An Eigenvector Problem Arising in the Study of Convergence of Walsh–Fourier Series

Abstract

This work establishes bounds for certain matrices that arise in the study of the convergence of expansions in Walsh functions, or Walsh–Fourier series. The matrices in question arise by “truncating” certain orthogonal matrices corresponding to expansions of dyadic step functions on the unit interval in the basis of Walsh functions. Here, truncation means, for each column, replacing all entries in that column below a column-dependent row by zero. The truncations correspond to partial sum operators in the Walsh basis. We study here a specific family of these truncated matrices that are shown elsewhere to have optimal norms among certain families of truncations. The main result here provides an approximate eigenvalue bound from which one can conclude that the norm of the truncation approaches a fixed value as the dimension of the truncation matrix approaches infinity. Its proof relies on the interplay between continuous and discrete sets. In particular, it is shown that integer samples of certain sinusoidal functions form approximate eigenvectors of a compressed version of the truncation. This bound plays an important role in a bigger new approach to the convergence of Walsh–Fourier series that this work is part of.

1. Introduction

The subject matter of this work is an eigenvalue problem that the authors stumbled upon as part of a larger study of the convergence of Fourier series in the Walsh setting. The broader work seeks uniform bounds on the norms of certain matrices obtained starting from a $2^N\times 2^N$ Hadamard matrix (in the Paley ordering) and setting, for each column k, all entries below a row $r\left(k\right)$ that depend on k, equal to zero (see Figure 1). These “truncated” matrices correspond to linearized partial sum operators for Walsh–Fourier series when operating on dyadic step functions on $[0,1]$ as explained in [1]. Uniform bounds on their norms imply a direct $L^2$-bound for the maximal partial sum operator for Walsh–Fourier series, and thereby almost everywhere convergence of these series on $L^2[0,1]$.

The study of almost everywhere convergence of trigonometric Fourier series has a long history. Seminal results include Carleson’s proof [2] of Lusin’s conjecture that the Fourier series of $f\in L^2(0,1)$ converges to f almost everywhere, Hunt’s extension [3] to $L^p$ ($1<p<\infty$) by means of interpolation, and Fefferman’s proof  [4] by means of linearized maximal partial sum operators. Some of the substantial body of related work on convergence of Fourier series in the trigonometric setting is documented in the references [5,6,7,8,9,10,11,12]. Almost everywhere convergence of Walsh–Fourier series in $L^2(0,1)$ was first established by Billard [13] not long after Carleson’s theorem was proved for the trigonometric case, and  extended to $L^p$ along the lines of Hunt’s approach for the trigonometric case by Tateoka in 1968 [14]. Gosselin provided a proof along the lines of Fefferman’s approach [15]. Some related work on the convergence of Walsh–Fourier series can be found in the references [16,17,18]. The Walsh setting corresponds to expansion in characters of the group ${\mathbb{Z}}_2^{\infty }$. This generalizes naturally to Vilenkin groups, and almost everywhere convergence has also been studied in this setting, e.g., [19,20,21]. Almost everywhere convergence in $L^2$ has been studied for other nontrigonometric orthogonal expansions, including Laguerre functions [22], orthogonal polynomials [23], and other orthogonal systems, e.g.,  [24,25,26]. The direct approach of our broader program to convergence in $L^2$ in the Walsh setting leverages the specific structure of dyadic partial sums, a type of lacunarity. Convergence of lacunary partial sums of trigonometric series has also been studied in the contexts of almost periodic functions [27], multiple Fourier series [28,29], and norms near $L^1$ in the trigonometric [30] and Walsh [31] settings. Together these references represent just a small fraction of the study of the convergence of Fourier series in the setting of function spaces. Pointwise (almost everywhere) convergence of Fourier expansions of $L^2$-functions in these settings typically follows from boundedness of associated maximal partial sum operators between certain endpoint function spaces. But $L^2$-boundedness of maximal partial sums is known indirectly, not by direct and more precise estimates. For example, Fefferman’s proof gives an $L^2\to L^1$-bound for a maximal partial sum operator. The goal of our broader work is to establish direct $L^2$-bounds for maximal partial sums in the Walsh setting. The specific matrices considered here have the largest norms among a family of partial sum operators in this setting as will be explained in forthcoming work. See [1] for further discussion. The purpose of the present paper is to identify a precise ${\mathcal{\ell }}^2\to {\mathcal{\ell }}^2$ operator norm bound on these specific matrices. While this represents just one piece of the bigger puzzle, it is a critical one. The methods required for this piece are particularly elementary, so we have decided to show it separately as an illustration of the interplay between continuous and discrete methods in harmonic analysis [32,33]. We will now leave this backstory and formulate the specific eigenvalue problem to be addressed here.

For each $N=2,3,\dots$ we define $(N+1)\times (N+1)$ matrices $D_N, M_N$, and $C_N$ with rows and columns indexed by $\{0,1,\dots ,N\}\times \{0,1,\dots ,N\}$ as follows.

$$\begin{array}{ccc}\hfill D_N& =& \mathrm{diag}(1,1,2,4,\dots ,2^{N-1}),\hfill \\ \hfill M_N(i,j)& =& \left\{\begin{array}{cc}1,\hfill & i+j\le N\hfill \\ 0,\hfill & \mathrm{otherwise},\hfill \end{array}\right.\hfill \\ \hfill C_N& =& 2^{-N/2}{D}^{1/2}{M}_N{D}^{1/2}.\hfill \end{array}$$

(1)

The matrix $C_{30}$ is illustrated in Figure 2a. The main result of this work can be stated as follows.

Theorem 1.

There is a constant $C>0$ such that for all $N=2,3,\dots$ there is an $\alpha >0$ and vector $\mathbf{s}=\{sin{\left(\alpha (1-k/N)\right\}}_{k=0}^N$ such that

$$\left|{\left(C_N\mathbf{s}\right)}_k-\left(1+{\displaystyle \frac{\sqrt{2}}{2}}\right){\left(\mathbf{s}\right)}_k\right|\le {\displaystyle \frac{C}{N}}.$$

If $Q={\displaystyle \frac{Nln2}{2}}$ then α can be taken to have the form $\alpha =\pi \left(1-{\displaystyle \frac{1}{Q}}+{\displaystyle \frac{1}{Q^2}}+O\left({\displaystyle \frac{1}{Q^3}}\right)\right)$ as $Q\to \infty$.

