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Matrix representation of operators using frames.


This paper addresses how to find a matrix representation of operators on a Hilbert space H with Bessel sequences, frames, and Riesz bases. In many applications these sequences are often preferable to orthonormal bases (ONBs). Therefore, it is useful to extend the known method of matrix representation by using these sequences instead of ONBs for these application areas. We will give basic definitions of the functions connecting infinite matrices defining bounded operators on [l.sup.2] and operators on H. We will show some structural results and give some examples. Furthermore, in the case of Riesz bases we prove that those functions are isomorphisms. Finally we apply this idea to the connection of Hilbert-Schmidt operators and Frobenius matrices.

Key words and phrases : frames, discrete expansion, operators, matrix representation, Hilbert-Schmidt operators, Frobenius matrices, Riesz bases.

1 Introduction

The relevance of signal processing in today's life is clearly evident. Without exaggeration it can be said that any advance in signal processing sciences directly leads to an application in technology and information processing. Without signal processing methods several modern technologies would not be possible, like mobile phones, UMTS, xDSL, or digital television.

The mathematical background for today's signal processing applications are Gabor [17] , wavelet [14] and sampling theory [7]. A signal is sampled and then analyzed using a Gabor wavelet system. Many applications use a modification on the coefficients obtained from the analysis operation [22, 23]. For them not only an analysis but also a synthesis operation is needed. If the coefficients are not changed, the result of the synthesis should be the original signal; i.e. perfect reconstruction. One way to achieve that is to analyze the signal using orthonormal bases (ONBs). In this case the analysis of a function is simply the correlation of the signal f with each basis element [e.sub.k], f [??] (<c.sub.k>) := (<f, [e.sub.k]>) . The synthesis that gives perfect reconstruction is simply the (possibly infinite) linear combination of the basis elements using the coefficients c = ([c.sub.k]), c [??] [summation over (k)] [c.sub.k] [e.sub.k].

From practical experience it soon has become apparent that the concept of an orthonormal basis is not always useful. Sometimes it is more important for a decomposing set to have other special properties rather than guaranteeing unique coefficients. For example, it is impossible to have good time-frequency localization for Gabor ONBs or a wavelet ONB with a mother wavelet which has exponential decay and is infinitely often differentiable with bounded derivatives [11]. Furthermore, suitable ONBs are often difficult to construct in a numerical efficient way. This led to the concept of frames, which was introduced by Duffin and Schaefer [15]. It was made popular by Daubechies [14], and today it is one of the most important foundations of Gabor [17], wavelet [2], and sampling theory [1]. In signal processing applications frames have received more and more attention [8, 29].

Models in physics [2] and other application areas, for example, in sound vibration analysis [5], are mostly continuous models. A lot of problems there can be formulated as operator theory problems, for example, in differential or integral equations. To be able to work numerically the operators must be discretized. One way to do this is to find (possibly infinite) matrices describing these operators using ONBs. In this paper we will investigate a way to describe an operator as a matrix using frames. This kind of 'sampling of operators' (compare to [24]) is especially important in application areas where frames are heavily used, so that the link between model and discretization is maintained. For implementation operator equations can be transformed in a finite, discrete problem with the finite section method [19] in the same way as in the ONB case.

The standard matrix description [13] of operators O using an ONB ([e.sub.k]) is by constructing an matrix M with the entries [M.sub.j,k] = <[O.sub.ek], [e.sub.j>. In [10] a concept was presented wherein an operator R is described by the matrix [(<R [[phi].sub.j], [[??].sub.i]>).sub.i,j] with ([phi].sub.i]) being a frame and ([??].sub.i]) its canonical dual. Recently such a representation was used for the description of operators in [21] using Gabor frames and [26] using linear independent Gabor systems. In this paper we will develop this idea in full generality for Bessel sequences, frames, and Riesz sequences, and also look at the dual function which assigns an operator to a matrix.

This paper is organized as follows: In Section 2 we collect results and notation we need. Section 3.1 gives the basic definitions and properties for Bessel sequences and frames. Matrix representation with Riesz bases is covered in Section 3.2. In Section 3.3 the connection of Frobenius matrices and Hilbert-Schmidt is investigated. Section 4 finishes the paper with perspectives.

