Convergence and integrability of trigonometric series with coefficients of bounded variation of order $(m,p)$

Abstract:

Let $\{ c(n)\}$ be a complex null sequence such that for some integer $m \geq 1$ and some $p \in (1,2]$ \[ \sum \limits _{|n| < \infty } {|{\Delta ^m}c(n){|^p} < \infty \quad {\text {and}}\quad \sum \limits _{n = 1}^\infty {|\Delta (c(n) - c( - n))|\lg n < \infty .} } \] It is shown that the series \[ ( * )\quad \sum \limits _{|n| < \infty } {c(n)} {e^{\operatorname {int} }},\quad t \in T = \frac {\mathbb {R}}{{2\pi \mathbb {Z}}}\] converges a.e. and that the well-known condition ${C_w}$ of J. W. Garrett and C. V. Stanojevic [4, 3] implies that the series (*) is the Fourier series of its sum. This generalizes results of W. O. Bray and C. V. Stanojevic [1]. An important consequence of the main result is that $n\Delta c(n) = 0(1),\quad |n| \to \infty$, implies that the condition ${C_w}$ is equivalent to the de la Vallee Poussin summability of partial sums $\{ {S_n}(c)\}$ as conjectured in [8].