Spatio-Spectral Limiting on Boolean Cubes

Abstract

The operator that first truncates to a neighborhood of the origin in the spatial domain then truncates to a neighborhood of the origin in the spectral domain is investigated in the case of Boolean cubes. This operator is self adjoint on the space of functions spanned by the Laplacian eigenvectors corresponding to small eigenvalues. The eigenspaces of this iterated projection operator are studied through reduced matrices based on certain invariant subspaces. They are shown to depend fundamentally on the neighborhood structure of the cube defined in terms of Hamming distance.

1 Introduction

This work continues an investigation begun in [11] of structure and properties of vertex functions on Boolean cubes ${\mathcal {B}}_N$ (also known as N-cubes, discrete cubes, Hamming cubes or hypercube graphs) that are in the span of low-eigenvalue eigenvectors of the Laplacian on ${\mathcal {B}}_N$, and also concentrated in a neighborhood of the origin in ${\mathcal {B}}_N$. In this work we identify the eigenspaces of the iterated projection operators that define these properties and provide an effective method to compute eigenvectors numerically. The N-cube is a graph whose vertices are the points in $\{0,1\}^N$. Two vertices are adjacent if they differ as elements of $\{0,1\}^N$ in a single coordinate. Symmetric neighborhoods of a vertex—Hamming balls of radius $r>0$—are defined as those vertices of Hamming distance at most r from the given vertex. For fixed K, let Q be the operator that truncates a vertex function f to the K-Hamming ball ${\mathbb {B}}_K$ centered at the origin in ${\mathcal {B}}_N$. For fixed $K'$, let P denote the operator that projects onto the span of the eigenvectors of the graph Laplacian of ${\mathcal {B}}_N$ with eigenvalues at most $2K'$. The operator PQ is an analogue on ${\mathcal {B}}_N$ of the time- and band-limiting operator $P_\Omega Q_T$ on ${\mathbb {R}}$ where $(Q_T f)(t)=(\mathbb {1}_{[-T,T]} f)(t)$ and $(P_\Omega f)(t)= ({\mathcal {F}}^{-1} \mathbb {1}_{[-\Omega /2,\Omega /2]} {\mathcal {F}}f)(t)$, where $({\mathcal {F}}f)(\xi )=\int f(t)\, e^{-2\pi i t\xi }\, dt$ denotes the Fourier transform on ${\mathbb {R}}$. The composition $P_\Omega Q_T$ is the object of study of the Bell Labs papers [16, 17, 28]. Euclidean, discrete, and finite analogues of these operators were studied by Slepian [26, 27], Grünbaum, [6, 7] and others [34], respectively.

Tsitsvero et al. [32] proposed analogues of time- and band-limiting operators in the context of (finite) graph signal processing. As is common in the graph signal processing literature (e.g., [4]), their spatial and spectrum limiting “Q” and “P”-type operators are defined in terms of limiting to fixed but arbitrary subsets of vertices and Laplacian eigenvectors—not just neighborhoods of a given vertex and smallest eigenvalues of the graph Laplacian. The aims in [32] were to establish (i) uncertainty principles, formulated in terms of possible pairs of norms ($\Vert P f\Vert$,$\Vert Qf\Vert$) of a unit-norm vertex function f depending on the spatial and spectral cutoff sets defining Q and P, and (ii) sampling properties, framed in terms of the operator norm of $P(I-Q)$.

While our approach on ${\mathcal {B}}_N$ relies fundamentally on graph adjacency structure, in contrast to studies like [32] that apply to general graphs, ours is a closer parallel to the Bell Labs theory that specifically studies spatial and spectral localizations to neighborhoods of the origin, leveraging specific geometry to identify and describe eigenfunctions of PQ-operators explicitly, and to analyze their spectra. One property of eigenvectors $\varphi$ of PQ on ${\mathcal {B}}_N$ established in [11] (for the case $K'=K$) is that the graph Fourier transforms of such $\varphi$ are equal to truncations of $\varphi$. This is also a property of Fourier transforms of the eigenfunctions of $P_\Omega Q_T$ (up to a unitary dilation) on ${\mathbb {R}}$ and their analogues on ${\mathbb {Z}}_N$. Such a property does not even make sense unless Fourier transforms are defined on a domain that is the same (or isomorphic to) that of the spatial signals themselves. For the groups ${\mathbb {R}}$ and ${\mathbb {Z}}_N$ (and any finite abelian group), Fourier transforms of signals are defined on isomorphic dual groups. Vertices of ${\mathcal {B}}_N$ are naturally identified with elements of the abelian group ${\mathbb {Z}}_2^N$, whose dual group elements correspond to eigenvectors of the Laplacian of ${\mathcal {B}}_N$. This allows Fourier transforms of vertex functions on ${\mathcal {B}}_N$ to be regarded as functions on (an isomorphic copy of) ${\mathcal {B}}_N$.

In [11] we also established other properties of eigenvectors of PQ-operators on ${\mathcal {B}}_N$ analogous to those of $P_\Omega Q_T$ on ${\mathbb {R}}$, including completeness and orthogonality on $\ell ^2({\mathbb {B}}_K)$, and so-called spectral accumulation: squares of Fourier transforms, weighted by eigenvalues, summing to $\mathbb {1}_{{\mathbb {B}}_{K'}}$ in the Fourier domain. One property of time and band limiting on ${\mathbb {R}}$ that does not appear to extend to ${\mathcal {B}}_N$ is commutation of PQ with a natural operator that gives rise to effective numerical computation of eigenvectors. On ${\mathbb {R}}$, $P_\Omega Q_T$ commutes with a certain second order differential operator with quadratic coefficients whose eigenfunctions—the prolate spheroidal wave functions—can be computed from polynomial approximations, e.g., [10, Sect. 1.2.4]. In [11] we showed that a formal analogue of this prolate differential operator does not commute with PQ, barring this as a direct route to effective computation of the eigenvectors of PQ. We established there a rudimentary method to compute the eigenvectors numerically, but that method is inefficient and, practically, is limited to cases of small N.

The principal purpose of this work is to identify eigenspaces of PQ-operators on ${\mathcal {B}}_N$ in a manner that leads to effective computation of eigenvectors. The new technique here is to identify adjacency-invariant spaces of vertex functions on the graph ${\mathcal {B}}_N$ that contain the (Fourier transforms of) eigenspaces of PQ. These spaces factor in terms of data on a fixed Hamming sphere, and coefficients of the images of that data under powers of an outer adjacency operator. Using this factorization in the spectral domain, the operator PQ on such an invariant space can be reduced to that of a matrix of size at most $N\times N$ (versus size $2^N\times 2^N$). We have extended some of these methods to certain other Cayley graphs of finite abelian groups, see [12, 13].

Much of the fairly extensive literature concerning Fourier methods on Boolean cubes specifically addresses problems related to learning Boolean functions—functions on N-cubes with values in $\{0,1\}$—that have sparse Fourier representations, not always associated with the smallest Laplacian eigenvalues, from a small number of samples, e.g., [1, 3, 15, 19, 20, 29, 33], cf. [22, 24]. The Fourier transform on ${\mathcal {B}}_N$ has a long history in terms of (discrete) Walsh functions. Statistical applications were addressed by Stoffer [30], see [14, 23] for more recent work. The transform is basic in Walsh codes used extensively in CDMA, e.g., [2]. See de Wolf [5] for an outline of some other applications of Fourier analysis on ${\mathcal {B}}_N$. Our approach here and in [11] was not driven by a specific application beyond finding mechanisms to identify how low-spectrum data might be spatially concentrated. It can be viewed as refinement of uncertainty principles on ${\mathcal {B}}_N$ [8] in the sense of identifying low-spectrum vertex functions that are spatially concentrated, just as the Bell Labs theory does so in terms of prolate functions on ${\mathbb {R}}$. Some potential uses of eigenvectors of PQ that would parallel applications of the Bell Labs theory are outlined in Sect. 7.

1.1 Contribution

Here is a description of the main results. We refer to Sects. 1.2 and 2 for further notation and technical background. The Boolean cube ${\mathcal {B}}_N$ is the undirected graph with $2^N$ vertices indexed by ${\mathbb Z}_2^N$. It is conventional to label vertices by subsets $S \subset \{1,\ldots , N\}$. Two vertices R, S are adjacent if the symmetric difference of R and S is a singleton, $|R\Delta S|=1$, in which case we write $R\sim S$. The path distance $d_H$ on ${\mathcal {B}}_N$ is Hamming distance, which can be expressed as $d_H (R,S)=|R\Delta S|$. Since $d_H(\emptyset ,S)=|S|$, the (discrete) K-ball in ${\mathcal {B}}_N$ consisting of those vertices of distance at most K from the origin (indexed by $\emptyset$) is ${\mathbb {B}}_K=\{S:|S|\le K\}$. We also denote the (Hamming) K-sphere by $\Sigma _K=\{S:|S|=K\}$. Observe that $|\Sigma _r|=\left( {\begin{array}{c}N\\ r\end{array}}\right)$ and $|{\mathbb {B}}_K|=\sum _{r=0}^K \left( {\begin{array}{c}N\\ r\end{array}}\right)$.