2 Notation and Preliminaries

2.1 Hilbert spaces and Operators

We will give only a short review; for details refer to [13]. We will denote infinite dimensional Hilbert spaces by H and their inner product with < .,. >, which is linear in the first coordinate. Let B([H.sub.1], [H.sub.2]) denote the set of all linear and bounded operators from [H.sub.1] to [H.sub.2]. With the operator norm, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], this set is a Banach space. We will denote the composition of two operators A : [H.sub.1] [right arrow] [H.sub.2] and B : [H.sub.2] [right arrow] [H.sub.3] by B o A : [H.sub.1] [right arrow] [H.sub.3] and the adjoint of an operator A by [A.sup.*], so that <Ax, y> = <x, [A.sup.*]y> for all x, y [member of] H.

Furthermore, we will denote the range of an operator A by ran(O) and its kernel by ker(A). An example for a Hilbert space is the sequence space [l.sup.2] consisting of all square-summable sequences in C with the inner product <c, d> = [summation over (k)] [c.sub.k] x [[bar.[d.sub.k]]. We will use the canonical basis [DELTA] = ([delta].sub.[kappa]) for sequence spaces,

where [([[delta].sub.k].sub.n] = [[delta].sub.k,n], using the Kronecker symbol: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Remember that a linear function between Banach algebras [phi]: [B.sub.1] [right arrow] [B.sub.2] is called a Banach algebra homomorphism if it is also multiplicative, i.e., for all x, y [member of] [B.sub.1] we have [phi](x x y) = [phi](x)x [phi](y). It is called a monomorphism if it is also injective.

Definition 2.1 Let X, Y, Z be sets, f : X [right arrow] Z, g : Y [right arrow] Z be arbitrary functions. The Kronecker product [[cross product].sub.o]: X x Y [right arrow] Z is defined by

(f [[cross product].sub.o] g) (x, y) = f(x)x g(y).

Let f [member of] [H.sub.1], g [member of] [H.sub.2] then define the inner tensor product as an operator from [H.sub.2] to [H.sub.1 by

(f [[cross product].sub.i][bar.g])(h) = <h, g> f for h [member of] [H.sub.2].

We will often write f [cross product] g instead of f [[cross product].sub.o] g or f [[cross product].sub.i] g if there is no chance of misinterpretation.

2.1.1 Hilbert Schmidt Operators

A bounded operator T [member of] B([H.sub.1], [H.sub.2]) is called a Hilbert-Schmidt (HS) operator if there exists an ONB ([e.sub.n]) [subset of equal to] [H.sub.1] such that


Let HS ([H.sub.1], [H.sub.2]) denote the space of Hilbert Schmidt operators from [H.sub.1] to [H.sub.2].

This definition is independent of the choice of the ONB. The class of Hilbert-Schmidt operators is a Hilbert space of compact operators with the following properties:

* [[parallel]T[parallel].sub.Op] [less than or equal to] [[parallel]T[parallel].sub.HS].

* [[parallel]T[parallel].sub.HS]=[[parallel][T.sup.*][parallel].sub.HS], and T [member of] HS [??] [T.sup.*] [member of] HS.

* If T [member of] HS and A [member of] B, then TA and AT [member of] HS. [[parallel]AT[parallel].sub.HS] [less than or equal to] [[parallel]A[parallel].sub.Op][[parallel]T[parallel].sub.HS] and [[parallel]TA[parallel].sub.HS] [less than or equal to] [[parallel]A[parallel].sub.Op] [[parallel]T[parallel].sub.HS].

For more details on this class of compact operators refer to [25] or [30].

2.2 Frames

For more details and proofs for this section refer, e.g., to [9, 11, 14, 20]. A sequence [PSI] = ([[psi].sub.k]|k [member of] K) is called a frame for the Hilbert space H if constants A, B > 0 exist such that

A x [[parallel]f[parallel].sup.2.sub.H] [less than or equal to] [summation over (k)] [[absolute value of (<f, [[psi].sub.k]>).sup.2] [less than or equal to B x [[parallel]f[parallel].sup.2.sub.H] [for all] f [member of] H. (1)

Here A is called a lower, B an upper frame bound. If the bounds can be chosen such that A = B the frame is called tight.

