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Density, overcompleteness, and localization of frames
 I. THEORY, J. FOURIER ANAL. APPL
, 2005
"... This work presents a quantitative framework for describing the overcompleteness of a large class of frames. It introduces notions of localization and approximation between two frames F = {fi}i∈I and E = {ej}j∈G (G a discrete abelian group), relating the decay of the expansion of the elements of F in ..."
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Cited by 62 (20 self)
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This work presents a quantitative framework for describing the overcompleteness of a large class of frames. It introduces notions of localization and approximation between two frames F = {fi}i∈I and E = {ej}j∈G (G a discrete abelian group), relating the decay of the expansion of the elements of F in terms of the elements of E viaamapa: I → G. A fundamental set of equalities are shown between three seemingly unrelated quantities: the relative measure of F, the relative measure of E—both of which are determined by certain averages of inner products of frame elements with their corresponding dual frame elements—and the density of the set a(I) inG. Fundamental new results are obtained on the excess and overcompleteness of frames, on the relationship between frame bounds and density, and on the structure of the dual frame of a localized frame. These abstract results yield an array of new implications for irregular Gabor frames. Various Nyquist density results for Gabor frames are recovered as special cases, but in the process both their meaning and implications are clarified. New results are obtained on the excess and overcompleteness of Gabor frames, on the relationship between frame bounds and density, and on the structure of the dual frame of an irregular Gabor frame. More generally, these results apply both to Gabor frames and to systems of Gabor molecules, whose elements share only a common envelope of concentration in the timefrequency plane.
TIMEFREQUENCY ANALYSIS OF SJÖSTRAND’S CLASS
, 2004
"... We investigate the properties an exotic symbol class of pseudodifferential operators, Sjöstrand’s class, with methods of timefrequency analysis (phase space analysis). Compared to the classical treatment, the timefrequency approach leads to striklingly simple proofs of Sjöstrand’s fundamental resu ..."
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Cited by 61 (14 self)
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We investigate the properties an exotic symbol class of pseudodifferential operators, Sjöstrand’s class, with methods of timefrequency analysis (phase space analysis). Compared to the classical treatment, the timefrequency approach leads to striklingly simple proofs of Sjöstrand’s fundamental results and to farreaching generalizations.
Adaptive frame methods for elliptic operator equations: the steepest descent approach
, 2007
"... ..."
Wiener’s lemma for infinite matrices
 Trans. Amer. Math. Soc
, 2006
"... Abstract. The classical Wiener lemma and its various generalizations are important and have numerous applications in numerical analysis, wavelet theory, frame theory, and sampling theory. There are many different equivalent formulations for the classical Wiener lemma, with an equivalent formulation ..."
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Cited by 31 (15 self)
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Abstract. The classical Wiener lemma and its various generalizations are important and have numerous applications in numerical analysis, wavelet theory, frame theory, and sampling theory. There are many different equivalent formulations for the classical Wiener lemma, with an equivalent formulation suitable for our generalization involving commutative algebra of infinite matrices W: = {(a(j − j ′)) j,j ′ ∈Zd: � j∈Zd a(j)  < ∞}. In the study of spline approximation, (diffusion) wavelets and affine frames, Gabor frames on nonuniform grid, and nonuniform sampling and reconstruction, the associated algebras of infinite matrices are extremely noncommutative, but we expect those noncommutative algebras to have a similar property to Wiener’s lemma for the commutative algebra W. In this paper, we consider two noncommutative algebras of infinite matrices, the Schur class and the Sjöstrand class, and establish Wiener’s lemmas for those matrix algebras. 1.
Modulation Spaces: Looking Back and Ahead
 SAMPL. THEORY SIGNAL IMAGE PROCESS
, 2006
"... This note provides historical perspectives and background on the motivations which led to the invention of the modulation spaces by the author almost 25 years ago, as well as comments about their role for ongoing research efforts within timefrequency analysis. We will also describe the role of mo ..."
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Cited by 26 (2 self)
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This note provides historical perspectives and background on the motivations which led to the invention of the modulation spaces by the author almost 25 years ago, as well as comments about their role for ongoing research efforts within timefrequency analysis. We will also describe the role of modulation spaces within the more general coorbit theory developed jointly with Karlheinz Gröchenig, and which eventually led to the development of the concept of Banach frames and more recently to the socalled localization theory for frames. A comprehensive bibliography is included.
Nonuniform average sampling and reconstruction in multiply generated shiftinvariant spaces
 Constr. Approx
"... Abstract. From an average (ideal) sampling/reconstruction process, the question arises whether and how the original signal can be recovered from its average (ideal) samples. We consider the above question under the assumption that the original signal comes from a prototypical space modelling signals ..."