The collection of real-valued vertex functions on ${\mathcal {B}}_N$ is denoted by $\ell ^2({{\mathcal {B}}}_N)$ with inner product

$$\begin{aligned} \langle f,g\rangle =\sum _{S\in {{\mathcal {B}}}_N}f(S)g(S). \end{aligned}$$

Any $2^N\times 2^N$ matrix M with entries indexed by vertex pairs determines a linear operator on $\ell ^2({{\mathcal {B}}}_N)$ via $(Mf)(S)=\sum _{R\in {{\mathcal {B}}}_N}M_{SR}f(R)$. The adjacency matrix A has entries

$$\begin{aligned} A_{RS}={\left\{ \begin{array}{ll} 1&{}\hbox { if}\ R\sim S\\ 0&{}\text { else.} \end{array}\right. } \end{aligned}$$

and provides a linear mapping of $\ell ^2({{\mathcal {B}}}_N)$ given by $(Af)(S)=\sum _{R\sim S}f(R)$. The (unnormalized) Laplacian on ${\mathcal {B}}_N$ is $L=NI-A$ (see Sect. 2.1). Its eigenvectors $h_S$ can also be indexed by $S\subset \{1,\ldots ,N\}$ with eigenvalue 2|S|. In fact, the spectrum of the ${\mathcal {B}}_N$-Laplacian can also be equipped with the structure of an isomorphic copy of ${\mathcal {B}}_N$. The graph Fourier transform is the unitary operator represented by the matrix H with orthonormal columns $h_S$ (note, also $H=H^T$, see also Sect. 2.1). For $K\ge 0$ fixed, denote by Q the operator on $\ell ^2({{\mathcal {B}}}_N)$ defined by

$$\begin{aligned} (Qf)(R)={\left\{ \begin{array}{ll} f(R)&{}\hbox { if}\ R\in {\mathbb {B}}_K\\ 0&{}\text { else}, \end{array}\right. } \end{aligned}$$

and let P be the operator that projects onto the Boolean Paley–Wiener space $\mathrm{PW}_{K'}=\mathrm{span}\,\{h_S:\ |S|\le K'\}$, which has dimension $\sum _{r=0}^{K'}\left( {\begin{array}{c}N\\ r\end{array}}\right)$. To simplify the presentation we study only the case $K'=K$. In this case we can write $P=HQH$ (again, $H=H^T$), keeping in mind that the truncation Q in this expression acts in the Fourier domain. For technical reasons, we assume that $K<N/2$ throughout.

The main result, Theorem 4, identifies a decomposition of $\mathrm{PW}_{K}$ in terms of invariant subspaces of PQ. In the spectral domain, these subspaces consist of vectors of the form $f=\sum _{k=0}^{K-r} c_k A_+^{k} g$ where $A_+$ is the lower triangular part of the adjacency matrix A of ${\mathcal {B}}_N$ in an appropriate ordering (detailed in Sect. 2.3), and $g\in {\mathcal {W}}_r$ where ${\mathcal {W}}_r$ is the collection of vertex functions that are supported in the sphere $\Sigma _r$ and lie in the kernel of the adjoint of $A_+$. Thus each such invariant subspace of $QP={H} PQ{H}$ is isomorphic to a product ${\mathcal {W}}_r\times {\mathbb {R}}^{K+1-r}$. In contrast to $P_\Omega Q_T$ and its ${\mathbb {Z}}_N$-analogues, which have simple eigenvalues, eigenspaces of QP are multi-dimensional. They are identified through the second component of the products defining the invariant subspaces: For each $r=0,\ldots , K-1$ we identify a matrix $M_{(K,r)}^{QP}$ of size $(K+1-r)\times (K+1-r)$ whose eigenvectors $\mathbf {d}=(d_0,\ldots , d_{K- r})^T$ define corresponding eigenspaces of QP with elements $f=\sum _{k=0} ^{K-r} d_k A_+^{k} g$, $g\in {\mathcal {W}}_r$. When $r<K< N/2$ the dimension of each eigenspace is that of ${\mathcal {W}}_r$, namely, $\left( {\begin{array}{c}N\\ r\end{array}}\right) -\left( {\begin{array}{c}N\\ r-1\end{array}}\right)$ (with the convention $\left( {\begin{array}{c}N\\ -1\end{array}}\right) =0$). The vectors in $\mathrm{PW}_K$ that are also the most concentrated in ${\mathbb {B}}_K$ are the inverse Fourier transforms of such f corresponding to the largest eigenvalues of QP. Entries of $M_{(K,r)}^{QP}$ are identified through Theorem 1 and Proposition 5.

Practically, numerical computation of the matrices $M_{(K,r)}^{QP}$ is problematic even for small N—of the order of $N=20$—because the magnitude of its entries can exceed the maximum machine integer. In the classical setting of the real line and corresponding discrete ${\mathbb {T}}\leftrightarrow {\mathbb {Z}}$ [27] and finite ${\mathbb {Z}}_N$ settings (e.g., [6, 7]), eigenfunctions, sequences or vectors of time- and band-limiting operators are obtained by identifying a commuting second order differential or difference operator that facilitates their computation. In [11] we showed that the operator in the Boolean setting defined by analogy with these settings, which we denote by BDO, does not in fact commute with PQ, although it commutes with P for appropriate parameter values, and then, numerically almost commutes with PQ, suggesting that corresponding eigenvectors of PQ and BDO are also close (the commutator of BDO and PQ was expressed in terms of hypergeometric functions in [11], but explicit norm bounds were not provided). As a consequence of Theorem 2, Lemma 3 and Proposition 5, BDO shares with PQ the invariant subspaces just described, which contain respective eigenspaces. Eigenspaces of BDO can be identified through eigenvectors of the tridiagonal matrices of size $(N-2r+1)\times (N-2r+1)$ identified in Theorem 2. These matrices admit efficient eigen-decomposition. The eigenspaces are quantified in Theorem 3. Starting from eigenvectors of $(K+1-r)\times (K+1-r)$ principal minors of these tridiagonal coefficient matrices of BDO as described in Proposition 4, Algorithm 1 describes a method to compute aforementioned eigenvectors $(d_0,\ldots , d_{K- r})^T$ of the corresponding coefficient matrices $M_{(K,r)}^{QP}$ of QP. Corresponding eigenvalues for a specific example (K, r) are given in Table 6. Eigenvalue distributions will be studied further elsewhere.

The remainder of this work is outlined as follows. After some notational remarks in Sect. 1.2, we review other necessary background pertaining to Boolean cubes in Sect. 2. Compositions of the operator $A_+$ and its adjoint $A_-$ are analyzed in Sect. 3. Properties established there are used to study BDO in Sect. 4 and PQ in Sect. 5. The numerical method to obtain eigenvectors of PQ from those of BDO is provided in Sect. 6. Some ancillary results are provided in Appendix A.

1.2 Survey of Notations for Spaces and Operators

We use capital letters to denote operators. We do not distinguish notationally between operators on $\ell ^2({\mathcal {B}}_N$) and their matrices defined in terms of the standard basis $\{\delta _S: S\subset \{1,\ldots , N\}\}$. Entries of a matrix M will be denoted using subscripts, e.g., $M_{RS}$ or $M_{k\ell }$, or ordered pairs, e.g., M(R, S), $M(k,\ell )$, and $M^T$ denotes the transpose of M. Since the domain of the Fourier transform on ${\mathcal {B}}_N$ is an isomorphic copy of its vertices (indexed by the power set ${\mathcal {P}}(\{1,\ldots , N\})$ of $\{1,\ldots , N\}$), we do not distinguish notationally between functions and operators on the primal graph ${\mathcal {B}}_N$ and on its Fourier dual. For example, Q can denote truncation to the K-ball in either the spatial or spectral domain copy of ${\mathcal {P}}(\{1,\ldots , N\})$. All matrices used have real entries and all functions are assumed real-valued. Specific notations used for sets, spaces and operators in multiple sections are listed in Table 1.

Table 1 Notations used

2 Boolean Cubes, Space Limiting and Spectrum Limiting

2.1 Boolean Cubes and Their Fourier Transforms

Here we recall some notation and results from [11]. The Boolean cube ${{\mathcal {B}}}_N$ described in Sect. 1 can be regarded as the Cayley graph of the group ${\mathbb {Z}}_2^N$ with generators $e_i$ having entry 1 in the ith coordinate and zeros in the other $N-1$ coordinates. With component-wise addition modulo one, vertices corresponding to elements of ${\mathbb {Z}}_2^N$ are adjacent precisely when their difference in ${\mathbb {Z}}_2^N$ is equal to $e_i$ for some i. It is common to label vertices of ${\mathcal {B}}_N$ corresponding to elements $\epsilon _S=(\epsilon _1,\ldots ,\epsilon _N)\in \{0,1\}^N$ by $S\subset \{1,\ldots , N\}$ according to $S=\{b_1,\ldots ,b_r\}=\{b_i: \epsilon _{b_i}=1\}$. Here, $\emptyset$ corresponds to the zero element $\mathbf {0}=(0,\ldots , 0)\in {\mathbb {Z}}_2^N$ and $|S|=d_H(\emptyset ,S)$. Hence $R,S\in {{\mathcal {B}}}_N$ are adjacent (written $R\sim S$) precisely when $|R\Delta S|=1$. The path distance $d_H(R,S)=|R\Delta S|$ is equal to the Hamming distance of the bit vectors $\epsilon _R$ and $\epsilon _S$ of R and S. The (unnormalized) graph Laplacian of ${\mathcal {B}}_N$ is the matrix $L=NI_{2^N}-A$ where A is the adjacency matrix and $I_{2^N}$ is the $2^N\times 2^N$ identity matrix. Its entries are

$$\begin{aligned} L_{RS}={\left\{ \begin{array}{ll} N&{}\hbox { if}\ R=S\\ -1&{}\hbox { if}\ R\sim S\\ 0&{}\text { else}\, . \end{array}\right. } \end{aligned}$$

Through matrix multiplication, the Laplacian is thought of as a linear operator on vertex functions, i.e., $L:\ell ^2({\mathcal B}_N)\rightarrow \ell ^2({{\mathcal {B}}}_N)$. Lemma 1 is standard, see e.g., [11] for a combinatorial proof.

Lemma 1

$h_S(R)=2^{-N/2}(-1)^{|R\cap S|}$ is an eigenvector of L with eigenvalue 2|S|.