A sequence [PSI] = ([[psi].sub.k]) is called a Bessel sequence with Bessel bound B if it fulfills the right inequality above:

[summation over (k)] [[absolute value of (<f, [[psi].sub.k]>).sup.2] [less than or equal to B x [[parallel]f[parallel].sup.2.sub.H] [for all] f [member of] H. (2)

The index set will be omitted in the following if no distinction is necessary. For a Bessel sequence, [PSI] = [[psi].sub.k]], let [C.sub.[PSI]]: H [right arrow] [l.sup.2](K) be the analysis operator [C.sub.[PSI]](f) = [(<f, [[psi].sub.k]>).sub.k]. Let [D.sub.[PSI]]: [l.sup.2](K)[right arrow] H be the synthesis operator [D.sub.[PSI]] (([c.sub.k])) = [summation over (k)] [c.sub.k] x [[psi].sub.k]. Let [S.sub.PSI]: H [right arrow] H be the (associated) frame operator [S.sub.[PSI]](f) = [summation over (k)] <f, [[psi].sub.k]>. To simplify notation we will write S for [S.sub.[PSI]], C for [C.sub.[PSI]], and D for [D.sub.[PSI]], if it is not necessary to distinguish different frames. We will use the notation [S.sub.[PSI]], [PHI] = [D.sub.[PSI]] o [C.sub.[PHI]]. C and D are adjoint to each other, D = [C.sup.*] with [[parallel]D[parallel].sub.Op] = [[parallel]C[parallel].sub.Op] [less than or equal to [square root of B]. The series [summation over (k)] [c.sub.k] x [[psi].sub.k] converges unconditionally for all ([c.sub.k]) [member of] [l.sup.2].

For a frame [PSI] = ([[psi].sub.k]) with bounds A, B, the operator C is a bounded, injective operator with closed range; and S = [C.sup.*]C = D [D.sup.*] is a positive invertible operator satisfying [AI.sub.H] [less than or equal to] S [less than or equal to] [BI.sub.H] and [B.sup.-1] [I.sub.H] [less than or equal to [S.sup.-l] [less than or equal to] [A.sup.-1][I.sub.H]. Even more we can find an expansion for every member of H: The sequence [PSI] = ([[??].sub.k]) = ([S.sup.-1.sub[psi]k) is a frame with frame bounds [B.sup.-1], [A.sup.-1] > 0, the so-called canonical dual frame. Every f [member of] H has the expansions f = [summation over (k [member of]K)] <(f, [[??].sub.k]> [[psi].sub.k] and f = [summation over (k [member of]K) <f, [[psi].sub.k]> [[??].sub.k] where both sums converge unconditionally in H.

Remember that a sequence ([e.sub.k]) is called a (Schauder) basis for H if for all f [member of] H there are unique coefficients ([c.sub.k]) such that f = [summation over (k)] [c.sub.k] [phi].sub.k]. Also, two sequences ([[psi].sub.k]), ([[phi].sub.k]) are called biorthogonal if <[[psi].sub.k], [[phi].sub.j]) = [[delta].sub.kj] for all h, j.

A complete sequence ([[psi].sub.k]) in H is called a Riesz basis if there exist constants A, B > 0 such that the inequalities


hold for all finite sequences ([c.sub.k]).

For a frame ([[psi].sub.k]) the following conditions are equivalent: (i) ([[psi].sub.k]) is a Riesz basis for H. (ii) The coefficients ([c.sub.k])[member of] [l.sup.2] for the series expansion with ([psi].sub.k) are unique. So the synthesis operator D is injective. (iii) The analysis operator C is surjective. (iv) ([[psi].sub.k]) and ([[??].sub.k]) are biorthogonal.

