DORIN ERVIN DUTKAY AND PALLE JORGENSEN. (Communicated by )

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1 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages S (XX) OVERSAMPLING GENERATES SUPER-WAVELETS arxiv:math/ v1 [math.fa] 16 Nov 2005 DORIN ERVIN DUTKAY AND PALLE JORGENSEN (Communicated by ) Abstract. We show that the second oversampling theorem for affine systems generates super-wavelets. These are frames generated by an affine structure on the space L 2 (R d )... L 2 (R d ). }{{} p times 1. Introduction We use the term sampling in the sense of Shannon. Shannon showed that a function f on the real line R which has its Fourier transform ˆf supported in a bounded interval can be reconstructed by interpolation of its values resulting from sampling at integral multiples of a certain rate ν, i.e., {f(nν) n Z}. (Functions f with Fourier transform ˆf of compact support are called band-limited. We shall adopt this convention even if f is a function in several variables.) Shannon s theorem has now found a variety of generalizations and refinements, and it is standard fare in both Fourier analysis and in applied mathematics. Given the finite support of ˆf, it is well known that there is an optimal rate ν (the Nyquist rate) which gives exact reconstruction of the function from its samples. Nonetheless there are instances where decreasing the rate from Nyquist, or rather increasing the chosen set of discrete sample points is desirable or unavoidable. Hence, the term oversampling! Shannons formula even has a formulation in Hilbert space in terms of reproducing kernels, but we shall be concerned here with wavelet bases. One way to view Shannon s interpolation is to think of the interpolation formula for the function f as an expansion into a generalized basis for L 2 (R), or analogously for L 2 (R d ) in higher dimensions. As is well known, the generalized bases take the form of frames (see [6] and [3]), and moreover the class of frames include wavelets, or rather wavelet bases which constitute frames, see eq (1.1) below. We shall be concerned with this framework for oversampling, and our results are in the context of the Hilbert space L 2 (R d ). This means that our sampling points will typically constitute a rank-d lattice, i.e., a discrete subgroup of R d of rank d. While the early results are based on Shannon s ideas, the subject received a boost from advances in wavelets and frames, and a number of authors have recently extended and improved the classical sampling and reconstruction results. Received by the editors November 15, Mathematics Subject Classification. 42C40, 47A20, 65T60, 94A20. Key words and phrases. wavelet, frame, sampling, oversampling, affine, scaling, lattice, interpolation, dilations, extensions, super wavelets, operators, frames, Hilbert space. 1 c 1997 American Mathematical Society

