Generalized variation, the Banach indicatrix, and the uniform convergence of Fourier series

K. I. Oskolkov

Matematičeskie zametki, Tome 12 (1972) no. 3, pp. 313-324.

Résumé

It is proved that if the continuous periodic function $f$ has bounded $\Phi$-variation, then the deviation of $f$ from the sum of $n$ terms of its Fourier series has the bound $$ ||f-S_n(f)||\leqslant c\int_0^{\omega(\pi n^{-1})}\log(v_\Phi(f)/\Phi(\xi))d\xi. $$ Here $c$ is an absolute constant, $\omega$ is the modulus of continuity, $v_\Phi(f)$ is the complete $\Phi$-variation of $f$ over a period. It is established that the Salem and Garsia–Sawyer criteria for the uniform convergence of the Fourier series in terms of the $\Phi$-variation and the Banach indicatrix respectively are definitive, and it is proved that the second of these variants is a corrolary of the first.