On $L^{1}$-Convergence of Fourier Series Under $MVBV$ Condition

TL;DR

This paper extends classical Fourier series convergence results by weakening the monotonicity condition to an $MVBV$ condition, providing new $L^{1}$-convergence and approximation results for functions in $L_{2 ext{-}pi}$.

Contribution

It introduces the $MVBV$ condition as a weaker alternative to monotonicity for Fourier coefficients, establishing $L^{1}$-convergence and approximation theorems.

Findings

$L^{1}$-convergence under $MVBV$ condition is established.

Classical results are generalized to complex function spaces.

Results include $L^{1}$-approximation of functions in $L_{2 ext{-}pi}$.

Abstract

Let $f\in L_{2\pi}$ be a real-valued even function with its Fourier series $ \frac{a_{0}}{2}+\sum_{n=1}^{\infty}a_{n}\cos nx,$ and let $S_{n}(f,x), n\geq 1,$ be the $n$-th partial sum of the Fourier series. It is well-known that if the nonnegative sequence $\{a_{n}\}$ is decreasing and $\lim\limits_{n\to \infty}a_{n}=0$, then $$ \lim\limits_{n\to \infty}\Vert f-S_{n}(f)\Vert_{L}=0 {if and only if} \lim\limits_{n\to \infty}a_{n}\log n=0. $$ We weaken the monotone condition in this classical result to the so-called mean value bounded variation ($MVBV$) condition. The generalization of the above classical result in real-valued function space is presented as a special case of the main result in this paper which gives the $L^{1}$% -convergence of a function $f\in L_{2\pi}$ in complex space. We also give results on $L^{1}$-approximation of a function $f\in L_{2\pi}$ under the $% MVBV$ condition.