(1/k)\over k\,\lambda_k}\leqno(*) \] converges and \(\lambda_k\) satisfies certain conditions, then for any \(f\in H_1^\omega\) \[ \sup_{n\in{\mathbb N}}{| S_n(f,x)| \over\lambda_n} \] is integrable, and \((*)\) is also necessary.
scientific article; zbMATH DE number 2177673