Abstract
Analogues of Slepian vectors are defined for finite-dimensional Boolean hypercubes. These vectors are the most concentrated in neighborhoods of the origin among bandlimited vectors. Spaces of bandlimited vectors are defined as spans of eigenvectors of the Laplacian of the hypercube graph with lowest eigenvalues. A difference operator that almost commutes with space and band limiting is used to initialize computation of the Slepian vectors.
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The second author would like to thank Robert Smits for pointing out Lemma 6.
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Communicated by Hans G. Feichtinger.
Appendix: Expressions for Entries of the Matrix HTLTH
Appendix: Expressions for Entries of the Matrix HTLTH
The purpose of this appendix is to derive explicit formulas for the entries of the matrix HTLTH that arises in the commutator \([\mathrm{BDO},\, {\bar{H}}[TLT,{\bar{H}}]]\). Besides providing a means to compute the commutator for large values of N, the formulas can be used as a basis for commutator estimates in terms of (N, K).
Of interest are the entries of \({\bar{H}}[TLT,\, {\bar{H}}]={\bar{H}} TLT{\bar{H}}-TLT\). As TLT is relatively uncomplicated, we focus on the entries of \(H TLTH=2^N{\bar{H}} TLT{\bar{H}}\). As before since \((TLT)_{RP} =0\) unless \(P=R\) or \(P\sim R\), writing \(|R|=r\),
As before, if P is an addition neighbor of R whose extension element is in S or if P is a subtraction neighbor of R whose subtraction element is in S then \(H_{PS}=-H_{RS}\); otherwise \(H_{PS}=H_{RS}\). This gives us the following.
Lemma 4
Write \(r=|R|\) and \(s=|S|\). One has
Equivalently, the coefficient of \(H_{RS}\) in \((TLT H)_{RS}\) is
Equation (4) follows from (3) by observing that \(|S^c\cap R^c|=N-s-r+|S\cap R|\), \(|S\cap R^c|=s-|S\cap R|\), etcetera. If one lets
and \(c_{RS}'=c_{RS}-c''_{RS}\), then
depends only on \(r=|R|\) and \(s=|S|\). Observing that \(H_{RP}H_{PS}=H_{S\Delta R,P}\), one has
where P runs over all subsets of \(\{1,\dots , N\}\).
Now write
Lemma 5
The entries of the matrix \(C'\) can be written
where \({}_2 F_1(a,b;c;z)\) denotes the Gaussian hypergeometric function.
Proof
For a fixed \(p\in \{0,1,\dots , S\Delta R\}\),
which follows by counting the number of ways that a p-element set can be divided into a subset of \(S\Delta R\) and a subset of the complement of \(S\Delta R\), and observing that \(H_{S\Delta R,P}=(-1)^k\) if P has k elements in common with \(S\Delta R\). The lemma then follows upon taking \(n=|S\Delta R|\), \(m=N-|S\Delta R|\) and \(z=-1\) in the identity (7) in Lemma 6 below. \(\square \)
Lemma 6
Define the Gaussian hypergeometric function
For nonnegative integers m, n, p such that \(p\le n\) one has
Proof
If n is a positive integer and \(0\le k\le n\) then
and \((-n)_k=0\) if \(k>n\). Thus, if n and p are nonnegative integers then
\(\square \)
In particular, if \(c=m-p+1\) where \(p\le m\) then \((m-p+1)_k=\frac{(m-p+k)!}{(m-p)!}\) and
This proves the lemma.
Lemma 7
Let \(c''_{RS}= |R\cap S| d''_{r}\) where \(d''_{r}=-4\sqrt{r(r+1)}-\sqrt{r(r-1)})\). The entries of the matrix \(C''\) in (6) can then be written
where \((p,r,s)=(|P|,|R|,|S|)\), \(|P\cap S|=k\), and \(|R\cap S|=q\).
The argument to prove Lemma 7 is similar to that of Lemma 5 but now one must take into account the distribution of values \(k=|P\cap S|\), counting the number of ways to decompose P, \(|P|=p\) according to its part that intersects R inside of S (or not) and its part that intersects R outside of S (or not), and recalling that \(q=|S\cap R|\) is fixed.
Up to this point we have derived formulas for entries of HTLTH in terms of binomial coefficients, values of hypergeometric functions, and coefficients involving factors \(\sqrt{p(p\pm 1)}\). The latter can be estimated in terms of Laurent series, in particular
For example, the coefficient \(c'_{RS}\) in (5) behaves as
for large r, while \(c''_{RS}= -4|R\cap S|(\sqrt{r(r+1)}-\sqrt{r(r-1)})\) behaves as
Therefore, in the expansion
each of the coefficients \(c_{PS}=c'_{PS}+c''_{PS}\) is essentially bounded by the maximum of N and the product ps. The nonzero entries of the commutator [Q, HTLTH] correspond to values \((HTLTH)_{RS}\) such that \(|R|\le K\) and \(|S|>K\) or \(|S|\le K\) and \(|R|>K\). As previously indicated, norm bounds for \([Q,\, \mathrm{BDO}]\) based on these approximations of \(C'\) and \(C''\) will appear in future work.
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Hogan, J.A., Lakey, J.D. An Analogue of Slepian Vectors on Boolean Hypercubes. J Fourier Anal Appl 25, 2004–2020 (2019). https://doi.org/10.1007/s00041-018-09654-w
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DOI: https://doi.org/10.1007/s00041-018-09654-w