[Paper Review] On $L^{1}$-Convergence of Fourier Series Under $MVBV$ Condition

Dan Sheng Yu, Ping Zhou+1 more|ArXiv.org|Apr 14, 2007

Approximation Theory and Sequence Spaces3 references4 citations

TL;DR

This paper establishes the $L^1$-convergence of Fourier series for complex-valued $2\pi$-periodic functions under the Mean Value Bounded Variation (MVBV) condition on Fourier coefficients. It generalizes classical results by replacing monotonicity with MVBV, proving that $\|f - S_n(f)\|_L \to 0$ if and only if $\hat{f}(n)\log|n| \to 0$, with the MVBV condition being the weakest possible for this convergence in the complex setting.

ABSTRACT

Let $f\in L_{2π}$ 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 $\lim\limits_{n o \infty}a_{n}=0$, then $$ \lim\limits_{n o \infty}\Vert f-S_{n}(f)\Vert_{L}=0 {if and only if} \lim\limits_{n o \infty}a_{n}\log n=0. $$ 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 $f\in L_{2π}$ in complex space. We also give results on $L^{1}$-approximation of a function $f\in L_{2π}$ under the $% MVBV$ condition.

Motivation & Objective

• To generalize classical $L^1$-convergence results for Fourier series by weakening the monotonicity condition on Fourier coefficients.
• To establish the $L^1$-convergence of Fourier series for complex-valued $2\pi$-periodic functions under the MVBV condition on their Fourier coefficients.
• To show that the MVBV condition is the weakest possible condition replacing monotonicity in the $L^1$-convergence result for even real-valued functions.
• To derive $L^1$-approximation rates under the MVBV condition, linking convergence to best approximation and modulus of continuity.

Proposed method

• Introduces the MVBVS (Mean Value Bounded Variation Sequence) condition for complex sequences, requiring bounded variation relative to mean values over dyadic intervals.
• Uses the key condition $\lim_{\mu \to 1^+} \limsup_{n \to \infty} \sum_{k=n}^{[\mu n]} |\Delta \hat{f}(k) - \Delta \hat{f}(-k)| \log k = 0$ to control oscillation in Fourier coefficients.
• Applies Lemma 1 to bound partial sums of differences using maximal values over dyadic intervals, leveraging the MVBV property.
• Employs Lemma 2 to relate $L^1$-norm of the difference $f - S_n(f)$ to the harmonic series of coefficient magnitudes.
• Establishes equivalence between $L^1$-convergence and $\hat{f}(n)\log|n| \to 0$ via necessity and sufficiency proofs using dyadic decomposition and norm estimates.
• Generalizes approximation theory by proving $\|f - S_n(f)\|_L = O(\psi_n)$ if and only if $E_n(f)_L = O(\psi_n)$ and $\hat{f}(n)\log|n| = O(\psi_{|n|})$, for decreasing $\psi_n \sim \psi_{2n}$.

Experimental results

Research questions

• RQ1 Under what conditions on Fourier coefficients does the $L^1$-norm of the difference $f - S_n(f)$ tend to zero for complex-valued $2\pi$-periodic functions?
• RQ2 Can the classical monotonicity condition in $L^1$-convergence results be weakened to a more general condition, and if so, what is the weakest such condition?
• RQ3 How does the MVBV condition relate to the rate of $L^1$-approximation of $f$ by trigonometric polynomials of degree $n$?
• RQ4 What is the precise equivalence condition for $L^1$-convergence in terms of the decay rate of Fourier coefficients under MVBV?

Key findings

• The $L^1$-convergence $\|f - S_n(f)\|_L \to 0$ holds if and only if $\hat{f}(n)\log|n| \to 0$, under the MVBV condition on $\{\hat{f}(n)\}_{n=0}^\infty$ and the auxiliary condition on differences $\Delta \hat{f}(k) - \Delta \hat{f}(-k)$.
• The MVBV condition is proposed as the weakest possible generalization of monotonicity for $L^1$-convergence, analogous to its role in uniform convergence of sine series.
• For real-valued even functions, the result reduces to the classical equivalence: $\|f - S_n(f)\|_L \to 0$ iff $a_n \log n \to 0$, provided $\{a_n\}$ satisfies the real MVBV condition.
• The $L^1$-approximation rate satisfies $\|f - S_n(f)\|_L = O(\psi_n)$ if and only if both the best approximation $E_n(f)_L = O(\psi_n)$ and $\hat{f}(n)\log|n| = O(\psi_{|n|})$ hold.
• Corollary 3 extends Jackson-type approximation theorems by showing $\|f - S_n(f)\|_L = O\left(\frac{1}{(n+1)^r}\omega(f^{(r)}, \frac{1}{n+1})_L\right)$ iff $\hat{f}(n)\log|n| = O\left(\frac{1}{(n+1)^r}\omega(f^{(r)}, \frac{1}{n+1})_L\right)$.
• The condition $\sum_{k=n}^{[\mu n]} |\Delta \hat{f}(k) - \Delta \hat{f}(-k)| \log k \to 0$ as $\mu \to 1^+$ is sufficient and can replace the full condition in Theorem 1.