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.

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. $\square$