ON SLANTED MATRICES IN FRAME THEORY AKRAM ALDROUBI, ANATOLY BASKAKOV, AND ILYA KRISHTAL
Abstract. In this paper we present a brief account of the use of the spectral theory of slanted matrices in frame and sampling theory. Some abstract results on slanted matrices are also presented.
1. Introduction. In this paper we study certain properties of so-called slanted matrices, which occur naturally in different fields of pure and applied analysis. A matrix is slanted if it has a decay property such that the coefficients vanish away from a diagonal, which is not necessarily the main diagonal; ideally, non-zero coefficients of such a matrix are contained between two parallel slanted lines. Potential applications of the theory of slanted matrices range through wavelet theory and signal processing [13, 14, 15, 17, 22], frame and sampling theory [1], differential equations [10, 11, 12], and even topology of manifolds [31]. Here we especially emphasize the use of slanted matrices in frame theory and related fields. In lieu of an introduction we provide a few explicit examples illustrating the appearance of slanted matrices. Example 1.1. Filter banks. In signal processing and communication, a sequence s (a discrete signal) is often split into a finite set of compressed sequences {s1 , . . . , sr } from which the original sequence s can be reconstructed or approximated. The compression is often performed with filter banks [17, 22] using the cascade algorithm. One way to introduce filters, in the simplest case, is to use the two-scale equation of the multiresolution analysis (MRA): X ϕ(x) = an ϕ(2x − n), n∈Z Z 2
where ϕ ∈ L (IR) is the so-called scaling function. The filter coefficients an , n ∈ ZZ, in the above equation are the Fourier coefficients of the low-pass filter m0 ∈ L2 (TT ), TT = IR/ZZ, which is a periodic function given by X m0 (ξ) = an e2πinξ , ξ ∈ IR. n∈Z Z
It is clear that the two-scale equation has the following equivalent form in the Fourier domain: ϕ(2ξ) ˆ = m0 (ξ)ϕ(ξ), ˆ ξ ∈ IR. An important role in the MRA theory is played by the periodization σϕ ∈ L∞ (TT ) of the scaling function ϕ, which is defined by X σϕ (ξ) = [ϕ, ˆ ϕ](ξ) ˆ = |ϕ(ξ ˆ + n)|2 . n∈Z Z
Key words and phrases. Slanted matrices, Banach frames, irregular sampling, non-uniform sampling. The first author was supported in part by NSF grants DM-0504788. The second author is supported in part by RFBR grant 07-01-00131.
It is a standard fact (see, e.g., [4, Lemma 2.11]) that this periodization satisfies σϕ (ξ/2) = |m0 (ξ)|2 σϕ (ξ) + |m0 (ξ + 1/2)|2 σϕ (ξ + 1/2). In fact, σϕ is the Perron-Frobenius eigen-vector of the transfer operator Rm0 which acts on different spaces of periodic functions via (1.1)
(Rm0 f )(ξ) = |m0 (ξ)|2 f (ξ) + |m0 (ξ + 1/2)|2 f (ξ + 1/2).