The family $\{h_S: S\subset {\mathcal {P}}(\{1,\ldots , N\})\}$ forms an orthonormal basis for $\ell ^2({\mathcal {B}}_N)$. The eigenvalue 2K has multiplicity $\left( {\begin{array}{c}N\\ K\end{array}}\right)$. The graph Fourier transform of ${\mathcal {B}}_N$, which is effectively the same as the group Fourier transform of ${\mathbb {Z}}_2^N$, is the linear operator $H:\ell ^2({{\mathcal {B}}}_N)\rightarrow \ell ^2({{\mathcal {B}}}_N)$ given by

$$\begin{aligned} (Hf)(S)=\sum _{R\in {{\mathcal {B}}}_N}h_S(R)f(R). \end{aligned}$$

(2.1)

Since $h_S(R)=h_R(S)$, the Fourier transform is self-adjoint. We use the symbol “H” because, up to normalization and reordering, the Fourier matrix H is the Hadamard matrix defined by the N-fold tensor product of $\bigl ({\begin{matrix} 1 &{} 1 \\ 1&{} -1\end{matrix}}\bigr )$. The vectors $\{2^{N/2} h_S\}$ form a group under pointwise multiplication that is isomorphic to ${\mathbb {Z}}_2^N$. In this way one can regard the Laplacian eigenvectors $h_S$ as defining vertices of a spectral dual graph, also labelled by ${\mathcal {P}}(\{1,\ldots , N\})$, with adjacency defined by the same condition $|R\Delta S|=1$ as for ${\mathcal {B}}_N$, so that this spectral dual graph is isomorphic to ${\mathcal {B}}_N$. We use the same notation A, P, and Q for the adjacency and localization operators on this spectral dual.

2.2 A Boolean Analogue of the Prolate Operator

Before addressing the eigenspace problem for PQ, we will consider the corresponding eigen-problem for a certain second-order difference operator analogue of the so-called prolate differential operator $\mathrm{PDO}(\Omega ,T)$ defined on twice-differentiable functions $f:[-T,T]\rightarrow {{\mathbb {R}}}$ by

$$\begin{aligned} \mathrm{PDO}(\Omega ,T)=\frac{\mathrm{d}}{\mathrm{d}t}(t^2-T^2) \frac{\mathrm{d}}{\mathrm{d}t}+ (\pi \Omega )^2 t^2.\, \end{aligned}$$

(2.2)

$\mathrm{PDO}(\Omega , T)$ commutes with the time-limiting operator $Q_T$ and the bandlimiting operator $P_\Omega$. Up to dilation, its eigenfunctions (and hence of time and band limiting) are the so-called prolate spheroidal wave functions (PSWFs) [10, 28]. The structure of $\mathrm{PDO}(\Omega , T)$ allows for efficient numerical computation of the PSWFs. It is reasonable to seek a parallel route to identify and compute eigenvectors of PQ in the Boolean case.

The identity $(\frac{\mathrm{d}}{\mathrm{d}t} f)\hat{\,}(\xi )= 2\pi \mathrm{i}\xi \hat{f}(\xi )$ means that, up to a complex constant, the differentiation operator may be written as the conjugation ${{\mathcal {F}}}^{-1}M{{\mathcal {F}}}$, where ${{\mathcal {F}}}$ is the Fourier transform and M is the multiplication operator $Mg(\xi )=\xi g(\xi )$. Let X be the $2^N\times 2^N$ diagonal matrix (indexed by the eigenvectors of the matrix L) whose diagonal elements are the square roots of the corresponding eigenvalues of L. The operator $D={H}X{H}$ that conjugates X by H (note that ${H}^{-1}={H}$) can thus be regarded formally, up to normalization, as a Boolean analogue of differentiation while the matrix X can also be regarded as an analogue, again up to normalization, of multiplication by t. The Boolean difference operator BDO defined by

$$\begin{aligned} \mathrm{BDO} = D(\alpha I-X^2) D+\beta X^2\, \end{aligned}$$

(2.3)

then can be regarded as a Boolean analogue of $\mathrm{PDO}(\Omega , T)$ in (2.2) where $\alpha$ and $\beta$ play normalizing roles related to the spatial- and spectral-limiting parameters. Specific values will be assigned to $\alpha ,\beta$ in terms of the truncation parameter K in Sect. 4.

The term difference is intended as an analogy—not literally. The analogy has limitations. For example, the Fourier transform ${\mathcal {F}}$ satisfies (in a distributional sense) $\mathrm{PDO}(2T, \Omega /2)\, {\mathcal {F}}= {\mathcal {F}}\, \mathrm{PDO}(\Omega , T)$. This fact boils down to the commutator between differentiation and multiplication by t equaling the identity, a property that does not hold for the pair X, D on ${\mathcal {B}}_N$. Consequently, unlike the case of the real line, BDO and PQ do not commute [11]. Nevertheless, just as $\mathrm{PDO}(\Omega , T)$ can be expressed in terms of a tri-diagonal matrix in an appropriate basis (of Legendre polynomials, e.g., [10, p. 16]), we will show that on certain adjacency-invariant spaces, the conjugation of BDO by the Fourier matrix is identified by a certain tri-diagonal matrix of size at most $N\times N$ (Theorem 2). Fourier conjugation of PQ on these invariant spaces has a similar matrix representation, a consequence of Theorem 4, but that matrix is full (not tri-diagonal), and subsequent eigenspace analysis of PQ is thus more complicated. In Sect. 6 a method is outlined to compute (numerically) eigenspaces of PQ starting from those of BDO.

2.3 Dyadic Lexicographic Order

In order to represent linear operators on $\ell ^2({\mathcal {B}}_N)$ by matrices one needs to fix an ordering of the elements of ${\mathcal {B}}_N$. The (dyadic) lexicographic ordering “$\prec$” of vertices of ${\mathcal {B}}_N$ stipulates that $R\prec S$ if the following conditions hold:

• (i) $|R|\le |S|$;

• (ii) if $|R|=|S|>0$ then the smallest index of $R\Delta S$ lies in R.

This ordering is consistent with that of the eigenvalues of L (Lemma 1). In this ordering, the adjacency matrix A is a symmetric matrix whose nonzero entries lie within rectangular blocks corresponding to products of Hamming spheres $\Sigma _r\times \Sigma _{r\pm 1}$. This ordering is used for the matrices and vectors displayed in Figs. 1, 2, and 3.

Fig. 1

Adjacency matrix for $N=8$ in lexicographic order

Fig. 2

Sample columns of the matrix of the projection from $\ell ^2(\Sigma _r)$ onto ${\mathcal {W}}_r$, $N=8$, $r=3$

Fig. 3

Eigenvectors of PQ, $(N,K,r)=(8,3,2)$. The dotted curves are plots of two different elements g of ${\mathcal {W}}_r$ obtained by projecting a delta in $\Sigma _r$ onto ${\mathcal {W}}_r$. The dashed curves are vertical shifts of eigenvectors f of QP of the form $\sum c_k A_+^k W$ where $c_k$ form the principal eigenvector of $M_{(K,r)}^{\mathrm{QP}}$. The solid curves are shifted multiples of Hf where H is the Fourier matrix. They are the corresponding eigenvectors of PQ

3 Hamming Sphere Analysis of Adjacency

The adjacency matrix A of ${\mathcal {B}}_N$ can be decomposed as $A=A_{+}+A_{-}$ where $A_-=A_+^T$ (the transpose of $A_+$) and $A_+$ is lower triangular when expressed in lexicographic order. The action of $A_+$ on a vertex function f can be expressed as $(A_+ f)(S)=\sum _{R\prec S, R\sim S} f(R)$. If f is supported in $\Sigma _N=\{\{1,\ldots , N\}\}$ then $A_+f=0$; similarly, $A_-f=0$ if f is supported in $\Sigma _0=\{\emptyset \}$. For $r\in \{0,1,\ldots ,N\}$, let $\ell ^2(\Sigma _r)$ be the subspace of $\ell ^2({{\mathcal {B}}}_N)$ consisting of vertex functions supported on the r-sphere $\Sigma _r$. If $r\in \{0,1,\ldots , N-1\}$ then $A_+$ maps $\ell ^2(\Sigma _r)$ to $\ell ^2(\Sigma _{r+1})$ while if $r\in \{1,2,\ldots ,N\}$ then $A_-$ maps $\ell ^2(\Sigma _r)$ to $\ell ^2(\Sigma _{r-1})$. We refer to $A_+$ and $A_-$ as the respective outer and inner adjacency maps.

The space $\ell ^2(\Sigma _r)$ ($r\in \{1,\ldots ,\lfloor N/2\rfloor \}$) has an orthogonal decomposition

$$\begin{aligned} \ell ^2(\Sigma _r)=A_+ \ell ^2(\Sigma _{r-1})\oplus {\mathcal {W}}_r \end{aligned}$$

(3.1)

where ${\mathcal {W}}_r$ is the orthogonal complement of $A_+ \ell ^2(\Sigma _{r-1})$ inside $\ell ^2(\Sigma _r)$. Equivalently,

$$\begin{aligned} {{\mathcal {W}}}_r=\text {null}(A_-)\Big |_{\ell ^2(\Sigma _{r})}=\{g\in \ell ^2(\Sigma _{r}):\, A_-g=0\}. \end{aligned}$$

Applying (3.1) on each sphere, one obtains a multi-level decomposition:

$$\begin{aligned} \ell ^2(\Sigma _r)=A_+ \ell ^2(\Sigma _{r-1})\oplus {\mathcal {W}}_r=\mathop {\oplus }\limits _{j=0}^rA_+^{r-j}{{\mathcal {W}}}_j \, . \end{aligned}$$

(3.2)

Since $\Sigma _0= \{\emptyset \}$ is a singleton, ${\mathcal {W}}_0$ is isomorphic to ${\mathbb {R}}$.

The outer and inner adjacencies do not commute. The following theorem quantifies composition of $A_-$ with a power of $A_+$ on ${{\mathcal {W}}}_r$.

