On various forms of convergence of trigonometric series

A short survey was given of some results on the convergence of trigonometric Fourier series (S. N. Bernstein, E. Fredholm, O. Szász, S. B. Stechkin, P. L. Ul'yanov and others). In the case of uniform convergence of trigonometric series and their conjugate series, a best-possible condition was given in terms of the moduli of continuity in the metric of $L_p(0,2\pi)$ with $p\in(1,\infty]$, of a type quite different from the well-known Dini–Lipschitz condition.