In [14] there is a detailed account of the relation between the spectral properties of the transfer operator on different function spaces and the properties of the corresponding MRA filters, scaling functions, and wavelets. Here we will just recall that the convergence rate of the above mentioned cascade algorithm is controlled by the second biggest eigen-value of Rm0 . The reason we use the transfer operator as an example is because of its matrix with respect to the Fourier basis in L2 (TT ). Following [14, Section 3.2], we let X cn = a ¯k an+k . k∈Z Z
Then the Fourier coefficients of Rm0 f and f are related via X (Rm0 f )n = c2n−k fk , k∈Z Z
and, hence, this is, indeed, a slanted matrix. In particular, if m0 (ξ) = a0 + a1 e2πiξ + a2 e2πi2ξ + a3 e2πi3ξ , a section of this matrix c3 c2 c1 0 0 c3 0 0 0 0 0 0 0 0 0
looks like c0 c−1 c−2 c−3 0 0 0 0 0 0 0 0 c2 c1 c0 c−1 c−2 c−3 0 0 0 0 0 0 0 c3 c2 c1 c0 c−1 c−2 c−3 0 0 0 0 0 0 0 c3 c2 c1 c0 c−1 c−2 c−3 0 0 0 0 0 0 0 c3 c2 c1 c0 c−1 c−2 c−3
Example 1.2. Sampling in shift invariant spaces. It is well known that the Paley-Wiener space P W1/2 = {f ∈ L2 (IR) : supp fˆ ⊂ [−1/2, 1/2]} can also be described as X ck φ(· − k), c ∈ `2 (ZZ)}, (1.2) P W1/2 = {f ∈ L2 (IR) : f = k∈Z Z sin π(x−k) π(x−k)
and the series converges in L2 (IR) (see e.g., [2]). Because of this equivalent where φ(x) = description of P W1/2 , the problem of reconstructing a function f ∈ P W1/2 from the sequence of its integer samples, {f (i)}i∈ZZ , is equivalent to finding the coefficients c ∈ `2 such that {f (i)} = Ac where A = (ai,j ) is the matrix with entries ai,j = φ(i − j). It is immediate, however, that A = I is the identity matrix and, therefore, X f= f (k)φ(· − k). k∈Z Z
If, instead, we sample a function f ∈ P W1/2 on 21 ZZ, then we obtain the equation {f ( 2i )} = Ac. In this case, the sampling matrix A is defined by ai,j = φ( 2i − j) and is no longer diagonal – π(x−k) it has constant values on slanted lines with slopes 1/2, for instance, a2j,j = 1. If φ = sinπ(x−k)
in (1.2) is replaced by a function ψ supported on [− M2 , M2 ], then the matrix A = (ai,j ) is zero outside the slanted band |j − i/2| ≤ M . Clearly, this matrix is not banded in the classical sense. Observe, also, that this matrix has the “opposite” slant compared to the matrix of the transfer operator in the previous example, while the Laurent-type structure is present in both cases. If we move to the realm of irregular sampling, however, the sampling matrix will be given by ai,j = φ(xi − j), where xi , i ∈ ZZ, are the sampling points. In this case, we no longer have constant values on slanted lines, i.e., the Laurent structure is gone, but the slanted structure is preserved if we have the same number of sampling points per period. An important fact [2] is that any function can be reconstructed from its samples at xi , i ∈ ZZ, if and only if the sampling matrix is bounded below and above. Example 1.3. Frame analysis operator. Let H be a separable Hilbert space. A sequence ϕn ∈ H, n ∈ ZZ d , is a frame for H if for some 0 < a ≤ b < ∞ X (1.3) a kf k2 ≤ |hf, ϕn i|2 ≤ b kf k2 n∈Z Zd
for all f ∈ H. The operator T : H → `2 , T f = {hf, ϕn i}n∈ZZ d , f ∈ H, is called an analysis operator. It is an easy exercise to show that a sequence ϕn ∈ H is a frame for H if and only if its analysis operator has a left inverse. The adjoint of the analysis operator, T ∗ : `2 → P PH, is given by T ∗ c = cn ϕn , c = (cn ) ∈ `2 . The frame operator is T ∗ T : H → H, T ∗ T f = hf, ϕn iϕn , n∈Z Zd
n∈Z Zd