Theorem 1

Let $r,k\in \{0,1,\ldots ,N\}$ with $r<N/2$ and $k+r<N$, and $g\in {\mathcal {W}}_r$. Then

$$\begin{aligned} A_-A_+^{k+1}g=m(r,k)A_+^kg, \end{aligned}$$

(3.3)

where the multiplier m(r, k) of $A_+^kg$ in (3.3) is given by

$$\begin{aligned} m(r,k)=\max \{(k+1)(N-2r-k),0\} . \end{aligned}$$

(3.4)

The proof of the case $k+2r<N$ uses the following lemma. Setting $m(r,k)=0$ in (3.4) for $k>N-2r$ is justified by Proposition 1 below.

Lemma 2

Let $C=[A_-,A_+]=A_-A_+-A_+A_-$ be the commutator of $A_-$ and $A_+$. Then for each r, $1\le r<N$, the restriction of C to $\ell ^2(\Sigma _r)$ is multiplication by $N-2r$.

Proof of Lemma 2

For $R\in \Sigma _r$ and $f\in \ell ^2(\Sigma _r)$, the value $(A_-A_+ f)(R)$ is the sum of the values f(S), $S\in \Sigma _r$ such that there is a two-edge path from S to R that passes through a vertex in $\Sigma _{r+1}$. This sum can be expressed as

$$\begin{aligned} (A_-A_+ f)(R)=(N-r) \, f(R)+\sum _{{b'}\notin R,\, b\in R} f(R\cup \{{b'}\}\setminus \{b\}), \end{aligned}$$

where the first term counts the number of elements not in R, corresponding to paths from R to a vertex in $\Sigma _{r+1}$ and back, and the second counts all other paths. Those latter paths necessarily have the form $S=R\cup \{{b'}\}\setminus \{b\}\mapsto R\cup \{{b'}\}\mapsto R$ where S is a vertex in $\Sigma _r$ differing from R by a one-element substitution. Adding and subtracting rf(R) from the two terms on the right, one obtains

$$\begin{aligned} (A_-A_+ f)(R)=(N-2r) \, f(R)+\sum _{b\in R}\Bigl [f(R)+\sum _{{b'}\notin R} f(R\cup \{{b'}\}\setminus \{b\})\Bigr ]. \end{aligned}$$

(3.5)

On the other hand, the value $(A_+A_- f)(R)$ is the sum of the values f(S), $S\in \Sigma _r$ such that there is a two-edge path from R to S that passes through a vertex in $\Sigma _{r-1}$. Any such path either has the form $R\mapsto R\setminus \{b\}\mapsto R$ (there are $r=|R|$ of these), or $R\mapsto R\setminus \{b\}\mapsto R\cup \{{b'}\}\setminus \{b\}$ where ${b'}\notin R$. This sum can be expressed as

$$\begin{aligned} (A_+A_- f)(R)=r \, f(R)+\sum _{{b'}\notin R,\, b\in R} f((R\setminus \{b\})\cup \{{b'}\}) \end{aligned}$$

where the first term counts the number of paths from R to $\Sigma _{r-1}$ and back to R and the second counts the number of paths that first delete an element of R then add an element not in R. Rearranging as above one can write

$$\begin{aligned} (A_+A_- f)(R)=\sum _{b\in R} \Bigl [f(R)+\sum _{{b'}\notin R} f(R\cup \{{b'}\}\setminus \{b\})\Bigr ] \, . \end{aligned}$$

(3.6)

Subtracting (3.6) from (3.5) gives

$$\begin{aligned} (Cf)(R)=(A_-A_+ f)(R)-(A_+A_- f)(R)=(N-2r)f(R)\,. \end{aligned}$$

This proves the result. $\square$

Proof of Theorem 1

The proof of Theorem 1 can be completed now by induction on k. The base case $k=0$ follows from Lemma 2: if $g\in {\mathcal {W}}_r$ then $A_-g=0$ so $A_-A_+g=Cg=(N-2r) g$ which agrees with (3.3) for the case $k=0$. Suppose that $A_-A_+^k=m(r,k-1) A_+^{k-1}$ has been established on ${\mathcal {W}}_r$. We have, for $g\in {\mathcal {W}}_r$,

$$\begin{aligned} A_-A_+^{k+1}g&=(A_-A_+ A_+^{k} -A_+A_- A_+^{k}+A_+A_- A_+^{k})g\\&=CA_+^{k}g+A_+A_- A_+^{k}g\\&=(N-2(k+r))A_+^k g+m(r,k-1)A_+^kg=m(r,k) A_+^k g, \end{aligned}$$

where we used Lemma 2, the inductive hypothesis, and the fact that $m(r,k-1)+(N-2(r+k))=m(r,k)$. This proves the result. $\square$

Remark

The theorem implies that $A_+^{k}$ is injective from ${\mathcal {W}}_r$ to $\ell ^2(\Sigma _{r+k})$ when $k+2r<N$. When $k=N-2r$ one has $m(r,k)=0$, that is $A_-A_+^{N-2r+1} g=0$ if $g\in {\mathcal {W}}_r$, $r<N/2$. The following refinement will be used in what follows. It is proved in Appendix A.3.

Proposition 1

If $g\in {\mathcal {W}}_r$, $0<r<N/2$, then $A_+^{N-2r+1}g=0$.

As a simple consequence of Theorem 1 we have the following result.

Corollary 1

Let $g_r\in {{\mathcal {W}}}_r$, $0\le r<N/2$ and $\ell \le N-2r$. Then

$$\begin{aligned} A_-^\ell A_+^\ell g_r =n(r,\ell )g_r \end{aligned}$$

where, for $1\le \ell \le N-2r$,

$$\begin{aligned} n(r,\ell )={\left\{ \begin{array}{ll} 1&{}\hbox { if}\ \ell =0\\ \prod _{j=0}^{\ell -1} m(r,\ell )=(\ell !)^2 \left( {\begin{array}{c}N-2r\\ \ell \end{array}}\right)&\text { else.} \end{array}\right. } \end{aligned}$$

(3.7)

Figure 2 plots projections of several vectors of the form $\delta _S$, $S\in \Sigma _r$, onto ${\mathcal {W}}_r$ in the case $N=8$ and $r=3$. The general method to express $f\in \ell ^2(\Sigma _r)$ as a sum of projections onto ${\mathcal {W}}_s$, $0\le s\le r$, is presented in Appendix A.1. The columns of the matrix of the orthogonal projection from $\ell ^2(\Sigma _r)$ onto ${\mathcal {W}}_r$ form a Parseval frame (defined e.g., in [9]) for ${\mathcal {W}}_r$. The norm of each column is $1-\left( {\begin{array}{c}N\\ r-1\end{array}}\right) /\left( {\begin{array}{c}N\\ r\end{array}}\right)$ and the inner product of columns R and S of this projection depends only on $|R\cap S|$.

4 Eigenspaces of Boolean Difference Operators

For $r\in \{0,1,\ldots ,\lfloor (N-1)/2\rfloor \}$ let ${\mathcal {V}}_r$ be the collection of vertex functions $f\in \ell ^2({{\mathcal {B}}}_N)$ of the form

$$\begin{aligned} f=\sum _{k=0}^{N-2r} c_k A_+^k g_r \end{aligned}$$

(4.1)

for some $g_r\in {{\mathcal {W}}}_r$. Since $A_+$ maps $\ell ^2(\Sigma _s)$ to $\ell ^2(\Sigma _{s+1})$, we see that $A_+^kg_r$ is supported on $\Sigma _{k+r}$ when $2r+k\le N$. As a consequence, the restriction of a vertex function f of the form (4.1) to $\Sigma _\rho$ is a fixed multiple of $A_+^{\rho -r}g_r$ ($\rho \ge r$), i.e., if $R\in \Sigma _\rho$ and f is the vertex function defined as in (4.1) then $f(R)=c_{\rho -r}(A_+^{\rho -r}g_r)(R)$. The following proposition, proved in Appendix A.2, will be used later.

Proposition 2

Let $0\le r<s<N/2$. The spaces ${\mathcal {V}}_r$ and ${\mathcal {V}}_s$ are mutually orthogonal.

Theorem 2 below describes the action of the conjugation of the Boolean difference operator BDO in (2.3) by the Fourier matrix H on ${{\mathcal {V}}}_r$. Define B to be the conjugation of BDO by H, that is $B={H}\text {BDO}{H}\,$. Since $D={H}X{H}$, B can be written as

$$\begin{aligned} B={H}\,\mathrm{BDO}\,{H}=X (\alpha -L)X+\beta L \end{aligned}$$

(4.2)

where, as before, X is the diagonal matrix with entries $X_{SS}=\sqrt{2|S|}$. Consider now the action of B on a vector $f=\sum _{k=r}^N f_k$ in ${\mathcal {V}}_r$ where $f_k$ is the restriction of f to $\Sigma _k$. Since $L=NI-A$, B maps $f_k$ to $\ell ^2(\Sigma _{k-1})\oplus \ell ^2(\Sigma _{k})\oplus \ell ^2(\Sigma _{k+1})$ with $(Lf_k)_k=Nf_k$, $(Lf_k)_{k-1}=-A_-f_k$ and $(Lf_k)_{k+1}=-A_+f_k$. One has

$$\begin{aligned} (Bf_k)_k= (X (\alpha -N I)X+\beta N I) f_k =[2k(\alpha -N)+\beta N] f_k\, . \end{aligned}$$

(4.3)

The restriction of $Bf_k$ to other spheres is given by

$$\begin{aligned} (X AX-\beta A)f_k=(X A_+X-\beta A_+)f_k+(X A_-X-\beta A_-)f_k\in \ell ^2(\Sigma _{k+1})\oplus \ell ^2(\Sigma _{k-1}). \end{aligned}$$

In fact, by considering the action of X, we have

$$\begin{aligned} (Bf_k)_{k+1}&=(XA_+X-\beta A_+)f_k=(2\sqrt{k(k+1)}-\beta )A_+f_k; \end{aligned}$$

(4.4)

$$\begin{aligned} (Bf_k)_{k-1}&=(XA_-X-\beta A_-)f_k=(2\sqrt{k(k-1)}-\beta )A_-f_k. \end{aligned}$$

(4.5)

As a consequence of these calculations, we can describe the action of B on ${{\mathcal {V}}}_r$.

