[Paper Review] Summability of Multi-Dimensional Trigonometric Fourier Series

Ferenc Weisz|arXiv (Cornell University)|Jun 8, 2012

Advanced Harmonic Analysis Research21 citations

TL;DR

This paper establishes boundedness and almost everywhere convergence results for various summability methods—Fejér, Riesz, θ-summability, and rectangular summability—of multi-dimensional trigonometric Fourier series and Fourier transforms. It proves that the maximal operators of these summability means are bounded from Hardy spaces $H^p$ to $L^p$ for $p > p_0$, and obtains weak-type estimates at $p=1$, ensuring a.e. convergence. The results extend to Fourier transforms and include sharp conditions on the summability kernels via the Wiener class and integrability of the Fourier transform of the kernel function.

ABSTRACT

We consider the summability of one- and multi-dimensional trigonometric Fourier series. The Fej{é}r and Riesz summability methods are investigated in detail. Different types of summation and convergence are considered. We will prove that the maximal operator of the summability means is bounded from the Hardy space $H_p$ to $L_p$, for all $p>p_0$, where $p_0$ depends on the summability method and the dimension. For $p=1$, we obtain a weak type inequality by interpolation, which ensures the almost everywhere convergence of the summability means. Similar results are formulated for the more general $θ$-summability and for Fourier transforms.

Motivation & Objective

• To establish $L^p$ boundedness and almost everywhere convergence of multi-dimensional trigonometric Fourier series under various summability methods.
• To characterize the conditions on summability kernels (e.g., $θ$-functions) that ensure convergence in $L^p$ and a.e. for $f \in L^p$ and $f \in L(\log L)^{d-1}$.
• To extend the theory of summability to rectangular and restricted summation, including cone- and cone-like sets.
• To analyze the behavior of maximal operators associated with summability means and relate them to Hardy spaces and maximal functions.
• To generalize results from periodic to non-periodic settings by studying $\theta$-summability of Fourier transforms on $\mathbb{R}^d$.

Proposed method

• Uses the theory of Hardy spaces $H^p(T^d)$ and $H^p(\mathbb{R}^d)$, particularly atomic decompositions and maximal functions.
• Applies real-variable techniques, including interpolation and weak-type estimates, to prove boundedness of maximal operators from $H^p$ to $L^p$.
• Employs kernel analysis: characterizes the $L^1$-norm and $L^\infty$-norm of summability kernels (e.g., Fejér, Riesz, Bochner-Riesz, and $\theta$-kernels) in terms of their Fourier transforms.
• Introduces and analyzes $\ell^q$-summability (triangular, circular, cubic) and rectangular summability, distinguishing between unrestricted and restricted convergence.
• Applies the Wiener class $W(C, \ell^1)$ and conditions on $\hat{\theta} \in E'_\infty$ to characterize convergence of $\theta$-means.
• Uses maximal function estimates and weak-type inequalities to derive a.e. convergence results, particularly at Lebesgue and strong Lebesgue points.

Experimental results

Research questions

• RQ1 For which $p > p_0$ is the maximal operator of $\ell^q$-summability means bounded from $H^p(T^d)$ to $L^p(T^d)$?
• RQ2 What conditions on the summability kernel $\theta$ ensure almost everywhere convergence of $\sigma_\theta^n f$ for $f \in L(\log L)^{d-1}(T^d)$?
• RQ3 How does the boundedness of the maximal operator $\sigma_\theta^*$ relate to the integrability and decay of $\hat{\theta}$?
• RQ4 Under what conditions does $\theta$-summability of the Fourier transform on $\mathbb{R}^d$ converge a.e. for $f \in L^1(\mathbb{R}^d)$?
• RQ5 What is the precise relationship between the Wiener class $W(C, \ell^1)$ and the convergence of $\theta$-means at Lebesgue points?

Key findings

• The maximal operator of $\ell^q$-summability means is bounded from $H^p(T^d)$ to $L^p(T^d)$ for all $p > p_0$, where $p_0$ depends on $q$ and $d$.
• For $p = 1$, a weak-type $(1,1)$ inequality is established via interpolation, ensuring almost everywhere convergence of the summability means.
• The $\theta$-summability means $\sigma_\theta^n f$ converge a.e. to $f$ for all $f \in L(\log L)^{d-1}(T^d)$ if $\theta \in W(C, \ell^1)(\mathbb{R}^d)$, $\theta(0) = 1$, and $\|K_\theta^n\|_{E'_\infty(T^d)} \leq C$, with uniform decay on compact sets away from the origin.
• For $\theta$-summability of Fourier transforms on $\mathbb{R}^d$, convergence a.e. holds at Lebesgue points of $f \in L^1(\mathbb{R}^d)$ if $\hat{\theta} \in E_\infty(\mathbb{R}^d)$.
• The converse holds: if $\sigma_\theta^n f \to f$ a.e. at Lebesgue points for all $f \in L^1(\mathbb{R}^d)$, then $\hat{\theta} \in E_\infty(\mathbb{R}^d)$, establishing a sharp condition.
• If $\theta = \prod_{j=1}^d \theta_j$ with each $\theta_j \in V^2_1(\mathbb{R})$ or $\theta_j \in Mv^1_1(\mathbb{R})$, then the $\theta$-means converge a.e. at Lebesgue points of $f \in L(\log L)^{d-1}(T^d)$.