2 2 DORIN ERVIN DUTKAY AND PALLE JORGENSEN An intriguing extension is to the non-uniform case, i.e., when the sampling points are not necessarily confined to a lattice in R d. The paper [1] and the book [2] contain a number of such new results, and they offer excellent overviews. In the real variable case, in one or several variables, and the d-dimensional Lebesgue measure, there is a separate generalizations of standard dyadic wavelets, again based on translation and scaling: And there is a powerful approach to the construction of wavelet bases in the Hilbert space L 2 (R d ), i.e., of orthogonal bases in L 2 (R d ), or just frame wavelet bases, but still in L 2 (R d ). Of course, the best know instance of this is d = 1, and dyadic wavelets [6]. In that case, the two operations on the real line R are translation by the group Z of the integers, and scaling by powers of 2, i.e., x 2 j, as j runs over Z. This is the approach to wavelet theory which is based on multiresolutions and filters from signal processing. In higher dimensions d, the scaling is by a fixed matrix, and the translations by the rank-d lattice Z d. Again we will need scaling by all integral powers. We view points x in R d as column vectors, and we then consider the group of scaling transformations, x M j x as j ranges over Z. Let M be a d d dilation matrix with integer entries, such that all the eigenvalues λ of M satisfy λ > 1. The dilation operator induced by M is Df(x) := detm f(mx), for f L 2 (R). For u R d, let T u denote the translation operator by T u f(x) := f(x u). Let H be a Hilbert space. A collection of vectors {e i i I} in H is called a frame if there are some constants A, B > 0 such that, for all f H, (1.1) A f 2 i I f e i 2 B f 2. The constants A and B are called the lower and the upper frame bounds. Our use of the term oversampling is motivated as follows: We start with a frame system in d real dimensions which is based on scaling by a fixed expansive matrix M; and we normalize the setting such that our initial set of sample points will be located on the standard rank-d lattice Z d. We then introduce and use a second matrix P to inflate the initial lattice of sample points. The two matrices M and P must satisfy a certain compatibility condition (generalizing the notion of relative prime for numbers); Definition 1.1. In our theorem (Theorem 2.1) we compare our two frame systems before and after oversampling. As a byproduct of our analysis, we are able to make precise and to quantify this kind of oversampling. We now turn to our affine frame systems, and their oversampled versions. Let Ψ = {ψ 1,..., ψ n } L 2 (R d ). The affine system generated by Ψ is X(Ψ) := {D j T k ψ i j Z, k Z d, l {1,..., n}}. Let P be a d d integer matrix. Denote by p := detp. The oversampled affine system generated by Ψ relative to P is X P (Ψ) := { 1 p D j T P 1 kψ i j Z, k Z d, i {1,..., n}}. It is helpful to view sampling in context of L 2 (R d )-wavelets. Start with an affine wavelet frame system in the Hilbert space L 2 (R d ). Our general idea is then to represent oversampling for such an affine frame basis in H as follows: While the initial frame system in H will have redundancy, we show that there is a specific inflated Hilbert space such that by passing to this ambient Hilbert space, we will

3 OVERSAMPLING GENERATES SUPER-WAVELETS 3 then get exact (Nyquist type) sampling. In fact we show that the inflated Hilbert space takes the form of an orthogonal sum of L 2 (R d ) with itself a finite number of times p say, where p depends on the amount of oversampling. Definition 1.1. The matrix P is called an admissible oversampling matrix for M (or simply admissible), if the matrix PMP 1 has integer entries and M 1 Z d P 1 Z d = Z d. The second oversampling theorem states that, when P is admissible, if the affine system X(Ψ) is a frame for L 2 (R d ) then the oversampled affine system X P (Ψ) is a frame for L 2 (R d ) with the same frame bounds. This type of oversampling was introduced by Chui and Shi in [5] for one dimension and scaling by 2. Since then, the result has been generalized and has become known as the second oversampling theorem ; see [11, 4, 10, 8]. For details on the history of the second oversampling theorem we refer to [8]. In this paper we will prove that, in fact more is true: if we oversample with a matrix P we will obtain frames in the larger space H := L 2 (R d )... L 2 (R d ). }{{} p times Specifically, we prove the following: Theorem 2.1. Let d be given, and let two integral d by d matrices M and P satisfy the conditions in Definition 1.1. Starting with an M-scale frame X(Ψ) in L 2 (R d ), let X P (Ψ) be the corresponding P-oversampled frame. Then there is an isometric and diagonal embedding of L 2 (R d ) in H and a third frame in H with the same frame bounds, such that X P (Ψ) arises from it as the projection onto the first component. These super-frames are also generated by an affine structure on H. In addition, the oversampled system X P (Ψ) can be recovered as the projection of the superframe onto the first component. Moreover, when the affine system is an orthonormal basis, the corresponding super-frame is also an orthonormal basis for H. These results generalize also Theorem 5.8 in [7] which treated the case of tight frames in dimension d = Statement of the results We will assume that P is an admissible oversampling matrix for M. We denote by M := PMP 1. Let {θ r 0 r p 1} be a complete set of representatives for P 1 Z d /Z d. We can take θ 0 = 0. The dual of this group is (P ) 1 Z d /Z d, and let {θq 0 q p 1} be a complete set of representatives for this group. We can take θ0 = 0. The duality is given by θr θq = e 2πiPθ r θ q. Since P is admissible, the matrix M induces a permutation σ of {r 0 r p 1}, Mθ r θ σ(r) mod Z d (see [8, Proposition 2.1]). The dual of this map induces a permutation σ of {q 0 q p 1}, M θ q θ σ (q) mod Z d. Indeed, e 2πiPMθr θ q = e 2πiM Pθ r θ q = e 2πiPθ r M θ q.