Theorem 2

Let $0\le r<N/2$. Let $f=\sum _{k=0}^{N-2r}c_kA_+^kg_r$ for some $g_r\in {{\mathcal {W}}}_r$. Then

$$\begin{aligned} Bf=\sum _{k=0}^{N-2r}(M_{(r)}^{B}{{\mathbf {c}}})_kA_+^kg_r \end{aligned}$$

with ${{\mathbf {c}}}=(c_0,c_1,\ldots ,c_{N-2r})^T\in {\mathbb R}^{N-2r+1}$ and $M_{(r)}^{B}\in {{\mathbb {R}}}^{(N-2r+1)\times (N-2r+1)}$ has entries

$$\begin{aligned} M_{(r)}^{B}(k,\ell )={\left\{ \begin{array}{ll} (2\sqrt{(r+k)(r+k+1)}-\beta )m(r,k)&{}\hbox { if}\ \ell =k+1\\ 2(r+k)(\alpha -N)+\beta N&{}\hbox { if}\ \ell =k\\ (2\sqrt{(r+k)(r+k-1)}-\beta )&{}\hbox { if}\ \ell =k-1\\ 0&{}\text { else} \end{array}\right. } \end{aligned}$$

for $0\le k,\ell \le N-2r$, where $\alpha ,\beta$ are as in (2.3) and (4.2).

Proof

Let f be as in the statement of the theorem. Then by (4.3), (4.4) and (4.5) we have

$$\begin{aligned} B f&=\sum _{k=0}^{N-2r}c_k[X(\alpha -L)X+\beta L]A_+^kg_r\\&=\sum _{k=0}^{N-2r}c_k[(X(\alpha -NI)X+\beta NI) +(XA_+X-\beta A_+)+(XA_-X-\beta A_-)]A_+^kg_r\\&=\sum _{k=0}^{N-2r}c_k[(2(r+k)(\alpha -N)+\beta N)A_+^kg_r\\&\quad +(\sqrt{2(r+k+1)(r+k)}-\beta )A_+^{k+1}g_r+(2\sqrt{(r+k-1)(r+k)}-\beta )A_-A_+^kg_r]\\&=\sum _{k=0}^{N-2r}c_k[(2(r+k)(\alpha -N)+\beta N)A_+^kg_r\\&\quad +(\sqrt{2(r+k+1)(r+k)}-\beta )A_+^{k+1}g_r+(2\sqrt{r+k-1)(r+k)}-\beta )m(r,k-1)A_+^kg_r]\\&=\sum _{k=0}^{N-2r}\Bigl \{[2(r+k)(\alpha -N)+\beta N] c_k\Bigr .\\&\quad \Bigl . +[2\sqrt{(r+k)(r+k-1)}-\beta ]c_{k-1} +[2\sqrt{(r+k)(r+k+1)}-\beta ]m(r,k) c_{k+1}\Bigr \}A_+^kg_r\\&=\bigg (\sum _{\ell =0}^{N-2r}M_{(r)}^{B}({k,\ell })c_\ell \bigg )A_+^kg_r \end{aligned}$$

where in the second last line we have applied Theorem 1, and $M_{(r)}^{B}({k,\ell })$ is as in the statement of the theorem. $\square$

We refer to the matrix $M_{(r)}^B$ as the coefficient matrix of B on ${\mathcal {V}}_r$. To relate the eigenfunctions of B with the eigenvectors of $M_{(r)}^{B}$, we first show that $M_{(r)}^{B}$ has a basis of real eigenvectors. Given a positive integer n and positive constants $\nu _0,\ldots ,\nu _{n-1}$, let $W\in {\mathbb R}^{n\times n}$ be the diagonal matrix with diagonal entries $W_{ii}=\nu _i>0$ and suppose $M\in {{\mathbb {R}}}^{n\times n}$ satisfies

$$\begin{aligned} M^T=W MW^{-1}. \end{aligned}$$

(4.6)

Let ${{\mathcal {H}}}={{\mathbb {R}}}^n$ be equipped with the weighted inner product $\langle \mathbf {v},\mathbf {w}\rangle _W :=\langle \mathbf {v},W \mathbf {w}\rangle$, where $\langle \cdot ,\cdot \rangle$ is the standard dot product on ${{\mathbb {R}}}^n$, i.e.,

$$\begin{aligned} \langle \mathbf {v},\mathbf {w}\rangle _{W}=\sum _{j=0}^{n-1} v_j w_j \nu _j\, . \end{aligned}$$

Then

$$\begin{aligned} \langle M\mathbf {v}, \mathbf {w}\rangle _W=\langle M\mathbf {v}, W \mathbf {w}\rangle =\langle \mathbf {v},M^T W \mathbf {w}\rangle =\langle \mathbf {v}, W M\mathbf {w}\rangle =\langle \mathbf {v}, M\mathbf {w}\rangle _W\, . \end{aligned}$$

Thus, M is a self-adjoint operator on ${{\mathcal {H}}}$, and it has a basis of real eigenvectors that are orthonormal with respect to the inner product $\langle \cdot ,\cdot \rangle _W$.

Proposition 3

Let $M_{(r)}^{B}$ be defined as in Theorem 2 and let W be the diagonal matrix with $W_{kk}=n(r,k)$ as defined in (3.7). Then $(M_{(r)}^{B})^T=W M_{(r)}^{B}W^{-1}$.

The proposition follows by direct calculation. We conclude that there is a basis $\{{{\mathbf {c}}}^{(0)},{{\mathbf {c}}}^{(1)},\ldots ,{{\mathbf {c}}}^{(N-r)}\}$ for ${{\mathbb {R}}}^{N-2r+1}$ consisting of eigenvectors of $M_{(r)}^{B}$ that is orthonormal with respect to the inner product (with n(r, k) as in (3.7))

$$\begin{aligned} \langle {{\mathbf {v}}},{{\mathbf {w}}}\rangle _W =\sum _{k=0}^{N-2r}n(r,k)v_kw_k \, . \end{aligned}$$

(4.7)

Theorem 3

Let $0\le r<N/2$.

• (i) A vertex function $f=\sum _{k=0}^{N-2r}c_kA_+^kg_r\in {{\mathcal {V}}}_r$ is an eigenfunction of B if and only if ${{\mathbf {c}}}=(c_0,c_1,\ldots ,c_{N-2r})^T$ is an eigenvector of $M_{(r)}^{B}$ with the same eigenvalue.

• (ii) Let $\{{{\mathbf {c}}}^{(i)}\}_{i=0}^{N-2r}$ be a basis for ${{\mathbb {R}}}^{N-2r+1}$ consisting of eigenvectors of $M_{(r)}^{B}$ that are orthogonal with respect to the inner product (4.7) and $\{g_r^{(j)}\}_{j=1}^{\mathrm{dim}({\mathcal {W}}_r)}$ be an orthonormal basis for ${{\mathcal {W}}}_r$. For each $0\le i\le N-r$ and $1\le j\le \mathrm{dim}({\mathcal {W}}_r)$, let

  $$\begin{aligned} f^{(i,j)}=\sum _{k=0}^{N-2r}{c}^{(i)}_kA_+^kg_r^{(j)}. \end{aligned}$$

  Then the collection ${{\mathcal {F}}}_r^{B}=\{f^{(i,j)}:\, 0\le i\le N-2r+1,\ 1\le j\le \mathrm{dim}({\mathcal {W}}_r)\}$ forms an orthonormal basis for ${{\mathcal {V}}}_r$.

Proof

The first part of the theorem is a direct consequence of Theorem 2. To prove the second part, let ${{\mathcal {F}}}_r^{B}$ be as in Theorem 3 (ii). Then

$$\begin{aligned} B f_j^{(i,j)}&=\sum _{k=0}^{N-2r}{c}_k^{(i)}B A_+^kg_r^{(j)}=\sum _{k=0}^{N-2r}(M_{(r)}^{B}{{\mathbf {c}}}^{(i)})_kA_+^kg_r^{(j)}\\&=\sum _{k=0}^{N-2r}\lambda ^{(i)}{c}_k^{(i)}A_+^kg_r^{(j)}=\lambda ^{(i)}f_r^{(i,j)} \end{aligned}$$

so that ${{\mathcal {F}}}_r^{B}$ consists of eigenfunctions of B. Also,

$$\begin{aligned} \langle f_r^{(i,j)},f_r^{(i',j')}\rangle =\left\langle \sum _{k=0}^{N-2r}{c}_k^{(i)}A_+^kg_r^{(j)},\sum _{\ell =0}^{N-2r}{c}_\ell ^{(i')}A_+^\ell g_r^{(j')}\right\rangle =\sum _{k=0}^{N-2r}\sum _{\ell =0}^{N-2r}{c}_k^{(i)}{c}_\ell ^{(i')}\langle A_+^kg_r^{(j)},A_+^\ell g_r^{(j')}\rangle . \end{aligned}$$

But $A_+^k g_r^{(j)}$ is supported on $\Sigma _{r+k}$, so $\langle A_+^kg_r^{(j)},A_+^\ell g_r^{(j')}\rangle =0$ when $k\ne \ell$, hence with $n(r,\ell )$ as in (3.7),

$$\begin{aligned} \langle f_r^{(i,j)},f_r^{(i',j')}\rangle&=\sum _{k=0}^{N-2r}{c}_k^{(i)}{c}_k^{(i')}\langle A_+^kg_r^{(j)},A_+^kg_r^{(j')}\rangle \\&=\sum _{k=0}^{N-2r}{c}_k^{(i)}{c}_k^{(i')}\langle g_r^{(j)},A_-^kA_+^kg_r^{(j')}\rangle =\sum _{k=0}^{N-2r}{c}_k^{(i)}{c}_k^{(i')}n(r,k)\delta _{j,j'}=\delta _{i,i'}\delta _{j,j'}. \end{aligned}$$

Since $|{{\mathcal {F}}}_r^{B}|=(N-2r+1)\, \mathrm{dim}({\mathcal {W}}_r)=\text {dim}({{\mathcal {V}}}_r)$, we conclude that ${{\mathcal {F}}}_r^{B}$ is an orthonormal basis for ${{\mathcal {V}}}_r$. $\square$

Theorem 3 applies to B defined in (4.2) without regard to the parameters $\alpha ,\beta$, which can be connected to spectrum-limiting properties. From this point onwards, we assume that $\alpha =\beta =2\sqrt{K(K+1)}$ in the definition of the operator B. In [11, Proposition 1] it was shown that in this case, the spectrum-limiting operator P commutes with BDO. Since conjugation of P by H is Q, this means that B commutes with Q. In this case, one can define the reduced space ${\mathcal {V}}_{r,K}$ of vectors of the form $\sum _{k=0}^{K-r} c_k A_+^{k}g_r$, $g_r\in {\mathcal {W}}_r$. Table 2 lists the entries of $M_{(r)}^{B}$ when $(N,K,r)=(8,3,2)$.

