Integrability of the Majorants of Fourier Series and Divergence of the Fourier Series of Functions with Restrictions on the Integral Modulus of Continuity

Abstract

We construct an example of a function from the class $$H_1^{\omega ^ * } $$ , where $$\omega ^ * (t) = \sqrt {\log \log (t^{ - 1} )/\log (t^{ - 1} )} $$ , $$0 < t \leqslant t_0 $$ , whose trigonometric Fourier series is divergent almost everywhere. We obtain sharp integrability conditions for the majorants of the partial sums of trigonometric Fourier series in terms of whether the functions in question belong to the classes $$H_1^\omega $$ .