[Paper Review] A note on the strong convergence of two-dimensional Walsh-Fourier series

George Tephnadze|arXiv (Cornell University)|Oct 5, 2014

Advanced Harmonic Analysis Research8 references5 citations

TL;DR

This paper establishes the sharpness of the convergence rate for quadratically averaged partial sums of two-dimensional Walsh-Fourier series in the dyadic Hardy space $ H_1(G^2) $. By constructing a specific function sequence and analyzing the $ L^1 $-norm of its rectangular partial sums, the authors prove that the weight $ \frac{1}{n\log^2(n+1)} $ in the convergence series cannot be improved, confirming its essential nature for strong $ L^1 $-convergence.

ABSTRACT

The main aim of this paper is to investigate the quadratical partial sums of the two-dimensional Walsh-Fourier series.

Motivation & Objective

• To investigate the sharpness of the convergence rate for quadratically averaged partial sums of two-dimensional Walsh-Fourier series in $ H_1(G^2) $.
• To determine whether the weight $ \frac{1}{n\log^2(n+1)} $ in the strong convergence estimate is optimal.
• To demonstrate that no nondecreasing function $ \Phi(n) \to \infty $ can improve the convergence rate.
• To extend known results on strong convergence of Walsh-Fourier series to the two-dimensional setting with sharp bounds.

Proposed method

• Construct a sequence of functions $ f_{n,n}(x,y) = (D_{2^{n+1}}(x) - D_{2^n}(x))(D_{2^{n+1}}(y) - D_{2^n}(y)) $ with known Walsh-Fourier coefficients.
• Use the identity $ S_{k,k}f_{n,n}(x,y) = w_{2^n}(x)w_{2^n}(y)D_{k-2^n}(x)D_{k-2^n}(y) $ for $ 2^n < k \leq 2^{n+1} $ to express partial sums.
• Apply the known estimate $ \|D_m\|_1 \geq \frac{1}{8}V(m) $, where $ V(m) $ is the variation of $ m $ in binary representation.
• Establish a lower bound on $ \|S_{k,k}f_{n,n}\|_1 \geq c V^2(k - 2^n) $ using the product structure of the kernel.
• Use the asymptotic behavior $ \frac{1}{n} \sum_{k=1}^n V(k) \sim \frac{1}{4\log 2} $ to estimate the average variation.
• Apply the Cauchy-Schwarz inequality and summation over dyadic intervals to show divergence of the weighted series.

Experimental results

Research questions

• RQ1 Is the weight $ \frac{1}{n\log^2(n+1)} $ in the strong $ L^1 $ convergence estimate for 2D Walsh-Fourier series sharp?
• RQ2 Can the convergence rate be improved by replacing $ \frac{1}{n\log^2(n+1)} $ with $ \frac{\Phi(n)}{n\log^2(n+1)} $ for any nondecreasing $ \Phi(n) \to \infty $?
• RQ3 What is the behavior of the $ L^1 $-norm of the rectangular partial sums $ S_{k,k}f $ for functions in $ H_1(G^2) $?
• RQ4 How does the variation $ V(m) $ of the binary representation of $ m $ influence the Lebesgue constant and partial sum norms?
• RQ5 Does the supremum of the weighted sum $ \sum \frac{\|S_{k,k}f\|_1 \Phi(k)}{k\log^2(k+1)} $ over $ \|f\|_{H_1} \leq 1 $ remain finite for any such $ \Phi $?

Key findings

• The weight $ \frac{1}{n\log^2(n+1)} $ in the strong convergence estimate for 2D Walsh-Fourier series is sharp, as it cannot be improved.
• For any nondecreasing function $ \Phi(n) \to \infty $, the supremum of $ \sum \frac{\|S_{k,k}f\|_1 \Phi(k)}{k\log^2(k+1)} $ over $ \|f\|_{H_1} \leq 1 $ is infinite.
• The constructed function $ f_{n,n} $ satisfies $ \|f_{n,n}\|_{H_1} = 1 $, confirming normalization in the Hardy space.
• The $ L^1 $-norm of the partial sum $ S_{k,k}f_{n,n} $ satisfies $ \|S_{k,k}f_{n,n}\|_1 \geq c V^2(k - 2^n) $ for $ 2^n < k \leq 2^{n+1} $.
• The average variation $ \frac{1}{n} \sum_{k=1}^n V(k) $ converges to $ \frac{1}{4\log 2} $, which is essential for the asymptotic lower bound.
• The divergence of the weighted series is established via Cauchy-Schwarz and dyadic interval summation, leading to $ \sim c \Phi(2^n) \to \infty $ as $ n \to \infty $.