f ∈ H. Again, a sequence ϕn ∈ H is a frame for H if and only if its frame operator is invertible. The canonical dual frame ϕ˜n ∈ H is then ϕ˜n = (T ∗ T )−1 ϕn and the (canonical) synthesis operator is T ] : `2 → H, T ] = (T ∗ T )−1 T ∗ , so that X X f = T ]T f = hf, ϕn iϕ˜n = hf, ϕ˜n iϕn n∈Z Zd
n∈Z Zd
for all f ∈ H. Traditionally (see [16, 19] and references therein), the frame properties are studied via the spectral properties of the frame operator. In this paper we show that some work can be done already at the level of the analysis operator. This makes extensions to Banach spaces easier since the analysis operator is more amenable to such. Connection with slanted matrices is readily illustrated if we consider a frame in `2 (ZZ) which consists of two copies of an orthonormal basis. Then a section of the matrix of the analysis operator with respect to that basis looks like 1 1 0 0 0 0
0 0 1 1 0 0
0 0 0 0 1 1
0 0 0 0 0 0
Clearly, the slant of the matrix may serve as a natural measure of redundancy of a frame. In the next section we will give a precise definition of slanted matrices and formulate some abstract results. In the third and final section we will interpret these results in view of the examples above. In particular, we will provide conditions under which a p-frame for some
p ∈ [1, ∞] is a Banach frame for all p ∈ [1, ∞] and a stable sampling set for some p ∈ [1, ∞] remains such for all p ∈ [1, ∞]. Proofs of most of the results below will appear in [1]. 2. Slanted matrices: Abstract results. We prefer to give a straightforward definition of slanted matrices with “rows” and “columns” indexed by the group ZZ d , d ∈ IN . Some of the results below extend to the cases of more general locally compact Abelian groups and even a wide class of metric spaces. For each n ∈ ZZ d we let Xn and Yn be (complex) Banach spaces and `p = `p (ZZ d , (Xn )) be ! p1 P p the Banach space of sequences x = (xn )n∈ZZ d , xn ∈ Xn , with the norm kxkp = kxn kXn n∈Z Zd
d
when p ∈ [1, ∞) and kxk∞ = sup kxn kXn . By c0 = c0 (ZZ , (Xn )) we denote the subspace n∈Z Zd
of `∞ of sequences vanishing at infinity, that is
lim kxn k = 0, where |n| = max |nk |, n =
|n|→∞
1≤k≤d
d
(n1 , n2 , . . . , nd ) ∈ ZZ . We will use this multi-index notation throughout the paper. Let amn : Xn → Ym be bounded linear operators. The symbol A will denote the operator matrix (amn ), m, n ∈ ZZ d . In this paper, we are interested only in those matrices that give rise to bounded linear operators that map `p into `p for all p ∈ [1, ∞] and c0 into c0 . We let kAkp be the operator norm of A in `p (ZZ d , (Yn )) and kAksup = sup kamn k. If Xn , Yn , n ∈ ZZ d , are separable Hilbert m,n∈Z Zd
spaces, we denote by A = the matrix defined by a?mn = a∗nm , where a∗nm : Yn → Xm are the (Hilbert) adjoints of the operators anm . Clearly, (A? )? = A. ?
(a?mn )
To define certain classes of operator matrices we use the polynomial weight functions ω = ωs : ZZ d → IR given by (2.1)
ωs (n) = (1 + |n|)s , s ≥ 0.
We also fix a slant α 6= 0 and denote by χS the characteristic function of a set S. (j)
Definition 2.1. For α 6= 0 and j ∈ ZZ d the j-th α-slant of A is the matrix Aj = Aαj = (amn ), m, n ∈ ZZ d , defined by d Y (j) amn = amn χ[jk ,jk +1) (αmk − nk ). k=1
Definition 2.2. Let ω = ωs be a weight as in (2.1).PThe class Σωα of matrices with ω-summable α-slants consists of matrices A such that kAkΣωα = j∈ZZ d kAj ksup ω(j) < ∞. Notice that when α = d = 1 we get the usual matrix diagonals as a special case of α-slants studied in this paper. The following lemma summarizes basic properties of slanted matrices. Lemma 2.1. For some p ∈ [1, ∞] we consider two operators A : `p (ZZ d , (Yn )) → `p (ZZ d , (Zn )) and B : `p (ZZ d , (Xn )) → `p (ZZ d , (Yn )). • If A ∈ Σωα and B ∈ Σωα˜ then we have AB ∈ Σωαα˜ .