Let [PSI] = ([[psi].sub.k]) and [PHI] = ([[phi].sub.k]) be two sequences in H. The Gram matrix [G.sub.[PSI], [PHI]] for these sequences is given by [([G.sub.[PSI], [PHI]).sub.j,m] = ([phi], [[psi].sub.j]), j, m [member of] K. We denote [G.sub.[PSI], [PSI]] by [G.sub.[PSI]]. We can look at the operator induced by the Gram matrix, defined for c [member of] [l.sup.2] formally as ([G.sub.PSI]], [[PSI].sup.c])j = [summation over (k)] [c.sub.k] <[[phi].sub.k], [psi].sub.j]). Clearly for two Bessel sequences it is well-defined as linear bounded operator because


and, therefore,[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. A frame is a Riesz sequence if and only if the Gram matrix defines a bounded and invertible operator on 12.

3 Representing Operators with Frames

Let ([[psi].sub.k]) be a frame in[H.sub.1]. An existing operator U [member of] B([H.sub.1], [H.sub.2]) is uniquely determined by its images of the frame elements. For f = [summation over (k)] [c.sub.k][[psi].sub.k]


On the other hand, contrary to the case for ONBs, we cannot simply choose a Bessel sequence ([[eta].sub.k]) and define an operator just by choosing V([psi].sub.k]):= [[eta].sub.k] and setting V([summation over (k)] [c.sub.k][[psi].sub.k]) = [summation over (k)] [c.sub.k][[eta].sub.k]. This is in general not well-defined. Only if


this definition is non-ambiguous, i.e., if ker ([D.sub.[psi]k) [subset or equal to] ker ([D.sub.[eta]k]). This condition is certainly fulfilled if [D.sub.[psi]k] is injective, i.e., for Riesz bases.

This problem can be avoided by using the following definition

V(f): = [summation over (k)<f,[[??].sub.k]>[[eta].sub.k]

As ([[eta].sub.k]) forms a Bessel sequence, the right hand side of Eq. (3) is well-defined. It is clearly linear, and it is bounded. The Bessel condition, Eq. (2), is necessary in the case of ONBs to have a bounded operator, too [13]. But contrary to the ONB case, here, in general, V([[psi].sub.k]) [not equal to] [[eta].sub.k].

Instead of changing the sequence with which the coefficients are resynthezised, an operator can also be described by changing the coefficients, as presented in the following sections.

3.1 Matrix Representation

For orthonormal sequence it is well known that operators can be uniquely described by a matrix representation [19]. The same can be constructed with frames and their duals. Recall the definition of the operator defined by a (possibly infinite) matrix: [(Mc).sub.j] = [summation over (k)] [M.sub.j,k][c.sub.k]. We will start with the more general case of Bessel sequences. Note that we will use the notation [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for the operator norm in B([H.sub.1], [H.sub.2]) to be able to distinguish between different operator norms.

Theorem 3.1 Let [PSI], = ([[psi].sub.k]) be a Bessel sequence in [H.sub.1] with bound B, [PHI] = ([[phi].sub.k]) one in [H.sub.2] with B'.

1. Let O : [H.sub.1] [right arrow] [H.sub.2] be a bounded, linear operator. Then the infinite matrix

[([[M.sup.[PHI],[PSI]]) (O)).sub.m,n] = <O[[psi].sub.n], [[phi].sub.m]

defines a bounded operator from [l.sup.2] to [l.sup.2] with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. As an operator [l.sup.2] [right arrow] [l.sup.2]

[M.sup.([PHI],[PSI])] (O) = [[C.sub.[PHI]] o O o [[D.sub.[PSI]].

This means the function [M.sup.([PHI],[PSI])]: B([H.sub.1], [H.sub.2]) [right arrow] B([l.sup.2,[l.sup.2]) is a well-defined bounded operator.

2. On the other hand let M be an infinite matrix defining a bounded operator from [l.sup.2] to [l.sup.2], [(Mc).sub.i] = [summation over (k)] [M.sub.i,k][C.sub.k]. Then the operator [O.sup.([PHI],[PSI])] defined by


is a bounded operator from [H.sub.1] to [H.sub.2] with


This means the function [O.sup.([PHI],[PSI])] : B([l.sup.2], [l.sup.2]) [right arrow] B([H.sub.1] [H.sub.2]) is a well- defined bounded operator.

Proof: Let M = [M.sup.([PHI],[PSI])] and O = [O.sup.([PHI],[PSI])]. Let O [member of] B([H.sub.1] [H.sub.2]); then


Equation (4) also shows us that as an operator we have

[M.sup.([PHI],[PSI])] (O) = [C.sub.[PHI]] o O o [D.sub.[PSI]].