4 4 DORIN ERVIN DUTKAY AND PALLE JORGENSEN Define the Hilbert space H := L 2 (R d )... L 2 (R d ). }{{} p times On this Hilbert space we define an affine structure generated by a dilation operator D, and the translation operators T P 1 k, k Z d. For the matrix M define the unitary operators: T P 1 k(f q ) 0 q p 1 = (e 2πik θ q TP 1 kf q ) 0 q p 1, (k Z d ); D(f q ) 0 q p 1 = (Df (σ ) 1 (q)) 0 q p 1. The operators satsify the commutation relation (2.1) DT MP 1 k = T P 1 kd, (k Z d ). For f L 2 (R d ), define Sf H S is an isometry. Sf = 1 p (f,..., f). Theorem 2.1. Let Ψ := {ψ 1,..., ψ n } L 2 (R d ). If the affine system X(Ψ) is a frame (orthonormal basis) for L 2 (R d ) then X(SΨ) := {D j T P 1 ksψ i j Z, k Z d, i {1,..., n}} is a frame (orthonormal basis) for H with the same frame bounds. The converse also holds. The projection of X(SΨ) onto the first component is the oversampled affine system X P (Ψ). We start with a lemma. Lemma 3.1. The matrix is unitary. 3. Proof of theorem 2.1 H := 1 p (e 2πiPθr θ q )0 r,q p 1 Proof. This is a well-known fact from harmonic analysis. The matrix is the matrix of the Fourier transform on the finite abelian group P 1 Z d /Z d. It will be convenient to make a change of variable x Px. Let D be the corresponding dilation operator D f(x) = detm f(m x), for f L 2 (R d ). For the matrix M, define the unitary operators: T k(f q ) 0 q p 1 = (e 2πik θ q Tk f q ) 0 q p 1, (k Z d ); They satisfy the commutation relation D (f q ) 0 q p 1 = (D f (σ ) 1 (q)) 0 q p 1. (3.1) D T M k = T kd, (k Z d ). The two affine structures {D,T P 1 k} and {D,T k } are conjugate. Define the unitary operator U P on L 2 (R d ) by U P f(x) = pf(px). Then U P D = DU P, U P T k = T P 1 ku P, (k Z d ).

5 OVERSAMPLING GENERATES SUPER-WAVELETS 5 Then Define the unitary operator U P on H Also if then S is an isometry and This shows that U P (f 0,..., f p 1 ) = (U P f 0,..., U P f p 1 ). U P D = DU P, U P T k = T P 1 ku P, (k Z d ). S f(x) = 1 p (f(p 1 x),..., f(p 1 x)), (f L 2 (R d )) U P S f = Sf, f L 2 (R). U P D j T k S f = D j T P 1 ksf, (j Z, k Z d, f L 2 (R d )). Therefore it is enough to prove that X(S Ψ) := {D j T ks ψ i j Z, k Z d, i {1,..., n}} is a frame with the same frame bounds. Lemma 3.2. Each f H can be written uniquely as for some f r L 2 (R d ). p 1 g = T Pθ r S f r Proof. Let g = (g q ) q H. We want ( p 1 ) (g q (x)) q = e 1 2πiPθr θ q p f r(p 1 (x Pθ r )) Equivalently, in matrix form, r=0 r=0 (g q (x)) q = H t ( 1 p f r (P 1 (x Pθ r ))) r. Using lemma 3.1, we get ( 1 p f r (P 1 (x Pθ r ))) r = (H t ) 1 (g q (x)) q, q and this uniquely determines f r, and f r L 2 (R d ). Lemma 3.3. For f, g L 2 (R d ), k Z d, j Z and r {0,..., p 1}, we have D j T ks f T Pθ r g = δ l,σ j (r) D j T P 1 kf T θr g, where l {0,..., p 1} is determined by k = Pθ l + Pm, with m Z d. Proof. First compute ( 1 (D j T ks f)(x) = Therefore e2πik θ p D j T k S f T Pθ r g p 1 = ) (σ ) (q) j (det M ) j f(p 1 (M j x k)) q=0 1 e2πi(k θ ) R p 2 (σ ) j (q) Pθr θ q d. q