If, moreover, Yn , Zn , n ∈ ZZ d , are Hilbert spaces, then we have A? : `p (ZZ d , (Zn )) → `p (ZZ d , (Yn )) and • A is invertible if and only if A? is invertible; • If A ∈ Σωα then A? ∈ Σωα−1 . Next we define the key matrix property investigated in this paper. Definition 2.3. We say that the matrix A is bounded below in `p or, shorter, p-bb, if (2.2)
kAxkp ≥ ℘p kxkp , for some ℘p > 0 and all x ∈ `p .
The following lemma is due to Pfander [25] (see also [26]). It shows, in particular, that a transfer operator in Example 1.1 cannot be bounded below. Lemma 2.2. Assume that Xn = Yn , n ∈ ZZ d , and that all these spaces are finite dimensional. If A ∈ Σα , for some α > 1, then 0 is an approximate eigen-value of A : `p → `p , p ∈ [1, ∞]. Equivalently, for any > 0 there exists x ∈ `p such that kxkp = 1 and kAxkp ≤ . We note that if Xn 6= Yn in the above lemma, then it is a simple exercise to provide a counterexample to the result. The following theorem presents our central theoretical result. Theorem 2.3. Let s > (d + 1)2 and ω = (1 + |j|)s . Then A ∈ Σωα is p-bb for some p ∈ [1, ∞] if and only if A is q-bb for all q ∈ [1, ∞]. Moreover, if A ∈ Σωα is p-bb for some p ∈ [1, ∞], then there exists ℘ > 0 such that for all q ∈ [1, ∞] kAxkq ≥ ℘ kxkq , for all x ∈ `q . We observe that to the best of our knowledge the above result has not been proved before even in the classical case of the slant α = 1. Our proof, as we mentioned above, will appear in [1]. Since it is somewhat similar to Sj¨ostrand’s proof of a non-commutative Wiener’s lemma [27], it is not surprising that it also leads to some extensions of that lemma. The classical Wiener’s Lemma [30] states that if a periodic function f has an absolutely convergent Fourier series and never vanishes then the function 1/f also has an absolutely convergent Fourier series. This result has many extensions (see [5, 7, 8, 9, 18, 20, 21, 23, 24, 27, 28, 29] and references therein), some of which have been used recently in the study of localized frames [6, 19]. Most of the papers just cited show how Wiener’s result can be viewed as a statement about the off-diagonal decay of matrices and their inverses. Using Lemma 2.1 and [9, Theorem 2] we obtain the following result about invertible slanted matrices. Theorem 2.4. Let Xn , Yn , n ∈ ZZ d , be Hilbert spaces and ω = ωs be a weight. If A ∈ Σωα is invertible for some p ∈ [1, ∞], then A is invertible for all q ∈ [1, ∞] and A−1 ∈ Σωα−1 . Moreover, if A ∈ Eα , then we also have A−1 ∈ Eα−1 . The following is a different, less trivial, extension of Wiener’s Lemma. Theorem 2.5. Let Xn = HX and Yn = HY be the same Hilbert (or Euclidean) spaces for all n ∈ ZZ d and A ∈ Σωα where ω(j) = (1 + |j|)s , s > (d + 1)2 . Let also p ∈ [1, ∞].