This uniform bound obviously also implies the ${\mathcal{\ell }}^2$-bound $\parallel C_N\mathbf{s}-(1+{\displaystyle \frac{\sqrt{2}}{2}}){\mathbf{s}\parallel }_2^2\le {\displaystyle \frac{C^2}{N}}$. Either the uniform or ${\mathcal{\ell }}^2$ bound allows us to state that $\{sin(\alpha (1-·/N)\left)\right\}$ is an approximate eigenvector of $C_N$. However, to be as precise as possible, it is more accurate to state that $\{sin(\alpha (1-·/N)\left)\right\}$ is an approximate eigenvector of $C_N$ corresponding to the largest eigenvalue of $C_N$. The norm of $C_N$ is increasing with N and is strictly smaller than $1+{\displaystyle \frac{\sqrt{2}}{2}}$ although evidently the numerical values of $\parallel C_N{\parallel }_{{\mathcal{\ell }}^2\to {\mathcal{\ell }}^2}$, which are the largest eigenvalues of $C_N$ (which is symmetric), approach $1+{\displaystyle \frac{\sqrt{2}}{2}}$ as N increases, as shown in Figure 2b. The proof of Theorem 1 is quite elementary, and the error terms leading to the bound are fairly explicit. The bound on $\parallel C_N\parallel$ provides a bound on norms of truncated Walsh–Hadamard matrices as described next.

1.1. Connection with “Truncated” Walsh–Hadamard Matrices

Figure 1a shows the Hadamard matrix $W{H}_5$ with columns in the “Paley order” (see [1] for a precise definition of $W{H}_N$). The middle plot in Figure 2b is the matrix $T{W}_5$ and the right plot shows a random “column truncation” of $W{H}_5$. By a column truncation here, we mean, for each column k, setting all entries equal to zero below a given row $r\left(k\right)$ that depends on k. The matrix $T{W}_N$ is such that the $(k,\mathcal{\ell })$-entry of $2^{N/2}T{W}_N$ is equal to one if $k\in I_i$ (i.e., $2^{i-1}\le k<2^i$), $\mathcal{\ell }\in I_j$, and $i+j\le N$, and otherwise $T{W}_N(k,\mathcal{\ell })=0$. In forthcoming work, it will be shown that for each N, $T{W}_N$ has maximal norm among all “dyadic column truncations” of $W{H}_N$, that is, column truncations for which the index of the last nonzero entry of a column is always a power of two, cf., [1]. As such, it is imperative to know what is the norm of $T{W}_N$, or at least to know a tight bound for this norm. Obtaining such a bound is the overall aim of the present work. The following lemma identifies $C_N$ as a compression of $T{W}_N$.

Lemma 1.

If $\mathbf{v}={[v_0,v_1,\dots ,v_{2^N-1}]}^T$ is an eigenvector of $T{W}_N$ then the entries of $\mathbf{v}$ are constant on dyadic intervals $I_j=2^{j-1},\dots ,2^j-1$, $j=1,\dots ,N$. That is, there is a vector $\mathbf{w}={[w_0,\dots ,w_N]}^T\in {\mathbb{R}}^{N+1}$ such that $v_i=w_j$ if $i\in I_j$. The compressed vector $\mathbf{c}=[c_0,\dots ,c_N]$ where $c_0=w_0$ and $c_j=2^{(j-1)/2}{w}_j$, $j=1,\dots ,N$ where $w_j$ is the value of $\mathbf{v}$ on $I_j$, is an eigenvector of $C_N$ with eigenvalue equal to that of $\mathbf{v}$ as an eigenvector of $T{W}_N$.

Proof of Lemma 1.

Any linear combination of columns of $T{W}_N$ is constant on each of the dyadic intervals $I_1,\dots ,I_N$ and the first statement follows from this observation. Suppose now that $\mathbf{v}=[w_0,w_1,\dots ]$ ($w_i$ is repeated $|I_i|=2^{i-1}$ times) is an eigenvector of $T{W}_N$ with eigenvalue $\lambda$. Since $w_j$ is repeated $2^{j-1}$ times for $j=1,\dots ,N$, by the definition of $T{W}_N$, if $k\in I_i$ then

$$\begin{array}{ccc}\hfill \lambda w_i=\lambda v_k={\left(T{W}_N\mathbf{v}\right)}_k& =& 2^{-N/2}[w_0+\sum _{j=1}^{N-i}{2}^{(j-1)}{w}_j]\hfill \\ & =& 2^{-N/2}[w_0+\sum _{j=1}^{N-i}{2}^{(j-1)/2}{c}_j]\hfill \\ & =& 2^{-(i-1)/2}{2}^{(i-1-N)/2}[c_0+\sum _{j=1}^{N-i}{2}^{(j-1)/2}{c}_j]\hfill \end{array}$$

where we used $c_j=2^{(j-1)/2}{w}_j$. On the other hand, direct calculation from the definition of $C_N$ gives

$${\left(C_N\mathbf{c}\right)}_i=2^{(i-1-N)/2}[c_0+\sum _{j=1}^{N-i}{2}^{(j-1)/2}{c}_j].$$

Since $k\in I_i$, it follows then that $2^{(i-1)/2}\lambda v_k=2^{(i-1)/2}\lambda w_i=\lambda c_i$, that is, $\mathbf{c}$ is a $\lambda$-eigenvector of $C_N$.    □

Remark 1.

The direct expression above for $C_N\mathbf{c}$ allows one to express a λ-eigenvector as a solution of the recurrence relation

$$\lambda c_{k+1}=\sqrt{2}\lambda c_k-{\displaystyle \frac{1}{\sqrt{2}}}{c}_{N-k}.$$

(2)

A simple derivation of this relation can be found in [1]. It can be of use in estimating the eigenvectors numerically when taking into account that entries of $C_N$ are below machine precision for large N. This relation, however, does not lead to any elegant analytical expression for values of eigenvectors of $C_N$. To get around this obstacle we introduce in the next section a continuous model for $C_N$.

1.2. Outline of Approach

Here is an outline of our approach to proving Theorem 1. In Section 2 we provide a continuous parameter model for the operators $C_N$ and denote the corresponding model operators by $K_Q$ where Q is related to N by $Q=Nln2/2$. In contrast to the finite-dimensional operator $C_N$, the spectrum of the operator $K_Q$ can be quantified precisely. In Section 4 we complete the proof of Theorem 1 by applying standard approximation methods in analysis to estimate the errors between coordinates of specific eigenvectors of $C_N$ and samples of corresponding eigenfunctions of $K_Q$. In Section 5, we provide plots of the numerically computed eigenvectors of $C_N$ having the largest eigenvalue alongside corresponding eigenfunctions of the model operators and the errors between them in order to illustrate how the quality of approximation depends on N. In Appendix A we prove a technical lemma introduced in Section 3 whose proof is less elementary than the other content.