Proposition 4

For each N and $r\le K<N/2$, with $\alpha =\beta =\sqrt{K(K+1)}$ in (4.2), the principal minor $M_{(K,r)}^{B}$ of size $(K+1-r)\times (K+1-r)$ of $M_{(r)}^{B}$ has eigenvalues equal to those of the conjugation of B by Q applied to the space ${\mathcal {V}}_{r,K}$.

Table 2 Matrix $M^{B}_{(2)}$ of B on ${\mathcal {V}}_2$, $N=8$, $K=3$

5 Eigenspaces of PQ

5.1 Spectrum Limiting as a Function of Adjacency

As before we assume that Q is multiplication by the characteristic function of the closed Hamming K-ball centered at the origin. Then ${H}Q{H}=P$ (see Sect. 1.1) and, since $H^2=I$, conjugation of PQ by H leaves the operator QP. That is, $HPQH=H(HQH)QH=QHQH=QP$. We study the eigenspaces of QP.

Lemma 3

For $0\le r<N/2$, $A_+$ and $A_-$ map ${\mathcal {V}}_r$ to itself.

Proof

For $g\in {\mathcal {W}}_r$, $A_+$ maps $\sum _{k=0}^{N-2r} c_k A_+^{k} g$ to $\sum _{k=0}^{N-2r-1} c_k A_+^{k+1} g = \sum _{k=1}^{N-2r} c_{k-1} A_+^{k} g$ which is an element of ${\mathcal {V}}_r$ whose coefficient of $A_+^0 g$ is zero, while $A_-$ maps $\sum _{k=0}^{N-2r} c_k A_+^{k} g$ to $\sum _{k=1}^{N-2r} c_k m(r,k-1) A_+^{k-1} g = \sum _{k=0}^{N-2r-1} c_{k+1} m(r,k) A_+^{k} g$ which is an element of ${\mathcal {V}}_r$ whose coefficient of $A_+^{N-2r}g$ is zero. $\square$

The proof of the lemma shows that the action of $A_+$ on ${\mathcal {V}}_r$ can be represented by the coefficient matrix $M_{(r)}^{A_+}$ that maps $(c_0,\ldots , c_{N-2r})^T$ to $(0, c_0,\ldots , c_{N-2r-1})^T$ while the action of $A_-$ on ${\mathcal {V}}_r$ can be represented by the matrix $M_{(r)}^{A_-}$ that maps $(c_0,\ldots , c_{N-2r})^T$ to $(m(r,0) c_1,\ldots , m(r,N-2r-1) c_{N-2r},0)^T$. Specifically,

$$\begin{aligned} M_{(r)}^{A_+}(k,\ell )={\left\{ \begin{array}{ll} 1&{}\hbox { if}\ \ell =k-1\\ 0&{}\text { else} \end{array}\right. };\qquad M_{(r)}^{A_-}(k,\ell )={\left\{ \begin{array}{ll} m(r,k)&{}\hbox { if}\ \ell =k+1\\ 0&{}\text { else}\, . \end{array}\right. } \end{aligned}$$

One can express A on ${\mathcal {V}}_r$ by $M_{(r)}^A=M_{(r)}^{A_+}+M_{(r)}^{A_-}$. Lemma 3 implies that $A=A_++A_-$ preserves ${\mathcal {V}}_r$ and so does any polynomial p(A) in A.

Proposition 5

The spectrum-limiting operator P can be expressed as a polynomial p(A) of degree N.

Proof

The proposition is actually a special case of a general fact about graph filters. We include the proof in order to provide an explicit expression for p(A). By Lemma 1 and since $A=NI-L$, one has $Ah_R=(N-2|R|)h_R$. One can then express P in terms of A by forming the Lagrange interpolating polynomial that maps the eigenvalues of A to those of P (which are one or zero). The Lagrange interpolating polynomial that maps $x_k$ to $y_k$, $k=0,\ldots , N$ is $\sum p_k$ where $p_k(x)=y_k\prod _{j=0,\,j\ne k}^N\frac{x-x_j}{x_k-x_j}$. We choose p so that $p(A) h_R=h_R$ if $|R|\le K$ and $p(A)h_R=0$ if $|R| >K$. Since $Ah_R=(N-2|R|) h_R$ this means one should have $p(N-2r)=1$ if $0\le r\le K$ and $p(N-2r)=0$ if $r>K$. Therefore, set

$$\begin{aligned} p_k(x)=\prod _{j=0,j\ne k}^N \frac{x-(N-2j)}{2(j-k)};\qquad p(x)=\sum _{k=0}^K p_k(x)\, . \end{aligned}$$

(5.1)

Then $P=p(A)$ as verified on the basis $\{h_S\}$ of $\ell ^2({\mathcal {B}}_N)$. This proves the proposition. $\square$

As a consequence of the proposition, the space ${\mathcal {V}}_r$ is invariant under P. Just as with the operators A and B, the action of P on ${\mathcal {V}}_r$ can be quantified by a coefficient matrix $M^P_{(r)}$ of size $(N-2r+1)\times (N-2r+1)$ such that if $g\in {\mathcal {W}}_r$ then $P (A_+^k g)=\sum _{\ell =0}^{N-2r} M^P_{(r)}(k,\ell ) A_+^\ell g$ in which $M^P_{(r)}(k,\ell )$ is independent of g. The entries of $M^P_{(r)}$ are determined by the polynomial p(A) defining P as follows. Since applying $A^n$ is the same as iterating A n-times, which on ${\mathcal {V}}_r$ is the same as iterating the coefficient matrix $M_{(r)}^A$ n-times, the coefficient matrix of $A^n$ on ${\mathcal {V}}_r$ is $M_{(r)}^{A^n}=(M_{(r)}^A)^n$, $n=0,1,2,\ldots$. Similarly, one can replace each occurrence of $A-(N-2j)I_{2^N}$ as a factor in (5.1) evaluated at A by $M_{(r)}^A-(N-2j)I_{N-2r+1}$ to obtain the matrix $M^P_{(r)}$. If $(d_0,\ldots , d_{N-2r})^T$ is an eigenvector of $M^P_{(r)}$ and $g\in {\mathcal {W}}_r$ then $f=\sum _{k=0}^{N-2r} d_k A_+^k g$ is an eigenvector of P. The following proposition establishes that $M^P_{(r)}$ has a basis of real eigenvectors orthogonal with respect to the inner product (4.7).

Proposition 6

For $0\le r<N/2$, the matrices $M_{(r)}^A$ and $M^P_{(r)}$ of the respective adjacency operator and spectrum-limiting operator on ${\mathcal {V}}_r$ as defined above satisfy $(M_{(r)}^{A})^T=W M_{(r)}^{A}W^{-1}$ and $(M^P_{(r)})^T=W M^P_{(r)}W^{-1}$ where the matrix W is as in Proposition 3.

The case of $M_{(r)}^A$ follows by direct calculation. The case of $M^P_{(r)}$ follows from the fact that the transpose of a polynomial of a square matrix is the polynomial evaluated on the transpose of the matrix.

5.2 Bases of Eigenvectors of QP in ${\mathcal {V}}_r$

We define a matrix $M^{QP}_{(K,r)}$ to be the $(K-r+1)$-principal minor of the matrix $M^P_{(K,r)}$ described in the preamble to Proposition 6. The eigenvectors of PQ have the form Hf where f is an eigenvector of QP (as before, $P=HQH$). Eigenvectors of QP in ${\mathcal {V}}_{r,K}$ can be obtained from those of the matrix $M^{QP}_{(K,r)}$. This is because if $f\in {\mathcal {V}}_{r,K} =Q{\mathcal {V}}_r$, $f=\sum _{k=0}^{K-r} c_k A_+^k g$, $g\in {\mathcal {W}} _r$ then $P f=\sum _{k=0}^{N-2r} d_k A_+^k g$ where $d_k=\sum _{\ell =0}^{K-r} M^P_{(r)} (k,\ell ) c_\ell$ and, finally, $QP f=\sum _{k=0}^{K-r} d_k A_+^k g$ for the same coefficients $d_k$. That, is, the coefficients of the powers of $A_+^k g$ that appear in QPf are obtained by applying the principal minor of $M^P_{(r)}$ to the vector of coefficients of $A_+^\ell g$ for $\ell \le K-r$. These observations, coupled with exactly the same arguments used in the proof of Theorem 3, give us the following.

Theorem 4

Let $0\le r\le K< N/2$.

• (i) A vertex function $f=\sum _{k=0}^{K-r}d_kA_+^kg_r\in {{\mathcal {V}}}_{r,K}$ is an eigenfunction of QP if and only if ${{\mathbf {d}}}=(d_0,d_1,\ldots ,d_{K-r})^T$ is an eigenvector of the matrix $M^{QP}_{(K,r)}$ defined above, with the same eigenvalue.