(i) If A is p-bb, then A is left invertible for all q ∈ [1, ∞] and a left inverse is given by A] = (A? A)−1 A? ∈ Σωα−1 . (ii) If A? is p-bb, then A is right invertible for all q ∈ [1, ∞] and a right inverse is given by A[ = A? (AA? )−1 ∈ Σωα−1 . Corollary 2.6. If A is as in Theorem 2.5(i) then Im A is a complementable subspace of `q , q ∈ [1, ∞]. 3. Slanted matrices: Applications. In this section we interpret Theorem 2.3 in the languages of Banach frames and sampling theory. 3.1. Banach Frames. The notion of a frame in a separable Hilbert space has already become classical. Its analogues in Banach spaces, however, are non-trivial (see [3, 6, 16, 19] and references therein). In this subsection we show that in case of certain localized frames the simplest possible extension of the definition remains meaningful. The difficulty of defining frames in general Banach spaces stems from the fact that one cannot use just the equivalence of norms similar to (1.3). The analysis operator is then only bounded below and not necessarily left invertible. As a result a “frame decomposition” remains possible but “frame reconstruction” no longer makes sense. Theorem 2.5(i) indicates, however, that often this obstruction does not exist. Definition 3.1. A sequence ϕn = (ϕnm )m∈ZZ d ∈ `1 (ZZ d , H), n ∈ ZZ d , is a p-frame (for `p (ZZ d , H)) for some p ∈ [1, ∞) if p X X p n (3.1) a kf k ≤ hfm , ϕm i ≤ b kf kp Zd n∈Z Z d m∈Z for some 0 < a ≤ b < ∞ and all f = (fm )m∈ZZ d ∈ `p (ZZ d , H). If X n (3.2) a kf k ≤ sup hfm , ϕm i ≤ b kf k n∈Z Zd m∈Z Zd for some 0 < a ≤ b < ∞ and all f = (fm )m∈ZZ d ∈ `∞ (ZZ d , H), then the sequence ϕn is called an ∞-frame. It is called a 0-frame if (3.2) holds for all f ∈ c0 (ZZ d , H). The operator Tϕ = T : `p (ZZ d , H) → `p (ZZ d ) = `p (ZZ d , C I ), given by X T f = hf, ϕn i := { hfm , ϕnm i}n∈ZZ d , f ∈ `p (ZZ d , H), m∈Z Zd
is called a p-analysis operator, p ∈ [1, ∞]. The 0-analysis operator is defined the same way for f ∈ c0 (ZZ d , H).
Definition 3.2. A p-frame ϕn with the p-analysis operator T , p ∈ {0}∪[1, ∞], is (s, α)-localized for some s > 1 and α 6= 0, if there exists an isomorphism J : `∞ (ZZ d , H) → `∞ (ZZ d , H) which leaves invariant c0 and all `q (ZZ d , H), q ∈ [1, ∞), and such that T J|`p ∈ Σωα , where ω(n) = (1 + |n|)s , n ∈ ZZ d . As a direct corollary of Theorem 2.5 and the above definition we obtain the following result. Theorem 3.1. Let ϕn , n ∈ ZZ d , be an (s, α)-localized p-frame for some p ∈ {0} ∪ [1, ∞] with s > (d + 1)2 . Then (i) The q-analysis operator T is well defined and left invertible for all q ∈ {0} ∪ [1, ∞], and the q-synthesis operator T ] = (T ∗ T )−1 T ∗ is also well defined for all q ∈ {0} ∪ [1, ∞]. (ii) The sequence ϕn , n ∈ ZZ d , and its dual sequence ϕ˜n = (T ∗ T )−1 ϕn , n ∈ ZZ d , are both (s, α)-localized q-frames for all q ∈ {0} ∪ [1, ∞]. (iii) In c0 and `q , q ∈ [1, ∞), we have the reconstruction formula X X f = T ]T f = hf, ϕn iϕ˜n = hf, ϕ˜n iϕn . n∈Z Zd
n∈Z Zd