On the other hand, let M be an infinite matrix; then


Definition 3.1 For an operator O and a matrix M as in Theorem 3.1, we call [M.sup.([PSI][PHI])](O) the matrix induced by the operator O with respect to the Bessel sequences [PSI] = ([[psi].sub.k]) and ([PHI]) = ([[phi].sub.k]) and [O.sup.[PSI][PHI])] (M) the operator induced by the matrix M with respect to the Bessel sequences [PSI] and [PHI]. (See Figure 1.)

To not stress the dependency on the frames and to avoid confusion, the notation M(O) and O(M) will be used.

For frames we can prove more properties:

Proposition 3.2 Let [PSI] = ([[psi].sub.k]) be a frame in [H.sub.1] with bounds A, B, [PHI] = ([[phi].sub.k]) in [H.sub.2] with [A', B'. Then


1. ([O.sup.([PHI],[PSI])] o [M.sup.([PHI],[PSI])] = Id = ([O.sup.([PHI],[PSI])] o [M.sup.([PHI],[PSI])]. And therefore for all O [member of] B([H.sub.1] [H.sub.2]):


2. [M.sup.([PHI],[PSI])] is injective and [O.sup.([PHI],[PSI])] is surjective.


4. Let [XI] = ([[xi].sub.k]) be any frame in [H.sub.3], and O : [H.sub.3] [right arrow] [H.sub.2] and P : [H.sub.1] [right arrow] [H.sub.3]. Then

[M.sup.([PHI],[PSI])] (O o P) = ([M.sup.([PHI],[XI])] (O) x [M.sup.([??],[PSI])] (P)).

Proof: 1. For f [member of] [H.sub.1] we get [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

For the other equality the roles of the flame and the dual must be switched.

2. From O M = Id we know that M is injective and O is surjective.


4. [M.sup.([PHI],[PSI])][(O o P).sub.p,q] = <O o P[[psi].sub.q], [[phi].sup.p]> = P[[psi].sub.q], [O.sup.*][[??].sup.p]>.

On the other hand,


As a direct consequence we arrive at the following corollary:

Corollary 3.3 For the frame [PHI] = ([[phi].sub.k]) the function [M.sup.([PHI],[??])] is a Banach-algebra monomorphism between the algebra of bounded operators (B([H.sub.1], [H.sub.1]), o) and the infinite matrices of (B([l.sup.2], [l.sup.2], *).

The other function O is in general not so "well-behaved." It is, however, well-behaved if the dual frames are biorthogonal. In this case these functions are isomorphisms; refer to Section 3.2.

Lemma 3.4 Let O : [H.sub.1] [right arrow] [H.sub.2] be a linear and bounded operator, let [PSI] = ([[psi].sub.k]) and [PHI] = ([[phi].sub.k]) be frames in [H.sub.1] resp. [H.sub.2]. Then [M.sup.([PHI],[PSI])] (O) maps ran ([C.sub.[PSI]]) into ran ([C.sub.[PHI]]) with

[(<(f, [[psi].sub.k]>).sub.k] [??] [(<(O f, [[phi].sub.k]>).sub.k].

If O is surjective, then [M.sup.([PHI],[PSI])] (O) maps ran ([C.sub.[PSI]]) onto ran ([C.sub.[PHI]]). If O is injective, [M.sup.([PHI],[??])] (O) is also injective.

Proof: Let c [member of] ran ([C.sub.[PSI]]), then there exists f [member of] [H.sub.1] such that [c.sub.k] = <f, [[psi].sub.k]>.


So [(<f, [[psi].sub.k]>).sub.k] [??] [(<(O f, [[phi].sub.k]>).sub.k].

If O is surjective, then for every f there exists a g such that Og = f, and therefore <g, [[psi].sub.k]> [??] <f, [[phi].sub.k]>.

If O is injective, then let us suppose that (O f, [[phi].sub.k]> = <O g, [[phi].sub.k]>. Because ([[phi].sub.k]>) is a frame [??] O f = O g [??] = f = g = <f, [[psi].sub.k]> = g, [[psi].sub.k].