• (ii) Let $\{{{\mathbf {d}}}^{(i)}\}_{i=0}^{K-r}$ be a basis for ${{\mathbb {R}}}^{K-r+1}$ consisting of eigenvectors of $M^{QP}_{(K,r)}$ that are orthogonal with respect to the inner product (4.7) (for $k=0,\ldots , K-r$) and $\{g_r^{(j)}\}_{j=1}^{\mathrm{dim}({\mathcal {W}}_r)}$ be an orthonormal basis for ${{\mathcal {W}}}_r$. For each $0\le i\le K-r$ and $1\le j\le \mathrm{dim}({\mathcal {W}}_r)$, let

  $$\begin{aligned} f^{(i,j)}=\sum _{k=0}^{K-r}{d}^{(i)}_kA_+^kg_r^{(j)}. \end{aligned}$$

  Then the collection ${{\mathcal {F}}}_r^{\mathrm{QP}}=\{f^{(i,j)}:\, 0\le i\le K-r,\ 1\le j\le \mathrm{dim}({\mathcal {W}}_r)\}$ forms an orthonormal basis for ${\mathcal {V}}_{r,K}=Q{{\mathcal {V}}}_r$.

• (iii) The union $\bigcup _{r=0}^K {{\mathcal {F}}}_r^{\mathrm{QP}}$ provides an orthonormal basis for $\ell ^2({\mathbb {B}}_K)$—the vertex functions supported in the ball ${\mathbb {B}}_K$ of vertices of Hamming distance at most K from the zero vertex—and their Fourier transforms form an orthonormal basis for $\mathrm{PW}_K$.

Proof

The proofs of (i) and (ii) follow the same lines as the corresponding parts of Theorem 3. To prove (iii) we observe that $\mathrm{dim}({\mathcal {W}}_r)=\left( {\begin{array}{c}N\\ r\end{array}}\right) -\left( {\begin{array}{c}N\\ r-1\end{array}}\right)$ and $|{\mathcal F}_r^{\mathrm{QP}}|=(K+1-r)\, \mathrm{dim}({\mathcal {W}}_r)$ ($1\le r\le K$) so that, summing by parts (again, $\left( {\begin{array}{c}N\\ -1\end{array}}\right) =0$),

$$\begin{aligned} \Bigl |\bigcup _{r=0}^K {{\mathcal {F}}}_r^{\mathrm{QP}}\Bigr |= \sum _{r=0}^K (K+1-r) \left( \left( {\begin{array}{c}N\\ r\end{array}}\right) -\left( {\begin{array}{c}N\\ r-1\end{array}}\right) \right) =\sum _{r=0}^K \left( {\begin{array}{c}N\\ r\end{array}}\right) =\bigl |{\mathbb {B}}_K\bigr |\, \end{aligned}$$

which is the dimension of $\mathrm{PW}_K$. Orthogonality of elements of ${{\mathcal {F}}}_r^{\mathrm{QP}}$ for different r follows from Proposition 2. The Fourier transforms of the elements of ${{\mathcal {F}}}_r^{\mathrm{QP}}$ lie in $\mathrm{PW}_K$, so (iii) follows from the orthogonality of H. $\square$

Computing the eigenspaces of PQ then boils down to computing the eigenvectors of the principal minors of the matrices $M^P_{(r)}$. We summarize how to compute eigenvectors and eigenvalues of the operator QP on ${\mathcal {V}}_{r,K}$, assuming that the truncation of $M^P_{(r)}$ to its size $(K+1-r)\times (K+1-r)$ principal minor has nondegenerate eigenvalues, in Sect. 6. Each ${\mathcal {V}}_{r,K}$-eigenspace of QP has dimension $\left( {\begin{array}{c}N\\ r\end{array}}\right) -\left( {\begin{array}{c}N\\ r-1\end{array}}\right)$, and there are $K+1-r$ of them. Once the eigenspaces of QP lying in ${\mathcal {V}}_{r,K}$ are identified for each r, a complete eigenspace decomposition of PQ is obtained by conjugating with the matrix H.

Decimal approximations of the entries of the matrices $M^P_{(r)}$ are given in Tables 3, 4 and 5 for $N=8$, $K=3$, and $r=1, 2, 3$. Sample eigenvectors of QP and corresponding eigenvectors of PQ, starting from different elements of ${\mathcal {W}}_2$, are plotted in Fig. 3 for $(N,K,r)=(8,3,2)$.

Table 3 Matrix $M^P_{(1)}$ of P on ${\mathcal {V}}_1$, $N=8$, $K=3$

Table 4 Matrix $M=M^P_{(2)}$ of P on ${\mathcal {V}}_2$, $N=8$, $K=3$

Table 5 Matrix $M=M^P_{(3)}$ of P on ${\mathcal {V}}_3$, $N=8$, $K=3$

6 Adapted Power Method to Compute Eigenvectors of QP

The adjacency matrix A has norm N. Numerical experiments not shown here suggest that the norm of $M^P_{(r)}$ scales like $N^{\alpha N}$ for some $\alpha >1/2$, and $M^P_{(r)}$ is poorly conditioned. For example, the condition number of the principal $(K+r-1)$-minor of $M^P_{(0)}$ when $N=2K+1$ grows faster than $10^{2K}$, as verified numerically for $K\le 100$. Consequently, direct numerical computation of the eigen-decomposition for this minor is problematic or impossible even for moderate N and K.

By Proposition 4, B can be represented on ${\mathcal {V}}_{r,K}$ by the size $(K-r+1)\times (K-r+1)$ matrix minor $M_{(K,r)}^{\mathrm{B}}$ when $\alpha =\beta =2\sqrt{K(K+1)}$ in (4.2). In [11] it was verified numerically that the operators PQ and BDO then almost commute in the sense that their commutator has a small norm relative to that of their constituents. This suggests that an eigenvector of one is nearly an eigenvector of the other, at least for larger eigenvalues. Since $M_{(K,r)}^{\mathrm{B}}$ is tri-diagonal, its eigen-decomposition is readily computed numerically. Algorithm 1 provides a method to compute eigenvectors of $M_{(K,r)}^{QP}$ numerically by first computing the corresponding eigenvectors of $M_{(K,r)}^{\mathrm{B}}$, then using these as inputs to a weight-adapted power method, using the weight W defined in (4.7) restricted to $k=0,\ldots , K-r$. Figure 4 plots the coefficient-wise weighted differences between the corresponding unit-weighted norm eigenvectors of the coefficient matrices $M_{(K,r)}^{B}$ and $M_{(K,r)}^{\mathrm{QP}}$ ($N=20$, $(K,r)=(6,1)$) when the former are used as seed vectors for a version of the power method in Algorithm 1. The weighted norm differences between the corresponding (unit-weighted norm) eigenvectors of these almost commuting matrices is on the order of $10^{-2}$. Eigenvalues of $M_{(K,r)}^{\mathrm{QP}}$, hence of QP, are computed as weighted norms of $M_{(K,r)}^{\mathrm{QP}}\mathbf {d}$ where $\mathbf {d}$ are numerically computed unit-weighted norm eigenvectors of $M_{(K,r)}^{\mathrm{QP}}$. Eigenvalues for $N=20$ and $K=6$ are listed in Table 6.

Technical Notes on Algorithm 1. The entries of $M_{(K,r)}^{B}$ are computed directly as in Theorem 2 with $\alpha =\beta =\sqrt{K(K+1)}$. The eigenvectors of $M_{(K,r)}^{B}$ can be computed directly, for example, using the matlab eig function. The application of $M_{(K,r)}^{\mathrm{P}}$ is accomplished by computing the outputs of the ${\mathcal {V}}_r$-coefficient matrices $M_{(r)}^{p_k(A)}$ corresponding to each term $p_k$ in (5.1) applied to the zero-padded input vector, truncating the output to the first $K+1-r$ entries, then summing over k. The application of $M_{(r)}^{p_k(A)}=\prod _{j=0,j\ne k}^N \frac{M_{(r)}^A-(N-2j)I_{N-2r+1}}{2(j-k)}$ is done by applying each successive factor in the product defining $M_{(r)}^{p_k(A)}$ to the output after applying the previous factor. For the example $N=20$, $K=6$, $r=1$ illustrated in Fig. 4, the error defined by weighted norms of differences of successive approximations $\mathbf {d}^{(i)}$ defined by successive iterations of the inner while loop was smaller than $10^{-9}$ in fewer than 600 iterations for each $i=0,\ldots , K-r$.

Table 6 Eigenvalues of PQ for $N=20$ and $K=6$

Fig. 4

Difference between unit-weighted norm seed eigenvectors of $M^{B}_{(K,r)}$ and computed eigenvectors of $M^{QP}_{(K,r)}$, $N=20$, $(K,r)=(6,1)$, weighted by $\sqrt{n(r,k)}$ defined in (3.7). The maximum weighted norm of the differences between the corresponding normalized eigenvectors of $M^{B}_{(K,r)}$ and $M^{QP}_{(K,r)}$ in this case is 0.0059

7 Conclusion and Outlook

We have identified eigenspaces of the iterated projection operator PQ that first truncates to a neighborhood of the origin in ${\mathcal {B}}_N$ then onto the span of the low-spectrum eigenvectors of the graph Laplacian on ${\mathcal {B}}_N$. The main technique was to identify certain adjacency-invariant spaces of vertex functions in which the eigenspaces lie and on which the action of the Fourier transform of PQ factors. To compute the eigenvectors we made use of an almost commuting auxiliary operator, BDO, that is analogous to the prolate differential operator that commutes with the time- and band-limiting operator on $L^2({\mathbb {R}})$. The invariant-space methods have been extended partly or fully to certain other Cayley graphs [12, 13]. The techniques rely on the fact that the spectra of these graphs can also be identified as (isomorphic) graphs. Further extensions may be possible in settings that admit a compliant spectral dual graph. See [18, 25] for ways to associate dual graphs to the Laplacian spectrum of a primal graph. The eigenvalues of PQ measure the concentration of an eigenvector on a Hamming ball in ${\mathcal {B}}_N$. The most concentrated eigenvectors can be used in analysis of functions that admit dyadic indexing in manners parallel to uses of prolate functions. For example, a spectral accumulation property established in [11, Sect. 5.5] means that these vectors can be used for a version of Thomson’s prolate-based multitaper spectrum estimation method (e.g., [31] and [10, Chap. 4]) that makes sense for locally dyadic stationary processes [14, 23]. In fact, in his comments [21] on Stoffer’s 1991 survey article on statistical applications of Walsh–Fourier analysis [30], Morettin stated, “I would have liked to have seen some mention of alternative estimator’s of the Walsh spectrum.” Shifted versions of the eigenvectors of PQ may also be of interest in CDMA applications using Walsh spreading codes for varying channels. See Zemen and Mecklenbräuker [35] regarding use of prolate sequences in such settings. Efficacy of such approaches will depend on the distribution of eigenvalues of PQ, which will be studied elsewhere.