2. A Continuous Model Operator for ${\mathit{C}}_{\mathit{N}}$

The sums defining eigenvectors of $C_N$ can be regarded as Riemann sum approximations of certain integrals. Specifically, think of $c_k$ as a value $f\left(k\right)$, $k=0,\dots ,N$ of a continuous function f and as the weight $2^{(k-1)/2}$ as an integer value of $2^{(t-1)/2}=2^{-1/2}{e}^{{\displaystyle \frac{ln2}{2}}t}$ for $t\in [0,N]$. Then,

$$2^{(k-1-N)/2}[c_0+\sum _{\mathcal{\ell }=1}^{N-k}{2}^{(\mathcal{\ell }-1)/2}{c}_{\mathcal{\ell }}]\approx {\displaystyle \frac{1}{2}}{e}^{(t-N){\displaystyle \frac{ln2}{2}}}{\int }_0^{N-t}{e}^{{\displaystyle \frac{ln2}{2}}s}f\left(s\right)ds.$$

Note that the value at zero is weighted differently from the other values—a matter that we will return to when discussing errors between continuous and discrete settings. In order to think of these sums as approaching some limit as $N\to \infty$ it is helpful to normalize and think of the samples as being taken on the fixed interval $[0,1]$. To do so one can set $s=Nu$, $ds=Ndu$ and $t=Nv$ so the integral can be rewritten

$${\displaystyle \frac{N}{2}}{e}^{N(v-1){\displaystyle \frac{ln2}{2}}}{\int }_{u=0}^{1-v}{e}^{{\displaystyle \frac{Nln2}{2}}u}f\left(Nu\right)du={\displaystyle \frac{N}{2}}{e}^{Q(v-1)}{\int }_{u=0}^{1-v}{e}^{Qu}f\left(Nu\right)du,$$

where $Q={\displaystyle \frac{Nln2}{2}}$. Define now the continuous model operator $K=K_Q$ associated with $C_N$ by

$$\left(Kg\right)\left(v\right)={\displaystyle \frac{Q}{ln2}}{e}^{Q(v-1)}{\int }_{u=0}^{1-v}{e}^{Qu}g\left(u\right)du,Q={\displaystyle \frac{Nln2}{2}}.$$

(3)

Our goal is to show that eigenfunctions of $K_Q$ are sinusoidal functions then, use this observation to show that the integer (or $k/N$) samples of sinusoidal eigenfunctions with largest eigenvalue form approximate $\parallel C_N\parallel$-eigenvectors of the matrix $C_N=D^{1/2}{M}_N{D}^{1/2}$. That the error in Theorem 1 is on the order of $1/N$ heuristically then follows from the entries of the eigenvectors being samples of the eigenfunctions of $K_Q$ at points $k/N$, $k=0,\dots ,N$.

3. Spectrum of the Continuous Model

The kernel of K is ${\displaystyle \frac{Q}{ln2}}{e}^{-Q}{e}^{Q(s+t)}{\mathbf{1}}_{s+t\le 1}(s,t)$ and K is therefore a compact, symmetric operator on $L^2[0,1]$ with discrete spectrum. From (3),

$$\begin{array}{ccc}\hfill {\displaystyle \frac{d\left(Kf\right)\left(t\right)}{dt}}& =& {\displaystyle \frac{Q}{ln2}}\left(Q{e}^{Q(t-1)}{\int }_0^{1-t}{e}^{Qu}f\left(u\right)du-e^{Q(t-1)}{e}^{Q(1-t)}f(1-t)\right)\hfill \\ & =& Q\left(Kf\right)\left(t\right)-{\displaystyle \frac{Q}{ln2}}f(1-t).\hfill \end{array}$$

If $Kf=\lambda f$ then, this can be expressed as

$$\lambda f^{\prime }\left(t\right)=Q\lambda f\left(t\right)-{\displaystyle \frac{Q}{ln2}}f(1-t)\mathrm{or}{f}^{\prime }\left(t\right)=Q\left(f\left(t\right)-{\displaystyle \frac{1}{ln2}}{\displaystyle \frac{1}{\lambda }}f(1-t)\right).$$

Differentiating the last equation again and substituting for $f^{\prime }$ gives

$$\begin{array}{ccc}\hfill f^{″}\left(t\right)& =& Q(f^{\prime }\left(t\right)+{\displaystyle \frac{1}{ln2}}{\displaystyle \frac{1}{\lambda }}{f}^{\prime }(1-t))\hfill \\ & =& Q^2\left(f\left(t\right)-{\displaystyle \frac{1}{ln2}}{\displaystyle \frac{1}{\lambda }}f(1-t)+{\displaystyle \frac{1}{ln2}}{\displaystyle \frac{1}{\lambda }}\left(f(1-t)-{\displaystyle \frac{1}{ln2}}{\displaystyle \frac{1}{\lambda }}f(1-(1-t))\right)\right)\hfill \\ & =& Q^{2}f\left(t\right)\left(1-{\displaystyle \frac{1}{{(\lambda ln2)}^2}}\right).\hfill \end{array}$$

Proposition 1.

If f is an eigenfunction of K on $(0,1)$ then, f satisfies

$$f^{″}\left(t\right)=Q^2\left(1-{\displaystyle \frac{1}{{(\lambda ln2)}^2}}\right)f\left(t\right),0<t<1.$$

In particular, the eigenvalues $\lambda$ of K and ${\alpha }^2$ of the Laplacian are related by $\lambda ={\displaystyle \frac{1}{ln2}}{\left({\displaystyle \frac{Q^2}{{\alpha }^2+Q^2}}\right)}^{1/2}$ so $\lambda \le {\displaystyle \frac{1}{ln2}}$. The eigenfunctions f of K have the form

$$f\left(t\right)=acos\left(\alpha t\right)+bsin\left(\alpha t\right),{\alpha }^2=Q^2\left({\displaystyle \frac{1}{{(\lambda ln2)}^2}}-1\right)$$

(4)

where $\lambda$ is the corresponding eigenvalue of K. We wish to emphasize here that for the bigger project of which this work is an important part, we are interested solely in the smallest positive value of $\alpha$ (largest value of $\lambda$) for which these relations hold when N is fixed, although we will not necessarily mention this in what follows when relationships between values of $\alpha$ and $\lambda$ appear.