For f ∈ `∞ the reconstruction formula remains valid provided the convergence is understood in the weak∗ -topology. Theorem 3.1(iii) shows that an (s, α)-localized p-frame is a Banach frame for c0 and all `q , q ∈ [1, ∞]. We observe that K. Gr¨ochenig shows in [19] how to extend a localized (Hilbert) frame to Banach frames for the associated Banach spaces. Slanted matrix theory, however, provides us with additional information which makes it possible to shift emphasis from the frame operator T ∗ T to the analysis operator T itself. 3.2. Sampling and Reconstruction Problems. The sampling and reconstruction problem includes devising efficient methods for representing a function f in terms of a discrete (finite or countable) set of its samples (values f (xj ) on a sampling set X) and reconstructing the original signal from the samples. Here we assume that the function f that belongs to a space X p V (Φ) = ck ϕk , d k∈Z Z
p
d
where c = (ck ) ∈ ` (ZZ ) when p ∈ [1, ∞], c ∈ c0 when p = 0, and Φ = {ϕk = ϕ(· − k)}k∈ZZ d ⊂ Lp (IRd ) is a countable collection of continuous functions. We impose the standard [2] assumptions on the generator ϕ to avoid convergence issues in the definition of V p (Φ) and ensure that it is a closed subspace of Lp . In particular, we assume that the shifts of ϕ generate an unconditional basis for V p (Φ) and belong to a Wiener-amalgam space Wω1 defined as follows. Definition 3.3. A measurable function ϕ belongs to Wω1 for a certain weight ω, if it satisfies X (3.3) kϕkWω1 = ω(k) · ess sup{|ϕ(x + k)| : x ∈ [0, 1]d } < ∞. k∈Z Zd
1 . When a function ϕ in Wω1 is continuous we write ϕ ∈ W0,ω
The following theorem is a more or less direct corollary of Theorem 2.3. 1 satisfy the standard Theorem 3.2. Let ω(n) = (1 + |n|)s , n ∈ ZZ d , s > (d + 1)2 , ϕ ∈ W0,ω assumptions, and
ap kf kLp ≤ k{f (xj )}k`p ≤ bp kf kLp , for all f ∈ V p (Φ), for some p ∈ [1, ∞] ∪ {0} and a separated set X = {xj , j ∈ ZZ d }. Then X is a stable set of sampling on V q (Φ) for all q ∈ [1, ∞] ∪ {0}. References [1] A. Aldroubi, A. Baskakov, and I. Krishtal, Slanted matrices, Banach frames, and sampling, submitted (2007). [2] A. Aldroubi and K. Gr¨ ochenig, Nonuniform sampling and reconstruction in shift-invariant spaces, SIAM Rev., 43 (2001), 585–620. [3] A. Aldroubi, Q. Sun, and W.-S. Tang, p-frames and shift-invariant subspaces of Lp , J. Fourier Anal. Appl., 7 (2001), 1–21. (2) [4] D. Baki´c, I. Krishtal, and E. Wilson, Parseval frame wavelets with En -dilations, Appl. Comput. Harmon. Anal., 19 (2005), no.3, 386–431. [5] R. Balan, A Noncommutative Wiener Lemma and A Faithful Tracial State on Banach Algebra of TimeFrequency Operators, to appear in Transactions of AMS (2007). [6] R. Balan, P. Casazza, C. Heil, and Z. Landau, Density, overcompleteness, and localization of frames. I,II, J. Fourier Anal. Appl., 12 (2006), no. 2, 105–143, no. 3, 309–344. [7] A.G. Baskakov, Wiener’s theorem and asymptotic estimates for elements of inverse matrices, Funct. Anal. Appl., 24 (1990), 222–224. [8] A.G. Baskakov, Estimates for the elements of inverse matrices, and the spectral analysis of linear operators, (Russian) Izv. Ross. Akad. Nauk Ser. Mat. 61 (1997), no. 6, 3–26; translation in Izv Math. 61 (1997), no. 6, 1113–1135. [9] A.G. Baskakov, Asymptotic estimates for elements of matrices of inverse operators, and harmonic analysis, (Russian) Sibirsk. Mat. Zh. 38 (1997), no. 1, 14–28; translation in Siberian Math. J. 38 (1997), no. 1, 10–22. [10] A.G. Baskakov, On correct linear differential operators, Sbornik: Mathematics 190 (1999), no. 3, 323–348. [11] A.G. Baskakov and I.A. Krishtal, Spectral analysis of operators with the two-point Bohr spectrum, J. Math. Anal. Appl. 308 (2005), no. 2, 420–439. [12] A.G. Baskakov and A.I. Pastukhov, Spectral analysis of a weighted shift operator