Particularly for O = Id the Gram matrix [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] maps ran ([C.sub.[PSI]]) bijectively on ran ([C.sub.[PHI]]). So we have a way to "switch" between frames by mapping from one analysis range into the other [3].

Let us give some examples:

Example 3.1 :

Let [PSI] = ([[psi].sub.k]) and [PHI] = ([[phi].sub.k]) be frames in H and [DELTA] = ([delta].sub.k]) the canonical basis of [l.sup.2]. Then

1. [S.sub.[PSI]] : H [right arrow] H [??] [M.sup.([PSI],[??])]([S.sub.[PSI]]) = [G.sub.[PSI]].

2. [S.sup.-1.sub.[PSI]] : H [right arrow] [??] [M.sup.([PSI],[??])]([S.sup.-1.sub.[PSI]]) = [G.sub.[??]].


4. Id : H [right arrow] H [??] [M.sup.([PHI],[PSI])] (Id) = [G.sub.[PHI],[PSI]].

5. Id : [l.sup.2] [right arrow] [l.sup.2] [??] [O.sup.[PHI],[PSI]] (Id) = [D.sub.[PHI]] o [C.sub.[??]] = [S.sub.[PSI],[??]].

3.1.1 Motivation: Solving Operator Equalities

Given an operator equality

O x f = g (5)

it is natural to discretize it to find a solution. Let [PHI] = ([[phi].sub.k]) be a frame. Let us suppose that for a given g with coefficients d = ([d.sub.k]) = (<g, [[phi].sub.k]>) and a matrix representation M of O there is an algorithm to find the least square solution of

M x c = d, (6)

for example, using the pseudoinverse [11]. Still, if using frames, we cannot expect to find a true solution for Eq. 5 just by applying [D.sub.[??]], on c as in general c is not in ran([C.sub.[PHI]]) even if d is. But rephrasing Eq. 5 we see the following:


It can be easily seen that this is equivalent to projecting c on ran(C), solving M [C.sub.[PHI]][D.sub.[??]] c = d, which is a common idea found in many algorithms; e.g., for a recent one see [27].

This gives us an algorithm for finding an approximative solution to the inverse operator problem Of = g.

1. Set M = [M.sup.([PHI],[??])] (O).

2. Find a good finite dimensional approximation [M.sub.N] of M by using the finite section method [19].

3. Then apply an algorithm like, e.g., the QR factorization [28] to find a solution for Eq. 6.

4. Synthezise with the dual frame [??].

Remark: It has been shown in [12] that the finite section is very useful in the case of frame theory. It would be very interesting to investigate the idea presented above further in this context.

3.2 Matrix representation using Riesz Bases

The coefficients using a Riesz basis are unique, so Theorem 3.1 can be extended to:

Theorem 3.5 Let [PHI] = ([[phi].sub.k]) be a Riesz basis for [H.sub.1], [PSI] = ([[psi].sub.k]) one for [H.sub.2]. The functions [M.sup.([PHI,[PSI])] and [O.sup.([??],[??])] between B([H.sub.1],[H.sub.2]) and the infinite matrices in B([l.sup.2],[l.sup.2]) are bijective. [M.sup.([PHI],[PSI]) and [O.sup.([??],[??])] are inverse to each other. For [H.sub.1] = [H.sub.2] the identity is mapped on the identity by [M.sup.([PHI],[PSI]) and [O.sup.([??],[??])]. If, furthermore, [PSI] = [PHI] then [M.sup.([PHI],[??]) and [O.sup.([PHI],[??])] are Banach algebra isomorphisms, respecting the identities [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and I [d.sub.H].

Proof: We know from Proposition 3.2 that (9 o All = Id. Let us consider the following:


So these functions are inverse to each other and, therefore, bijective:


We know that [M.sup.([PHI],[??]]) is a Banach algebra homomorphism and so is its inverse.

Peter Balazs

Austrian Academy of Sciences, Acoustics Research Institute, Reichsratsstrasse 17

A-1010 Vienna, Austria
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Author:Balazs, Peter
Publication:Sampling Theory in Signal and Image Processing
Article Type:Technical report
Geographic Code:4EUAU
Date:Jan 1, 2008
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