The integral structure of K forces (continuous) eigenfunctions to vanish at $t=1$. This implies that

$$acos\alpha +bsin\alpha =0,\mathrm{or}{\displaystyle \frac{a}{b}}=-tan\alpha .$$

Up to a normalizing factor, this is satisfied by taking $a=sin\alpha$ and $b=-cos\alpha$ and hence $f\left(t\right)=sin\left(\alpha \right(1-t\left)\right)$. Also, the equation $f^{\prime }\left(t\right)=Q[f\left(t\right)-{\displaystyle \frac{1}{\lambda ln2}}f(1-t)]$ for $f\left(t\right)=sin\alpha cos\left(\alpha t\right)-cos\alpha sin\left(\alpha t\right)$ implies

$$\begin{array}{ccc}\hfill f^{\prime }\left(0\right)& =& -\alpha cos\alpha =Qsin\alpha \mathrm{or}{\displaystyle \frac{\alpha }{Q}}=-tan\alpha \hfill \end{array}$$

(5)

$$\begin{array}{ccc}\hfill f^{\prime }\left(1\right)& =& -\alpha =-sin\alpha {\displaystyle \frac{Q}{\lambda ln2}}\mathrm{or}\lambda ={\displaystyle \frac{sin\alpha }{\alpha }}{\displaystyle \frac{Q}{ln2}}=-{\displaystyle \frac{cos\alpha }{ln2}}.\hfill \end{array}$$

(6)

Corollary 1.

If $f\left(t\right)=acos\left(\alpha t\right)+bsin\left(\alpha t\right)$ is an eigenfunction of K in (3) then,

$${\displaystyle \frac{a}{b}}=-tan\alpha ={\displaystyle \frac{\alpha }{Q}}$$

with α as in (4). Consequently, up to normalization, f can be written as $f\left(t\right)=sin\left(\alpha \right(1-t\left)\right)$, $t\in [0,1]$. The corresponding eigenvalue λ of K is associated with α via (6).

Relations Between $\lambda$ and $\alpha$

We will freely invoke relationships (4)–(6) and the ones we mention presently in what follows, sometimes without reference. The identity (4) implies that $\lambda ={\displaystyle \frac{1}{ln2}}{\left({\displaystyle \frac{Q^2}{{\alpha }^2+Q^2}}\right)}^{1/2}$. Combining this with (6) allows elimination of $\lambda$ and identification of the values of $\alpha$ via $-cos\left(\alpha \right)={\left({\displaystyle \frac{Q^2}{{\alpha }^2+Q^2}}\right)}^{1/2}$ which in turn implies ${cos}^2\alpha ={\displaystyle \frac{Q^2}{{\alpha }^2+Q^2}}$ and ${sin}^2\alpha ={\displaystyle \frac{{\alpha }^2}{{\alpha }^2+Q^2}}$. From (4)–(6) one also has $\lambda =-{\displaystyle \frac{sec\alpha }{ln2}}\left({\displaystyle \frac{Q^2}{{\alpha }^2+Q^2}}\right)$. Setting this equal to the value $\lambda ={\displaystyle \frac{sin\alpha }{\alpha }}{\displaystyle \frac{Q}{ln2}}$ in (6) one obtains

$$sin\left(2\alpha \right)=2sin\alpha cos\alpha =2\lambda \alpha {\displaystyle \frac{ln2}{Q}}\left(-{\displaystyle \frac{1}{\lambda ln2}}\left({\displaystyle \frac{Q^2}{{\alpha }^2+Q^2}}\right)\right)=-{\displaystyle \frac{2\alpha }{Q}}\left({\displaystyle \frac{Q^2}{{\alpha }^2+Q^2}}\right).$$

Starting from this identity one can derive the approximation of its smallest positive solution stated in Lemma 2. Because it requires less elementary techniques we delay the proof to Appendix A. We provide a numerical example below for illustration.

Lemma 2.

For Q sufficiently large (e.g., $Q>2$), the smallest positive solution ${\alpha }_0$ of $sin\alpha ={\left({\displaystyle \frac{{\alpha }^2}{{\alpha }^2+Q^2}}\right)}^{1/2}$ has the form

$${\alpha }_0=\pi \left(1-{\displaystyle \frac{1}{Q}}+{\displaystyle \frac{1}{Q^2}}+O\left({\displaystyle \frac{1}{Q^3}}\right)\right).$$

Example 1.

Standard root finding involving transcendental functions follows the method just outlined: one approximates the function by a polynomial, then applies, for example, Newton’s method to find a root of the derived equation. If one enters

solve sin(t) = sqrt (t^2/(t^2 + q^2)), q = 1000, 0 < t < 10

into Wolfram alpha the smallest nonzero solution produced is $t=3.13845\dots$. For this value, $1-t/\pi \approx 9.98\times {10}^{-4}\approx {10}^{-3}$. Moreover for this value one has $1-t/\pi -{10}^{-3}=-1.002\times {10}^{-6}$ and $1-t/\pi -{10}^{-3}+{10}^{-6}=-2.27\times {10}^{-9}$. Similarly, taking $q={10}^4$ the positive value returned is $t=3.141278\dots$ where $1-t/\pi \approx 9.998\times {10}^{-5}\approx {10}^{-4}$ while $1-t/\pi -{10}^{-4}=-1.0002\times {10}^{-8}$ and $1-t/\pi -{10}^{-4}+{10}^{-8}=-2.288\times {10}^{-12}$. These cases confirm that the smallest positive value of α solving $sin\alpha ={\left({\displaystyle \frac{{\alpha }^2}{{\alpha }^2+Q^2}}\right)}^{1/2}$ has the form $\alpha =\pi \left(1-{\displaystyle \frac{1}{Q}}+{\displaystyle \frac{1}{Q^2}}+O\left({\displaystyle \frac{1}{Q^3}}\right)\right)$ for these values of Q.