with unbounded operator coefficients, (Russian) Sibirsk. Mat. Zh. 42 (2001), no. 6, 1231–1243; translation in Siberian Math. J. 42 (2001), no. 6, 1026–1035. [13] L. Berg and G. Plonka, Spectral properties of two-slanted matrices, Results Math. 35 (1999), no. 3-4, 201–215. [14] O. Bratteli and P. Jorgensen, Wavelets through a looking glass: The world of the spectrum, Applied and Numerical Harmonic Analysis, Birkh¨ auser, 2002. [15] M. Buhmann, and C. Micchelli, Using two-slanted matrices for subdivision, Proc. London Math. Soc. (3) 69 (1994), no. 2, 428–448. [16] O. Christensen, An introduction to Riesz Bases, Birkh¨auser, 2003. [17] P. Flandrin, P. Goncalv´es, and G. Rilling, EMD equivalent filter banks, from interpretation to applications. Hilbert-Huang transform and its applications, 57–74, Interdiscip. Math. Sci., 5, World Sci. Publ., Hackensack, NJ, 2005. [18] I. Gohberg, M. A. Kaashoek, H.J. Woerderman, The band method for positive and strictly contractive extension problems: an alternative version and new applications, Integral Equations Operator Theory, 12 (1989), no. 3, 343–382. [19] K. Gr¨ochenig, Localization of frames, Banach frames, and the invertibility of the frame operator, J. Fourier Anal. Appl., 10 (2004), no. 2, 105–132.
[20] K. Gr¨ochenig and M. Leinert, Wiener’s lemma for twisted convolution and Gabor frames, J. Amer. Math. Soc., 17 (2004), no. 1, 1–18 (electronic). [21] S. Jaffard, Propri´et´es des matrices “bien localis´ees” pr`es de leur diagonale et quelques applications, Ann. Inst. H. Poincar´e Anal. Non Lin´eaire, 7 (1990), no. 5, 461–476. [22] J. Kovaˇcevi´c, P.L. Dragotti, and V. Goyal, Filter bank frame expansions with erasures. Special issue on Shannon theory: perspective, trends, and applications. IEEE Trans. Inform. Theory 48 (2002), 1439–1450. [23] V.G. Kurbatov, Algebras of difference and integral operators Funct. Anal. Appl., 24 (1990), no. 2, 156–158. [24] L.H. Loomis, An introduction to abstract harmonic analysis. D. Van Nostrand Company, Inc., Toronto-New York-London, 1953. x+190 pp. [25] G. Pfander, Two results on the invertibility of bi-infinite matrices with applications to Gabor analysis. Preprint. [26] G. Pfander, and D. Walnut, Operator identification and Feichtinger algebra, STPIP, 1 (2002), 1–18. ´ [27] J. Sj¨ostrand, Wiener type algebras of pseudodifferential operators, S´eminaire sur les Equations aux D´eriv´ees ´ Partielles, 1994–1995, Exp. No. IV, 21 pp., Ecole Polytech., Palaiseau, 1995. [28] Q. Sun, Wiener’s lemma for infinite matrices with polynomial off-diagonal decay, C. R. Math. Acad. Sci. Paris 340 (2005), no. 8, 567–570. ˇ [29] M.A. Subin [Shubin], Almost periodic functions and partial differential operators. (Russian) Uspehi Mat. Nauk 33 (1978), no. 2 (200), 3–47; translation in Russian Math. Surveys 33 (1978), no. 2, 1–52. [30] N. Wiener, Tauberian theorems, Ann. of Math. (2) 33 (1932), no. 1, 1–100. [31] G. Yu, Higher index theory of elliptic operators and geometry of groups, Proc. International Congress of Mathematicians, Madrid, Spain 2006, 1624–1939. Department of Mathematics, email:
[email protected]
Vanderbilt
University,
Nashville,
TN
37240
Department of Applied Mathematics and Mechanics, Voronezh State University, Voronezh, Russia 394693 email:
[email protected] Department of Mathematical Sciences, Northern Illinois University, DeKalb, IL 60115 email:
[email protected] 