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Cited by 23 (12 self)
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Abstract. From an average (ideal) sampling/reconstruction process, the question arises whether and how the original signal can be recovered from its average (ideal) samples. We consider the above question under the assumption that the original signal comes from a prototypical space modelling signals with finite rate of innovation, which includes finitelygenerated shiftinvariant spaces, twisted shiftinvariant spaces associated with Gabor frames and Wilson bases, and spaces of polynomial splines with nonuniform knots as its special cases. We show that the displayer associated with an average (ideal) sampling/reconstruction process, that has welllocalized average sampler, can be found to be welllocalized. We prove that the reconstruction process associated with an average (ideal) sampling process is robust, locally behaved, and finitely implementable, and thus we conclude that the original signal can be approximately recovered from its incomplete average (ideal) samples with noise in real time. Most of our results in this paper are new even for the special case that the original signal comes from a finitelygenerated shiftinvariant space. average sampling, ideal sampling, signals with finite rate of innovation, shiftKey words. invariant spaces
Pseudodifferential operators on locally compact abelian groups and Sjöstrand’s symbol class
 J. Reine Angew. Math
, 2006
"... We investigate pseudodifferential operators on arbitrary locally compact abelian groups. As symbol classes for the KohnNirenberg calculus we introduce a version of Sjöstrand’s class. Pseudodifferential operators with such symbols form a Banach algebra that is closed under inversion. Since “hard ana ..."
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Cited by 21 (2 self)
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We investigate pseudodifferential operators on arbitrary locally compact abelian groups. As symbol classes for the KohnNirenberg calculus we introduce a version of Sjöstrand’s class. Pseudodifferential operators with such symbols form a Banach algebra that is closed under inversion. Since “hard analysis ” techniques are not available on locally compact abelian groups, a new timefrequency approach is used with the emphasis on modulation spaces, Gabor frames, and Banach algebras of matrices. Sjöstrand’s original results are thus understood as a phenomenon of abstract harmonic analysis rather than “hard analysis ” and are proved in their natural context and generality. 1
Frames in spaces with finite rate of innovations
 Adv. Comput. Math
"... Abstract. Signals with finite rate of innovation are those signals having finite degrees of freedom per unit of time that specify them. In this paper, we introduce a prototypical space Vq(Φ, Λ) modelling signals with finite rate of innovation, such as stream of (different) pulses found in GPS applic ..."
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Cited by 20 (14 self)
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Abstract. Signals with finite rate of innovation are those signals having finite degrees of freedom per unit of time that specify them. In this paper, we introduce a prototypical space Vq(Φ, Λ) modelling signals with finite rate of innovation, such as stream of (different) pulses found in GPS applications, cellular radio and ultra wideband communication. In particular, the space Vq(Φ, Λ) is generated by a family of welllocalized molecules Φ of similar size located on a relativelyseparated set Λ using ℓ q coefficients, and hence is locally finitelygenerated. Moreover that space Vq(Φ, Λ) includes finitelygenerated shiftinvariant spaces, spaces of nonuniform splines, and the twisted shiftinvariant space in Gabor (Wilson) system as its special cases. Use the welllocalization property of the generator Φ, we show that if the generator Φ is a frame for the space V2(Φ, Λ) and has polynomial (subexponential) decay, then its canonical dual (tight) frame has the same polynomial (subexponential) decay. We apply the above result about the canonical dual frame to the study of the Banach frame property of the generator Φ for the space Vq(Φ, Λ) with q = 2, and of the polynomial (subexponential) decay property of the mask associated with a refinable function that has polynomial (subexponential) decay. Advances in Computational Mathematics, to appear 1.
Generalized coorbit theory, Banach frames, and the relation to αmodulation spaces
 Proceedings of the London Mathematical Society
, 2008
"... This paper is concerned with generalizations and specific applications of the coorbit space theory based on group representations modulo quotients that has been developed quite recently. We show that the general theory applied to the affine Weyl–Heisenberg group gives rise to families of smoothness ..."
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Cited by 19 (8 self)
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This paper is concerned with generalizations and specific applications of the coorbit space theory based on group representations modulo quotients that has been developed quite recently. We show that the general theory applied to the affine Weyl–Heisenberg group gives rise to families of smoothness spaces that can be identified with αmodulation spaces. Key Words: Square integrable group representations, time–frequency analysis, atomic decompositions, (Banach) frames, homogeneous spaces, weighted coorbit
Stability of Localized Operators
, 2008
"... Let ℓ p,1 ≤ p ≤ ∞, be the space of all psummable sequences and Ca be the convolution operator associated with a summable sequence a. It is known that the ℓ p stability of the convolution operator Ca for different 1 ≤ p ≤ ∞ are equivalent to each other, i.e., if Ca has ℓ pstability for some 1 ≤ p ..."
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Cited by 18 (9 self)
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Let ℓ p,1 ≤ p ≤ ∞, be the space of all psummable sequences and Ca be the convolution operator associated with a summable sequence a. It is known that the ℓ p stability of the convolution operator Ca for different 1 ≤ p ≤ ∞ are equivalent to each other, i.e., if Ca has ℓ pstability for some 1 ≤ p ≤ ∞ then Ca has ℓ qstability for all 1 ≤ q ≤ ∞. In the study of spline approximation, wavelet analysis, timefrequency analysis, and sampling, there are many localized operators of nonconvolution type whose stability is one of the basic assumptions. In this paper, we consider the stability of those localized operators including infinite matrices in the Sjöstrand class, synthesis operators with generating functions enveloped by shifts of a function in the Wiener amalgam space, and integral operators with kernels having certain regularity and decay at infinity. We show that the ℓ p stability (or L pstability) of those three classes of localized operators are equivalent to each other, and we also prove that the left inverse of those localized operators are well localized.