4. Spectrum Approximation of ${\mathit{C}}_{\mathit{N}}$

We show now that samples of the vectors $sin\left(\alpha \right(1-·/N\left)\right)$, where $\alpha$ is the smallest positive value such that $sin\left(\alpha \right(1-t\left)\right)$ is an eigenfunction of $K=K_Q$, form approximate eigenvectors of $C_N$ defined by (1) ($Q={\displaystyle \frac{Nln2}{2}}$). That is, we will show that there is a “discrete” approximate eigenvalue ${\lambda }_{\mathrm{discr}}$ such that the vectors $\mathbf{s}={\mathbf{s}}^{\alpha }$ with entries ${\left({\mathbf{s}}^{\alpha }\right)}_k=sin\left(\alpha (1-k/N)\right)$, $k=0,\dots ,N$, form approximate eigenvectors of $C_N$ in the sense that $|{\left(C_N{\mathbf{s}}^{\alpha }\right)}_k-{\lambda }_{\mathrm{discr}}\left(N\right){\left({\mathbf{s}}^{\alpha }\right)}_k|\le C/N$ for some constant $C>0$ independent of N. When $\alpha$ is the smallest positive solution of $sin\alpha ={\left({\displaystyle \frac{{\alpha }^2}{{\alpha }^2+Q^2)}}\right)}^{1/2}$, which corresponds to the largest eigenvalue of K, the specific value that we provide for the approximate eigenvalue of $C_N$ is ${\lambda }_{\mathrm{discr}}=1+{\displaystyle \frac{\sqrt{2}}{2}}$, which is an upper bound for $\parallel C_N\parallel$ for any fixed $N=2,3,\dots$.

It will help to return to the simple observation that with $v=Nt$ and $u=Ns$,

$${\int }_0^{1-t}{e}^{Qs}f\left(s\right)ds={\int }_0^{N-v}{e}^{Qu/N}f(u/N){\displaystyle \frac{du}{N}}$$

so that

$$\begin{array}{ccc}\hfill \left(Kf\right)\left({\displaystyle \frac{v}{N}}\right)& =& {\displaystyle \frac{Q}{ln2}}{e}^{Q\left({\displaystyle \frac{v}{N}}-1\right)}{\int }_0^{N-v}{e}^{Qu/N}f\left({\displaystyle \frac{u}{N}}\right){\displaystyle \frac{du}{N}}\hfill \\ & =& {\displaystyle \frac{1}{2}}{e}^{{\displaystyle \frac{ln2}{2}}\left(v-N\right)}{\int }_0^{N-v}{e}^{{\displaystyle \frac{ln2}{2}}u}f\left({\displaystyle \frac{u}{N}}\right)du.\hfill \end{array}$$

When applied to the function $sin\left(\alpha \right(1-u/N\left)\right)$ one then has

$$\left(Ksin\left(\alpha \left(1-{\displaystyle \frac{·}{N}}\right)\right)\right)\left({\displaystyle \frac{k}{N}}\right)={\displaystyle \frac{1}{2}}{2}^{{\displaystyle \frac{k-N}{2}}}\sum _{\mathcal{\ell }=0}^{N-k-1}{2}^{{\displaystyle \frac{\mathcal{\ell }}{2}}}{\int }_0^1{e}^{{\displaystyle \frac{ln2}{2}}u}sin\left(\alpha \left(1-{\displaystyle \frac{(\mathcal{\ell }+u)}{N}}\right)\right)du.$$

(7)

To compute the integrals in the sum over the parameter ℓ one can use integration by parts (twice) to obtain the following.

Proposition 2.

The integrals in (7) satisfy

$$\begin{array}{c}\left(1+{\left({\displaystyle \frac{Nln2}{2\alpha }}\right)}^2\right){\int }_0^1{e}^{{\displaystyle \frac{ln2}{2}}u}sin\left(\alpha (1-(\mathcal{\ell }+u)/N)\right)du\hfill \\ \hfill ={\displaystyle \frac{N}{\alpha }}[e^{{\displaystyle \frac{ln2}{2}}u}cos\left(\alpha (1-(\mathcal{\ell }+u)/N)\right){|}_0^1+{\displaystyle \frac{N^2}{{\alpha }^2}}{\displaystyle \frac{ln2}{2}}sin\left(\alpha (1-(\mathcal{\ell }+u)/N)\right)e^{{\displaystyle \frac{ln2}{2}}u}{|}_0^1.\end{array}$$

Proof of Theorem  1.

Continuing to write $Q={\displaystyle \frac{Nln2}{2}}$, evaluating the terms on the right of the equation in Proposition 2 at the endpoints and dividing both sides by $1+{\left({\displaystyle \frac{Q}{\alpha }}\right)}^2$ allows us to write

$$\begin{array}{cc}& {\int }_0^1{e}^{{\displaystyle \frac{ln2}{2}}u}sin(\alpha (1-(\mathcal{\ell }+u)/N)du\hfill \\ =& {\displaystyle \frac{\alpha N}{{\alpha }^2+Q^2}}\left[\sqrt{2}cos\left(\alpha \left(1-{\displaystyle \frac{(\mathcal{\ell }+1)}{N}}\right)\right)-cos\left(\alpha \left(1-{\displaystyle \frac{\mathcal{\ell }}{N}}\right)\right)\right]\hfill \\ +& {\displaystyle \frac{2}{ln2}}{\displaystyle \frac{Q^2}{{\alpha }^2+Q^2}}\left[\sqrt{2}sin\left(\alpha \left(1-{\displaystyle \frac{(\mathcal{\ell }+1)}{N}}\right)\right)-sin\left(\alpha \left(1-{\displaystyle \frac{\mathcal{\ell }}{N}}\right)\right)\right]\hfill \\ =& C\left(\mathcal{\ell }\right)+S\left(\mathcal{\ell }\right)\hfill \end{array}$$

where $C\left(\mathcal{\ell }\right)$ denotes the cosine difference term and $S\left(\mathcal{\ell }\right)$ the sine difference term. For the cosine difference terms we have

$$\begin{array}{ccc}\hfill \sum _{\mathcal{\ell }=0}^{N-k-1}{2}^{{\displaystyle \frac{\mathcal{\ell }}{2}}}{C}_{\mathcal{\ell }}& =& {\displaystyle \frac{N{sin}^2\alpha }{\alpha }}\sum _{\mathcal{\ell }=0}^{N-k-1}\left[2^{{\displaystyle \frac{\mathcal{\ell }+1}{2}}}cos\left(\alpha \left(1-{\displaystyle \frac{(\mathcal{\ell }+1)}{N}}\right)\right)-2^{{\displaystyle \frac{\mathcal{\ell }}{2}}}cos\left(\alpha \left(1-{\displaystyle \frac{\mathcal{\ell }}{N}}\right)\right)\right]\hfill \\ & =& {\displaystyle \frac{N{sin}^2\alpha }{\alpha }}\left[2^{{\displaystyle \frac{N-k}{2}}}cos\left(\alpha \left(1-{\displaystyle \frac{(N-k)}{N}}\right)\right)-cos\left(\alpha \right)\right]\hfill \\ & =& {\displaystyle \frac{N{sin}^2\alpha }{\alpha }}\left[2^{{\displaystyle \frac{N-k}{2}}}cos\left(\alpha {\displaystyle \frac{k}{N}}\right)-cos\left(\alpha \right)\right].\hfill \end{array}$$

