Results 1  10
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11
Framelets: MRABased Constructions of Wavelet Frames
, 2001
"... We discuss wavelet frames constructed via multiresolution analysis (MRA), with emphasis on tight wavelet frames. In particular, we establish general principles and specific algorithms for constructing framelets and tight framelets, and we show how they can be used for systematic constructions of spl ..."
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Cited by 127 (50 self)
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We discuss wavelet frames constructed via multiresolution analysis (MRA), with emphasis on tight wavelet frames. In particular, we establish general principles and specific algorithms for constructing framelets and tight framelets, and we show how they can be used for systematic constructions of spline, pseudospline tight frames and symmetric biframes with short supports and high approximation orders. Several explicit examples are discussed. The connection of these frames with multiresolution analysis guarantees the existence of fast implementation algorithms, which we discuss briefly as well.
The Construction Of Single Wavelets In DDimensions
 J. GEOM. ANAL
, 1999
"... Sets K in ddimensional Euclidean space are constructed with the property that the inverse Fourier transform of the characteristic function 1K is a single dyadic orthonormal wavelet. The construction is characterized by its generality in terms of a quantitative iterative procedure, by its computat ..."
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Cited by 16 (2 self)
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Sets K in ddimensional Euclidean space are constructed with the property that the inverse Fourier transform of the characteristic function 1K is a single dyadic orthonormal wavelet. The construction is characterized by its generality in terms of a quantitative iterative procedure, by its computational implementation, and by its simplicity. The general case in which the inverse Fourier transforms of the characteristic functions 1 K 1 ; : : : ; 1 K L are a family of orthonormal wavelets is treated in [Leo99].
Generalized shiftinvariant systems and frames for subspaces
, 2005
"... Let Tk denote translation by k ∈ Zd. Given countable collections of functions {φj}j∈J, { φj ˜}j∈J ⊂ L2 (Rd) and assuming that {Tkφj} j∈J,k∈Zd and {Tk ˜ φj} j∈J,k∈Zd are Bessel sequences, we are interested in expansions f = ∑ ∑ j∈J k∈Zd 〈 f, Tk ˜φj Tkφj, ∀f ∈ span {Tkφj} k∈Zd,j∈J. Our main result ..."
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Cited by 15 (14 self)
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Let Tk denote translation by k ∈ Zd. Given countable collections of functions {φj}j∈J, { φj ˜}j∈J ⊂ L2 (Rd) and assuming that {Tkφj} j∈J,k∈Zd and {Tk ˜ φj} j∈J,k∈Zd are Bessel sequences, we are interested in expansions f = ∑ ∑ j∈J k∈Zd 〈 f, Tk ˜φj Tkφj, ∀f ∈ span {Tkφj} k∈Zd,j∈J. Our main result gives an equivalent condition for this to hold in a more general setting than described here, where translation by k ∈ Z d is replaced by translation via the action of a matrix. As special cases of our result we find conditions for shiftinvariant systems, Gabor systems, and wavelet systems to generate a subspace frame with a corresponding dual having the same structure.
A Characterization Of Dimension Functions Of Wavelets
, 1999
"... . This paper is devoted to the study of the dimension functions of (multi)wavelets, which was introduced and investigated by P. Auscher in [1]. Our main result provides a characterization of functions which are dimension functions of a (multi)wavelet. As a corollary, we obtain that for every functio ..."
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Cited by 10 (3 self)
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. This paper is devoted to the study of the dimension functions of (multi)wavelets, which was introduced and investigated by P. Auscher in [1]. Our main result provides a characterization of functions which are dimension functions of a (multi)wavelet. As a corollary, we obtain that for every function D that is the dimension function of a (multi)wavelet, there is an MSF (multi)wavelet whose dimension function is D. In addition, we show that if a dimension function of a wavelet not associated with an MRA attains the value K, then it attains all integer values from zero to K. Moreover, we prove that every expansive matrix which preserves Z N admits an MRA structure with an analytic (multi)wavelet. 2 1. Introduction The dimension function of an orthonormal wavelet 2 L 2 (R) is dened as D () = 1 X j=1 X k2Z j ^ (2 j ( + k))j 2 : The importance of the dimension function was discovered by P. G. Lemarie, who used it to prove that certain wavelets are associated with ...
Generalized Shift Invariant Systems
"... A countable collection X of functions in L 2 (IR ) is said to be a Bessel system if the associated analysis operator T # X : L 2 (IR (#f, x#) x#X is welldefined and bounded. A Bessel system is a fundamental frame if T # X is injective and its range is closed. ..."
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Cited by 8 (2 self)
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A countable collection X of functions in L 2 (IR ) is said to be a Bessel system if the associated analysis operator T # X : L 2 (IR (#f, x#) x#X is welldefined and bounded. A Bessel system is a fundamental frame if T # X is injective and its range is closed.
