Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Towards direct $L^2$-bounds for maximal partial sums of Walsh--Fourier series: The case of dyadic partial sums

Joseph D. Lakey

Abstract

We outline an approach to obtain direct $L^2$ estimates not requiring interpolation for so-called linearized partial sums operators associated with expansions in Walsh functions. We focus specifically on a simpler case of dyadic partial sums but also outline a second approach to proving bounds on general linearized partial sums.

Towards direct $L^2$-bounds for maximal partial sums of Walsh--Fourier series: The case of dyadic partial sums

Abstract

We outline an approach to obtain direct estimates not requiring interpolation for so-called linearized partial sums operators associated with expansions in Walsh functions. We focus specifically on a simpler case of dyadic partial sums but also outline a second approach to proving bounds on general linearized partial sums.
Paper Structure (18 sections, 3 theorems, 38 equations, 9 figures, 1 table)

This paper contains 18 sections, 3 theorems, 38 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction and background
  2. Convergence of Fourier series
  3. Partial sums of Walsh--Fourier series
  4. Walsh functions
  5. Walsh--Hadamard matrices and finite-dimensional representations of dyadic step functions
  6. Partial sum operators and truncated WH-matrices
  7. Norm bounds for dyadically truncated Walsh--Hadamard matrices
  8. Branching in DTWH matrices
  9. Branching reduces norm in DTWH matrices
  10. Evidence for optimality of the standard truncation
  11. Analysis of $F=F_2+F_1$
  12. Subclaim: $A^2(N-1,K)$ increases moderately as $K$ ranges from null to $K=N-1$.
  13. Subclaim: $x(N-1,K)=2^{-N/2}\|\mathbf{x}(N-1,K)\|_1$ decreases in $K$.
  14. Subclaim: $\gamma(N-1,K)/\alpha(N-1,K)$ is decreasing in $K$.
  15. Eigenvectors of $W_N^{\rm opt}$
  16. ...and 3 more sections

Key Result

Lemma 2

Let $\Phi\in D$. Let $h,g$ be a pair of columns of $W_\Phi$. Then either $\langle h,g\rangle=0$ or $(h)_n=(g)_n$ for $0\leq n<\min(\ell(h),\ell(g))$.

Figures (9)

  • Figure 1: Left: Walsh--Hadamard matrix $\emph{WH}_N$ of size $32\times 32$ ($N=5$). Middle: DTWH matrix of size $32\times 32$. Column truncation length $\ell=2^r$ where $r\in \{0,\dots, 5\}$ is uniformly randomly generated. Right: Correlation matrix of DTWH matrix in the middle
  • Figure 2: Norm of the standard truncation matrix $W_N^{\rm opt}$ as a function of $N$. The limiting value as $N\to \infty$ appears to equal $1+\frac{\sqrt{2}}{2}=1.7071\dots$
  • Figure 3: Left: The $32\times 32$ matrix $W_N^{\rm opt}$ ($N=5$). Right: $32\times 32$ two-branch matrix $B_{N-1,K}$ with $N=5$ and $K=2$
  • Figure 4: Left: Plot of $\|\mathbf{y}\|_1/\|\mathbf{x}\|_1$ where $\mathbf{x}$ and $\mathbf{y}$ optimize $\|B_{N-1,K}(\mathbf{u})\|$ as in (\ref{['eq:bnk_norm']}) for $K=1$ to $K=23$ and $N=25$. Right: $\log_2(\|\mathbf{y}\|_1/\|\mathbf{x}\|_1)$ for same range. The ratio is approximately proportional to $2^{K/2}$ for small $K$
  • Figure 5: Left: The level norm eigenvector $\mathbf{c}$ of the matrix $M$ in (\ref{['eq:level_matrix']}) for $N=24$ corresponding to $W_{24}^{\rm opt}$. Right: Plot of $1/(N c_0(N))$ for coefficient $c_0$ of level eigenvector corresponding to $W_N^{\rm opt}$ vs $N$ for $N=1,\dots, 1000$. It appears that $1/c_0$ grows faster than $N$ but slower than $N^2$ for large $N$
  • ...and 4 more figures

Theorems & Definitions (9)

  • Conjecture 1
  • Lemma 2
  • Conjecture 3
  • Definition 4
  • Definition 5
  • Conjecture 6
  • Proposition 7
  • Conjecture 8
  • Proposition 9