(8)

Denote by EC the “cosine error” equal to $2^{(k-N)/2}$ times this last expression, that is,

$$\mathrm{EC}\left(k\right)={\displaystyle \frac{N{sin}^2\alpha }{\alpha }}\left[cos\left(\alpha {\displaystyle \frac{k}{N}}\right)-2^{{\displaystyle \frac{k-N}{2}}}cos\left(\alpha \right)\right].$$

By (6), $sin\alpha /\alpha =2\lambda /N$, hence $N{sin}^2\alpha /\alpha =4\alpha {\lambda }^2/N$. Then, since $cos\left(t\right)\le 1$ we have

$$2^{{\displaystyle \frac{N-k}{2}}}\left|\mathrm{EC}\left(k\right)\right|=\left|\sum _{\mathcal{\ell }=0}^{N-k-1}{2}^{{\displaystyle \frac{\mathcal{\ell }}{2}}}{C}_{\mathcal{\ell }}\right|\le 2^{{\displaystyle \frac{N-k}{2}}}{\displaystyle \frac{(\sqrt{2}+1)4\alpha {\lambda }^2}{N}}.$$

(9)

In addressing the terms $S\left(\mathcal{\ell }\right)$, it will help to recall that the eigenvalue $\lambda$ of K is $\lambda =-{\displaystyle \frac{sec\alpha }{ln2}}{\displaystyle \frac{Q^2}{{\alpha }^2+Q^2}}$ from which it follows that the coefficient ${\displaystyle \frac{2}{ln2}}{\displaystyle \frac{Q^2}{{\alpha }^2+Q^2}}$ in $S\left(\mathcal{\ell }\right)$ is equal to $-2cos\left(\alpha \right)\lambda$. Although the terms $S\left(\mathcal{\ell }\right)$ can also form a telescoping series, we treat those terms differently in order to compare to the sine-term outputs of the operator K. For this reason we write  

$$\begin{array}{ccc}\hfill S\left(\mathcal{\ell }\right)& =& -2\lambda cos\alpha \left[\sqrt{2}sin\left(\alpha \left(1-{\displaystyle \frac{(\mathcal{\ell }+1)}{N}}\right)\right)-sin\left(\alpha \left(1-{\displaystyle \frac{\mathcal{\ell }}{N}}\right)\right)\right]\hfill \\ & =& -2\lambda cos\alpha \left[\sqrt{2}\left(sin\left(\alpha \left(1-{\displaystyle \frac{(\mathcal{\ell }+1)}{N}}\right)\right)-sin\left(\alpha \left(1-{\displaystyle \frac{\mathcal{\ell }}{N}}\right)\right)\right)\right.\hfill \\ & & \left.+(\sqrt{2}-1)sin\left(\alpha \left(1-{\displaystyle \frac{\mathcal{\ell }}{N}}\right)\right)\right]\hfill \\ & =& -2\lambda cos\alpha \left[-\sqrt{2}{\displaystyle \frac{\alpha }{N}}cos\left(\alpha \left(1-{\displaystyle \frac{t_{\mathcal{\ell }}}{N}}\right)\right)+(\sqrt{2}-1)sin\left(\alpha \left(1-{\displaystyle \frac{\mathcal{\ell }}{N}}\right)\right)\right]\hfill \\ & =& SC\left(\mathcal{\ell }\right)+SS\left(\mathcal{\ell }\right)\hfill \end{array}$$

where $t_{\mathcal{\ell }}\in (\mathcal{\ell }/N,(\mathcal{\ell }+1)/N)$ is determined by the mean value theorem and where $SC\left(\mathcal{\ell }\right)$ is the “cosine” term and $SS\left(\mathcal{\ell }\right)$ the corresponding “sine” term in the preceding line. Define $\mathrm{ESC}\left(k\right)=2^{(k-N)/2}{\sum }_{\mathcal{\ell }=0}^{N-k-1}{2}^{\mathcal{\ell }/2}SC\left(\mathcal{\ell }\right)$. Since $|cos(t\left)\right|\le 1$ it follows that for each ℓ

$$SC\left(\mathcal{\ell }\right)=2\lambda cos\alpha \sqrt{2}{\displaystyle \frac{\alpha }{N}}cos\left(\alpha \left(1-{\displaystyle \frac{t_{\mathcal{\ell }}}{N}}\right)\right)\le {\displaystyle \frac{2^{3/2}\lambda \alpha }{N}}$$

so it follows that

$$2^{{\displaystyle \frac{N-k}{2}}}\left|\mathrm{ESC}\left(k\right)\right|\le {\displaystyle \frac{2^{\frac{3}{2}}\lambda \alpha }{N}}\sum _{\mathcal{\ell }=0}^{N-k-1}{2}^{{\displaystyle \frac{\mathcal{\ell }}{2}}}={\displaystyle \frac{2^{\frac{3}{2}}\lambda \alpha }{N}}{\displaystyle \frac{2^{\frac{N-k}{2}}-1}{\sqrt{2}-1}}\le 2^{{\displaystyle \frac{N-k}{2}}}{\displaystyle \frac{(2+\sqrt{2})2\lambda \alpha }{N}}.$$

(10)