6 6 DORIN ERVIN DUTKAY AND PALLE JORGENSEN But (det M ) j f(p 1 (M j x k))g(p 1 (x Pθ r ))dx. p 1 1 e 2πi(k θ (σ ) j (q) Pθr θ q ) = 1 p p q=0 p 1 e 2πi(Pθ l θ (σ ) j (q) Pθr θ q ) = q=0 1 p 1 e 2πi(Pθ σ j (l) θ q Pθr θ q ) = δ p σ j (l),r = δ l,σ j (r). q=0 We used Lemma 3.1 is the second to last equality. Then 1 (detm ) L 2 (R d ) p j f(p 1 (M j x k))g(p 1 (x Pθ r ))dx = 1 (detm)j f(m j P 1 x P 1 k)g(p 1 x θ r )dx = L 2 (R d ) p (det M)j f(m j y P 1 k)g(y θ r )dx = D j T P 1 kf T θr g. L 2 (R d ) This proves the lemma. Lemma 3.3 and Lemma 3.2 show that the Hilbert space H can be written as an orthogonal sum H = H 0 H 1... H p 1, H r = T rs L 2 (R d ). Moreover the set X(S Ψ) splits into p mutually orthogonal subsets X r (S Ψ) := {D j T Pθ σ j (r) +Pm S ψ i j Z, m Z d, i {1,..., n}} H r. Also note, as another consequence of Lemma 3.3, that the unitary D J maps H r onto H σ J (r). Therefore it is enough to show that X r (S Ψ) is a frame with the same frame bounds for H r. We have, with Lemma 3.3, for f L 2 (R d ): (3.2) i,j,m D j T Pθ σ j (r) +Pm S ψ i T Pθ r S f 2 = i,j,m D j T θσ j (r) +mψ i T θr f 2. But according to the proof of Theorem 3.2 in [8], this quantity is bigger than A T θr 2 = A f 2 = A T Pθr S f 2. This yields the lower bound. For the upper bound, we have, using (3.1): Q := i,m j 0 i,m j 0 i,m j 0 D j T Pθ σ j (r) +Pm S ψ i T Pθ r S f D j T Pθ σ j (r) M j Pθr +Pm S ψ i S f 2 = 2 = D j T Pθ σ j (r) PM j θ r+pm S ψ i S f 2 But Pθ σ j (r) PM j θ r = Ph for some h Z d, therefore, with Lemma 3.3, Q = D j T Pm S ψ i S f 2 = D j T m ψ i f 2 B f 2 = B T Pθr S f 2. i,m i,m j 0 j 0

