Fourier series of functions of $\Lambda$-bounded variation

Abstract:

It is shown that the Fourier coefficients of functions of $\Lambda$-bounded variation, $\Lambda = \{ {\lambda _n}\}$, are $O({\lambda _n}/n)$. This was known for ${\lambda _n} = {n^{\beta + 1}}, - 1 \leqslant \beta < 0$. The classes L and HBV are shown to be complementary, but L and $\Lambda {\text {BV}}$ are not complementary if $\Lambda {\text {BV}}$ is not contained in HBV. The partial sums of the Fourier series of a function of harmonic bounded variation are shown to be uniformly bounded and a theorem analogous to that of Dirichlet is shown for this class of functions without recourse to the Lebesgue test.