As for the terms $SS\left(\mathcal{\ell }\right)$ one has

$$\begin{array}{ccc}\hfill \sum _{\mathcal{\ell }=0}^{N-k-1}{2}^{{\displaystyle \frac{\mathcal{\ell }}{2}}}SS\left(\mathcal{\ell }\right)& =& -{\displaystyle \frac{2\lambda cos\alpha }{\sqrt{2}+1}}\sum _{\mathcal{\ell }=0}^{N-k-1}{2}^{{\displaystyle \frac{\mathcal{\ell }}{2}}}sin\left(\alpha (1-\mathcal{\ell }/N)\right)\hfill \\ & =& -{\displaystyle \frac{2\lambda cos\alpha }{\sqrt{2}+1}}\left[\sqrt{2}sin\alpha +\sqrt{2}\left(\sum _{\mathcal{\ell }=1}^{N-k-1}{2}^{{\displaystyle \frac{\mathcal{\ell }-1}{2}}}{s}_{\mathcal{\ell }}^{\alpha }\right)-(\sqrt{2}-1)sin\alpha \right]\hfill \\ & =& -{\displaystyle \frac{2^{3/2}\lambda cos\alpha }{\sqrt{2}+1}}\left[sin\alpha +\left(\sum _{\mathcal{\ell }=1}^{N-k}{2}^{{\displaystyle \frac{\mathcal{\ell }-1}{2}}}{s}_{\mathcal{\ell }}^{\alpha }\right)-2^{{\displaystyle \frac{N-k-1}{2}}}{s}_{N-k}^{\alpha }-{\displaystyle \frac{\sqrt{2}-1}{\sqrt{2}}}sin\alpha \right]\hfill \\ & =& -{\displaystyle \frac{2^{3/2}\lambda cos\alpha }{\sqrt{2}+1}}\left[2^{{\displaystyle \frac{N-k+1}{2}}}{\left(C_N{\mathbf{s}}^{\alpha }\right)}_k-2^{{\displaystyle \frac{N-k-1}{2}}}{s}_{N-k}^{\alpha }-{\displaystyle \frac{\sqrt{2}-1}{\sqrt{2}}}sin\alpha \right]\hfill \end{array}$$

where, as before, ${\mathbf{s}}^{\alpha }=\{sin\left(\alpha (1-·/N)\right)\}$. Now define $\left|\mathrm{ESS}\left(k\right)\right|=2^{(k-N)/2}{\displaystyle \frac{2\lambda |cos\alpha |}{\sqrt{2}+1}}(\sqrt{2}-1)$$sin\alpha$. By Lemma 4, the smallest nonzero $\alpha$ corresponding to an eigenvalue of K, which is the value of interest here, satisfies $\alpha =\pi (1-1/Q+1/Q^2+O(1/Q^3))$. Because of this we have $0<sin\left(\alpha \right)<{\displaystyle \frac{\pi }{Q}}={\displaystyle \frac{2\pi }{Nln2}}$. Therefore, we have the bound

$$\left|\mathrm{ESS}\left(k\right)\right|\le 2^{{\displaystyle \frac{k-N}{2}}}4{\displaystyle \frac{\sqrt{2}-1}{\sqrt{2}+1}}\lambda |cos\alpha |{\displaystyle \frac{\pi }{Nln2}}.$$

(11)

Assembling all of these pieces one has  

$$\begin{array}{ccc}& & (Ksin\left(\alpha \left(1-{\displaystyle \frac{·}{N}}\right)\right)\left({\displaystyle \frac{k}{N}}\right)\hfill \\ & =& {\displaystyle \frac{1}{2}}{2}^{{\displaystyle \frac{k-N}{2}}}\sum _{\mathcal{\ell }=0}^{N-k-1}{2}^{{\displaystyle \frac{\mathcal{\ell }}{2}}}{\int }_0^1{e}^{{\displaystyle \frac{ln2}{2}}u}sin\left(\alpha \left(1-{\displaystyle \frac{(\mathcal{\ell }+u)}{N}}\right)\right)du\hfill \\ & =& {\displaystyle \frac{1}{2}}{2}^{{\displaystyle \frac{k-N}{2}}}\sum _{\mathcal{\ell }=0}^{N-k-1}{2}^{{\displaystyle \frac{\mathcal{\ell }}{2}}}(C\left(\mathcal{\ell }\right)+S\left(\mathcal{\ell }\right))\hfill \\ & =& {\displaystyle \frac{1}{2}}\left(\mathrm{EC}\left(k\right)+\mathrm{ESC}\left(k\right)+\mathrm{ESS}\left(k\right)\right.\hfill \\ & & \left.+{\displaystyle \frac{2^{3/2}\begin{array}{c}\hfill \lambda (-cos\alpha )\end{array}}{\sqrt{2}+1}}{2}^{{\displaystyle \frac{k-N}{2}}}\left(2^{{\displaystyle \frac{N-k+1}{2}}}{\left(C_N{\mathbf{s}}^{\alpha }\right)}_k-2^{{\displaystyle \frac{N-k-1}{2}}}{s}_{N-k}^{\alpha }\right)\right)\hfill \\ & =& {\displaystyle \frac{1}{2}}\left(\mathrm{EC}\left(k\right)+\mathrm{ESC}\left(k\right)+\mathrm{ESS}\left(k\right)+{\displaystyle \frac{2\lambda (-cos\alpha )}{\sqrt{2}+1}}\left(2{\left(C_N{\mathbf{s}}^{\alpha }\right)}_k-s_{N-k}^{\alpha }\right)\right)\hfill \end{array}$$

where the terms $\mathrm{EC}\left(k\right)$, $\mathrm{ESC}\left(k\right)$, and $\mathrm{ESS}\left(k\right)$ are bounded by multiples of $1/N$ according to (9), (10), and (11), respectively.

Since $(Ksin\left(\alpha \left(1-{\displaystyle \frac{·}{N}}\right)\right)\left({\displaystyle \frac{k}{N}}\right)=\lambda sin\left(\alpha \left(1-{\displaystyle \frac{k}{N}}\right)\right)$ and the terms $\mathrm{EC}\left(k\right)$, $\mathrm{ESC}\left(k\right)$, and $\mathrm{ESS}\left(k\right)$ are $O(1/N)$ the relationship just derived can be written

$$\lambda s_k^{\alpha }=O\left({\displaystyle \frac{1}{N}}\right)+{\displaystyle \frac{\lambda (-cos\alpha )}{\sqrt{2}+1}}\left(2{\left(C_N{\mathbf{s}}^{\alpha }\right)}_k-s_{N-k}^{\alpha }\right).$$

For $\lambda \ne 0$, canceling $\lambda$ from both sides this becomes

$$\left(2{\left(C_N{\mathbf{s}}^{\alpha }\right)}_k-s_{N-k}^{\alpha }\right)=(-sec\alpha )(\sqrt{2}+1)s_k^{\alpha }+O\left({\displaystyle \frac{1}{N}}\right)$$

or

$${\left(C_N{\mathbf{s}}^{\alpha }\right)}_k={\displaystyle \frac{1}{2}}\left((-sec\alpha )(\sqrt{2}+1)s_k^{\alpha }+s_{N-k}^{\alpha }\right)+O\left({\displaystyle \frac{1}{N}}\right).$$