7 OVERSAMPLING GENERATES SUPER-WAVELETS 7 Now take J 0. Using the fact that D J permutes the subspaces H r, we can write D J T Pθ r S f = T Pθ σ J (r) S f for some f L 2 (R d ), with f = f. Then i,m j 0 i,m j J D j T Pθ σ j (r) +PmS ψ i T Pθ r S f 2 = D j T Pθ σ j (r) +Pm S ψ i T Pθ σ J (r) S f 2 B f = B T Pθ r S f 2. Letting J we obtain the upper bound. This proves the first statement of the theorem. If X(Ψ) is an orthonormal basis then X(SΨ) is a frame with bounds A = B = 1. Also the norm of the SΨ is 1, so X(SΨ) is an orthonormal basis. For the converse, take r = 0 in (3.2). We know that X 0 (S Ψ) is a frame for H 0, because the families X r (S Ψ) are mutually orthogonal. Then we use (3.2) and conclude that X(Ψ) is also a frame with the same frame bounds. The projection onto the first component P 0 corresponds to θ0 = 0. Since σ (θ0 ) = θ 0, we obtain P 0(X(SΨ)) = X P (Ψ). This concludes the proof of Theorem 2.1. Note that from the proof of Theorem 2.1, and equation (3.2), we obtain also the following corollary Corollary 3.4. If X(ψ) is a frame for L 2 (R) then for all r {0,..., p 1} {D j T θσ j (r) +mψ i j Z, m Z d, i {1,..., n}} is a frame with the same bounds for L 2 (R d ). Remark 3.5. In our discussion above, we have stressed uniform sampling, i.e., the case when the sample points are located on a suitably chosen lattice in R d, and we then analyzed refinements of lattices as an instance of oversampling. But the operator theory going into our method applies also to the more general and perhaps more interesting case of non-uniform case, also called irregular sampling; see e.g., [9], and [2] for related modern results. Since the results are more clean in the case of lattice-refinement, we have stated our theorem in this context. Acknowledgements. The authors are pleased to thank Professor Akram Aldroubi for enlightening discussions about sampling. Both authors were supported by a grant from the National Science Foundation. References 1. Akram Aldroubi, Karlheinz Gröchenig, Nonuniform sampling and reconstruction in shiftinvariant spaces. SIAM Rev. 43 (2001), no. 4, John J. Benedetto, Paulo J. S. G. Ferreira. Modern sampling theory. Mathematics and applications. Applied and Numerical Harmonic Analysis. Birkhuser Boston, Inc., Boston, MA, xvi+417 pp. ISBN: O. Christensen, An introduction to frames and Riesz bases. Applied and Numerical Harmonic Analysis. Birkhuser Boston, Inc., Boston, MA, xxii+440 pp. 4. C. Chui, W.Czaja, M. Maggioni, G. Weiss, Characterization of tight-frame wavelets with arbitrary dilation and general tightness preserving oversampling, J. Fourier Anal. Appl., 8 (2002), pp

8 8 DORIN ERVIN DUTKAY AND PALLE JORGENSEN 5. C. Chui, X. Shi, n oversampling preserves any tight-affine frame for odd n, Proc. Amer. Math. Soc., 121 (1994), pp I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Regional Conf. Ser. in Appl. Math., vol. 61, Society for Industrial and Applied Mathematics, Philadelphia, D.E. Dutkay, The local trace function for super-wavelets. Wavelets, frames and operator theory, , Contemp. Math., B.D. Johnson, On the oversampling of affine wavelet frames, SIAM J. Math. Anal. 35 (2003), no. 3, P.E.T. Jorgensen, An optimal spectral estimator for multidimensional time series with an infinite number of sample points. Math. Z. 183 (1983), no. 3, R. Laugesen, Translational averaging for completeness, characterization, and oversampling of wavelets, Collect. Math., 53 (2002), pp A. Ron, Z. Shen, Affine systems in L 2 (R d ): The analysis of the analysis operator, J. Funct. Anal. 148 (1997), pp Department of Mathematics, Hill Center-Busch Campus, Rutgers, The State University of New Jersey, 110 Frelinghuysen Rd, Piscataway, NJ , USA address: ddutkay@math.rutgers.edu Department of Mathematics, The University of Iowa, 14 MacLean Hall, Iowa City, IA, 52242, USA address: jorgen@math.uiowa.edu

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