Appendix: Proofs of Some Ancillary Facts

1.1 Projection Onto ${\mathcal {W}}_r$

Theorem 1 can be used to compute the projection of $\ell ^2(\Sigma _r)$ onto ${{\mathcal {W}}}_r$, the orthogonal complement of $A_+(\ell ^2(\Sigma _{r-1}))$ in $\ell ^2(\Sigma _r)$. The iterative procedure for achieving this is detailed in Proposition 7 below.

Lemma 4

Suppose $f_0$ is the vertex function given by $f_0(S)=\delta _{S,\emptyset }$, i.e., $f_0(\emptyset )=1$ while $f_0(S)=0$ for all $S\ne \emptyset$. Then for all $0\le r\le N$,

$$\begin{aligned} A_+^rf_0(R)={\left\{ \begin{array}{ll} r!&{}\hbox { if}\ R\in \Sigma _r\\ 0&{}\text { else.}\end{array}\right. } \end{aligned}$$

(A.1)

Proof

We have already seen that $A_+(\ell ^2(\Sigma _r))\subset \ell ^2(\Sigma _{r+1})$, so that $A_+f_0\in \ell ^2(\Sigma _1)$. Note that if $R\in \Sigma _1$,

$$\begin{aligned} A_+f_0(R)=\sum _{S\prec R, S\sim R}f_0(S)=f_0(\emptyset )=1 \end{aligned}$$

so that (A.1) is verified when $r=1$. Suppose now that (A.1) holds for $r=\ell$. Then $A_+^{\ell +1}f_0(R)=0$ if $R\notin \Sigma _{\ell +1}$ while if $R\in \Sigma _{\ell +1}$, we have

$$\begin{aligned} A_+^{\ell +1}f_0(R)=\sum _{S\prec R, S\sim R}A_+^\ell f_0(S)&=\sum _{S\in \Sigma _\ell ,\,S\sim R}f_0(S)\\&=\ell !|\{S\in \Sigma _\ell :\ S\sim R\}|=\ell !(\ell +1)=(\ell +1)!, \end{aligned}$$

i.e., (A.1) is verified for $r=\ell +1$. We conclude that (A.1) holds for all $0\le r\le N$. $\square$

Proposition 7

Let $f\in \ell ^2(\Sigma _r)$ $(r<N/2)$. By (3.2), f has a unique decomposition of the form

$$\begin{aligned} f=\sum _{\ell =0}^rA_+^{r-\ell }g_\ell \quad (g_\ell \in {\mathcal W}_\ell ). \end{aligned}$$

(A.2)

The vertex functions $\{g_\ell \}_{\ell =0}^r$ are given iteratively by:

• (i) $g_0=\dfrac{A_-^rf}{n(0,r)}$;

• (ii) $g_\ell =\dfrac{A_-^{r-\ell }\big [f-\sum _{j=0}^{\ell -1}A_+^{r-j}g_j\big ]}{n(\ell , r-\ell )}$ $(1\le \ell \le r)$

where the constants $n(r,\ell )$ are defined in (3.7).

Proof

An application of $A_-^r$ to both sides of (A.2) yields

$$\begin{aligned} A_-^rf=\sum _{j=0}^rA_-^jA_-^{r-j}A_+^{r-j}g_j=\sum _{j=0}^rn(j,r-j)A_-^jg_j=n(0,r)g_0 \end{aligned}$$

where we have applied Corollary 1 and the fact that $A_-g_j=0$ for all j. Dividing both sides of this equation by n(0, r) gives definition (i) of $g_0$. Similarly, applying $A_-^{r-1}$ to both sides of (A.2) yields

$$\begin{aligned} A_-^{r-1}f&=A_-^{r-1}A_+^rg_0+\sum _{j=1}^rA_-^{r-1}A_+^{r-j}g_j\\&=A_-^{r-1}A_+^rg_0+A_-^{r-1}A_+^{r-1}g_1=A_-^{r-1}A_+^rg_0+n(1,r-1)g_1 \end{aligned}$$

so that $g_1=\dfrac{A_-^{r-1}(f-A_+^rg_0)}{n(1,r-1)}$, thus verifying (ii) in the case $\ell =1$. Now suppose that $g_0,g_1,\ldots ,g_{\ell -1}$ have been determined. Applying $A_-^{r-\ell }$ to both sides of (A.2) gives

$$\begin{aligned} A_-^{r-\ell }f&=A_-^{r-\ell }\big (\sum _{j=0}^{\ell -1}A_+^{r-j}g_j+A_+^{r-\ell }g_\ell +\sum _{j=\ell +1}^rA_+^{r-j}g_j\big )\\&=\sum _{j=0}^{\ell -1}A_-^{r-\ell }A_+^{r-j}g_j+A_-^{r-\ell }A_+^{r-\ell }g_\ell =\sum _{j=0}^{\ell -1}A_-^{r-\ell }A_+^{r-j}g_j+n(\ell ,r-\ell )g_\ell . \end{aligned}$$

Note that $n(\ell ,r-\ell )\ne 0$ for $\ell \le r<N/2$. Rearranging this equation gives (ii). This proves the result. $\square$

Remark

The projections must be computed iteratively starting with the innermost projections in order to ensure that components extracted at each step are orthogonal to components from smaller spheres.

1.2 Proof of Proposition 2

We restate Proposition 2 as follows.

Proposition 8

Let $0\le r<s<N/2$. Let $f_r=\sum _{\ell =0}^{N-2r} c_k A_+^k g_r\in {\mathcal {V}}_r$ and $f_s=\sum _{\ell =0}^{N-2s} d_k A_+^k g_s\in {\mathcal {V}}_s$. Then $f_r$ and $f_s$ are orthogonal.

Proof

It suffices to show that the restrictions of $f_r$ and $f_s$ to any Hamming sphere $\Sigma _\rho$ are orthogonal. This is automatically true if $\rho <s$ or $\rho >N-s$ since $f_s=0$ on such spheres. If $s\le \rho \le N-s$ then $f_r|_{\Sigma _\rho } =c_{\rho -r} A_+^{\rho -r}g_r$ and $f_s|_{\Sigma _\rho } =d_{\rho -s} A_+^{\rho -s}g_s$. Therefore

$$\begin{aligned}&\langle f_r,\, f_s\mathbb {1}_{\Sigma _\rho }\rangle =c_{\rho -r} d_{\rho -s} \langle A_+^{\rho -r}g_r , A_+^{\rho -s}g_s\rangle \\&\quad =c\langle A_-^{\rho -s}A_+^{\rho -r} g_r, g_s \rangle =c\, m(r,\rho -r-1)\cdots m(r,s-r) \langle A_+^{s-r} g_r,\, g_s\rangle =0 \end{aligned}$$

by virtue of (3.2). $\square$

1.3 Proof of Proposition 1

Lemma 5

If $h\in \ell ^2(\Sigma _r)\setminus {\mathcal {W}}_r$, $0<r<N/2$, then $A_+A_- h\ne 0$.

Proof

Theorem 1 implies that $A_+$ is injective on $\ell ^2(\Sigma _{r-1})$ ($1<r<N/2$). Since $A_-h\in \ell ^2(\Sigma _{r-1})\setminus \{0\}$, the lemma follows. $\square$

Proof of Proposition 1

Consider the involution $\widetilde{g}(S)=g(\widetilde{S})$ where $\widetilde{S}$ is the complement of S in $\{1,\ldots N\}$. This involution is an isomorphism between $\ell ^2(\Sigma _r)$ and $\ell ^2(\Sigma _{N-r})$. In addition, $(A_-g)^{\sim }=A_+ \widetilde{g}$. To see this, if ${b'}\in S$ then $(A_+ \widetilde{g})(\widetilde{S}\cup {b'})=\sum _{b\in (\widetilde{S}\cup {b'})} \widetilde{g}((\widetilde{S}\cup {b'})\setminus b)= \sum _{b\notin (S\setminus {b'})}g((S\setminus {b'})\cup b)=(A_-g)(S\setminus {b'})=(A_-g)^{\sim }(\widetilde{S}\cup {b'})$.

By Theorem 1, if $g\in {\mathcal {W}}_r$ then $A_-A_+^{N-2r+1}g=0$. We claim that, in fact, $A_+^{N-2r+1}g=0$. Set $\widetilde{u}= A_+^{N-2r}g$. The claim then is that $A_+\widetilde{u}=0$, that is, $A_-u=0$, so $u\in {\mathcal {W}}_r$. Suppose not. Since $\widetilde{u}\in \ell ^2(\Sigma _{N-r})$ and ${u}\in \ell ^2(\Sigma _{r})$, u has a unique orthogonal decomposition $u=u_1+h$ where $u_1\in {\mathcal {W}}_r$ and $h=A_+u_2$ for some nonzero $u_2\in \ell ^2(\Sigma _{r-1})$. The condition $A_-A_+^{N-2r+1}g=0$ becomes $A_-A_+ \widetilde{h}=0$. Involuting gives $A_+A_-h=0$ which contradicts Lemma 5. This contradiction proves Proposition 1. $\square$