We will see momentarily (see Lemmas 3 and 4) that for the smallest $\alpha >0$ satisfying (4) one has $s_{N-k}^{\alpha }=-cos\alpha s_k^{\alpha }+O(1/N)$. With this additional estimate, finally, one has

$${\left(C_N\mathbf{s}\right)}_k={\displaystyle \frac{1}{2}}\left(-sec\left(\alpha \right)(\sqrt{2}+1)-cos\left(\alpha \right)\right)s_k^{\alpha }+O\left({\displaystyle \frac{1}{N}}\right)=(1+{\displaystyle \frac{\sqrt{2}}{2}})s_k^{\alpha }+O\left({\displaystyle \frac{1}{N}}\right)$$

where we have used that for $\pi -\alpha =\pi (1/Q-1/Q^2+O(1/Q^3))$ one has $-sec\alpha =1+{\displaystyle \frac{{\pi }^2}{2Q^2}}+O\left(Q^{-4}\right)$ while $-cos\alpha =1-{\displaystyle \frac{{\pi }^2}{2Q^2}}+O\left(Q^{-4}\right)$. This completes the proof of Theorem 1, pending Lemmas 3 and 4 below.    □

Remark 2.

As observed in Figure 2b, the norms of $C_N$ are increasing with N and bounded above by $1+{\displaystyle \frac{\sqrt{2}}{2}}$. Further numerical analysissuggests that $1+{\displaystyle \frac{\sqrt{2}}{2}}-\parallel C_N\parallel$ decays like $1/N^2$, just as ${\displaystyle \frac{1}{ln2}}-\parallel K_Q\parallel$ decays like $1/Q^2$.While the estimates established here will suffice for bounds on sums of Walsh–Fourier series to be established elsewhere, identifying, for fixed N, a more precise approximation of $\parallel C_N\parallel$ that is strictly smaller than the value $1+{\displaystyle \frac{\sqrt{2}}{2}}$ identified by Theorem 1 would require a more detailed analysis of each of the error terms $\mathrm{EC}$, $\mathrm{ESC}$, and $\mathrm{ESS}$.

Lemma 3.

When $\alpha =\pi (1-{\displaystyle \frac{1}{Q}}+O(1/Q^2))$ ($Q=Nln2/2$) one has

$$sin(\alpha k/N)=-cos\alpha sin\left(\alpha \right(1-k/N\left)\right)+O(1/N).$$

Proof. 

$$\begin{array}{ccc}\hfill sin(\alpha k/N)& =& sin(\pi -\alpha k/N)=sin(\pi -\alpha +\alpha (1-k/N\left)\right)\hfill \\ & =& sin(\pi -\alpha )cos\left(\alpha \right(1-k/N\left)\right)+cos(\pi -\alpha )sin\left(\alpha \right(1-k/N\left)\right)\hfill \\ & =& \left({\displaystyle \frac{\pi }{Q}}+O\left({\displaystyle \frac{1}{Q^2}}\right)\right)cos\left(\alpha (1-k/N)\right)-cos\alpha sin\left(\alpha (1-k/N)\right)\hfill \end{array}$$

where we have used the Taylor approximation for sine after Lemma 2. We observe also that $cos(\pi -\alpha )=1-{\displaystyle \frac{{\pi }^2}{2Q^2}}+O(1/Q^4)$ which allows us to conclude also that $sin(\alpha k/N)-sin\left(\alpha \right(1-k/N\left)\right)=O(1/N)$.    □

Lemma 4.

When $\alpha =\pi (1-{\displaystyle \frac{1}{Q}}+O(1/Q^2))$ ($Q=Nln2/2$) one has $sin\left(\alpha \right)={\displaystyle \frac{2\pi }{Nln2}}+O(1/N^2)$.

This lemma follows immediately from the Taylor approximation of $sint$.

5. Plots

Figure 3 plots the $\parallel C_N\parallel$-eigenvectors of $C_N$ for $N=32,180$ and 1000, the last value being close to the maximum size that matlab (R2025b) can handle due to entries of $C_N$ being below machine precision beyond this range. Note that 180 is close to being the geometric mean of 32 and 1000. The eigenvectors of $C_N$ are plotted as solid curves. In comparison, the dashed plots are the curves $sin\left(\gamma \right(1-k/N\left)\right)$ where $\gamma =\pi (1-1/Q+1/Q^2)$ is an approximation of the value $\alpha$ defined by (4) for the corresponding eigenvalue, see Lemma 2. The curves near the bottom of each plot are the differences between the corresponding (normalized) eigenvector and $sin\left(\gamma \right(1-k/N\left)\right)$ curves. These error curves are themselves approximately sinusoidal. This suggests that the $\parallel C_N\parallel$-eigenvectors of $C_N$ themselves are approximately sinusoidal (except in the zeroth coordinate) and that the error comes mainly from the approximation of the value $\alpha$ itself. The ${\mathcal{\ell }}^2$-norms of these errors are $0.1475$ ($N=32$), $0.0344$ ($N=180$) and $0.0065$ ($N=1000$), respectively.

6. Conclusions

We have proved that the vectors ${\mathbf{s}}^{\alpha }={\{sin\left(\alpha (1-k/N)\right)\}}_{k=0}^N$ form approximate eigenvectors of $C_N$ with approximate eigenvalue $1+{\displaystyle \frac{\sqrt{2}}{2}}$ when $\alpha$ is the smallest nonzero solution of $sin\alpha ={\left({\displaystyle \frac{{\alpha }^2}{{\alpha }^2+Q^2}}\right)}^{1/2}$. The methods do not strictly prove that $1+{\displaystyle \frac{\sqrt{2}}{2}}$ is an upper bound on the eigenvalues of $C_N$ but this can be confirmed by running the arguments backwards. That is, starting with a sequence of norm-maximizing eigenvectors of $C_N$, one can work back to showing that these eigenvectors approximately correspond to samples of eigefunctions of the continuous model operator K, which are sinusoidal. That the norm of $C_N$ is bounded by $1+{\displaystyle \frac{\sqrt{2}}{2}}$ has been verified numerically directly for computable values of N, and the numerical norm-maximizing vectors are evidently sinusoidal (with the exception of values at the zeroth coordinate where a correction term has to be introduced due to the boundary weighting). The estimates presented here, using elementary techniques from calculus and a bit of linear algebra, form an important component of a study of convergence of Fourier series in the setting of Walsh functions on $L^2[0,1]$.