Wavelets On General Lattices, Associated With General Expanding Maps Of R^n
 of R n . AMS Research Announcements
, 1999
"... . In the context of a general lattice \Gamma in R n and a strictly expanding map M which preserves the lattice, we characterize all the wavelet families, all the MSF wavelets, all the multiwavelets associated with a Multiresolution Analysis (MRA) of multiplicity d 1; and all the scaling functi ..."
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Cited by 6 (1 self)
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. In the context of a general lattice \Gamma in R n and a strictly expanding map M which preserves the lattice, we characterize all the wavelet families, all the MSF wavelets, all the multiwavelets associated with a Multiresolution Analysis (MRA) of multiplicity d 1; and all the scaling functions. Moreover, we give several examples: in particular, we construct a single, MRA and C 1 (R n ) wavelet, which is nonseparable and with compactly supported Fourier transform. 1. Introduction An orthonormal wavelet is a function / 2 L 2 (R) such that the set \Phi / j;k j 2 j=2 /(2 j x \Gamma k) : j; k 2 Zg (1) is an orthonormal basis for L 2 (R): A complete characterization of these wavelets is given by the equations a) X j2Z j b /(2 j )j 2 = 1 a.e. 2 R; b) 1 X j=0 b /(2 j ) b /(2 j ( + 2k)) = 0 a.e. 2 R; k 2 2Z + 1; (I) together with the assumption k/k 2 1: These two equations have been known since the beginning of the theory of wavelets (see [L1] ...
Translational averaging for completeness, characterization, and oversampling of wavelets, Collect
 Math
"... The single underlying method of “averaging the wavelet functional over translates” yields first a new completeness criterion for orthonormal wavelet systems, and then a unified treatment of known results on characterization of wavelets on the Fourier transform side, on preservation of frame bounds b ..."
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Cited by 6 (2 self)
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The single underlying method of “averaging the wavelet functional over translates” yields first a new completeness criterion for orthonormal wavelet systems, and then a unified treatment of known results on characterization of wavelets on the Fourier transform side, on preservation of frame bounds by oversampling, and on the equivalence of affine and quasiaffine frames. The method applies to multiwavelet systems in all dimensions, to dilation matrices that are in some cases not expanding, and to dual frame pairs. The completeness criterion we establish is precisely the discrete Calderón condition. In the single wavelet case this means we take invertible matrices a and b and a function ψ ∈ L 2 (R d), and assume either a is expanding or else a is amplifying for ψ. We prove that the system {  det a  j/2 ψ(a j x − bk):j ∈ Z,k ∈ Z d} is an orthonormal basis for L 2 (R d) if and only if it is orthonormal and ∑ j∈Z  ˆ ψ(ξa j)  2 =  det b  for almost every row vector ξ ∈ R d. 1.
PAIRS OF FREQUENCYBASED NONHOMOGENEOUS DUAL WAVELET FRAMES IN THE DISTRIBUTION SPACE
"... Abstract. In this paper, we study stationary and nonstationary nonhomogeneous dual wavelet frames with an arbitrary real dilation factor in the frequency domain by introducing and investigating a pair of frequencybased nonhomogeneous dual wavelet frames in the distribution space. This notion of a p ..."
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Cited by 3 (3 self)
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Abstract. In this paper, we study stationary and nonstationary nonhomogeneous dual wavelet frames with an arbitrary real dilation factor in the frequency domain by introducing and investigating a pair of frequencybased nonhomogeneous dual wavelet frames in the distribution space. This notion of a pair of frequencybased nonhomogeneous dual wavelet frames in the distribution space enables us to completely separate its perfect reconstruction property from its stability property in function spaces. The results in this paper lead to a natural explanation for the oblique extension principle for constructing dual wavelet frames from refinable functions without any a priori condition on the generating wavelet functions and refinable functions. A pair of frequencybased nonhomogeneous dual wavelet frames in the distribution space, that is not necessarily derived from refinable functions via a multiresolution analysis, has a natural multiresolutionlike structure, which is closely linked to the fast wavelet frame transform. Moreover, nonhomogeneous dual wavelet frames in the distribution space play a basic role in understanding dual wavelet frames in various function spaces and have a close relation to nonstationary dual wavelet frames, which are of interest in applications. To illustrate the flexibility and generality of the results in this paper, we characterize a pair of fully nonstationary dual wavelet frames in the distribution space. Our results naturally link a nonstationary dual wavelet frame filter bank having the perfect reconstruction property to a pair of nonstationary dual wavelet frames in the distribution space.
A characterization of wavelet families arising from biorthogonal MRA's of multiplicity d
"... In this paper we give a necessary and sufficient condition for a pair of wavelet families \Psi = f/ 1 ; : : : ; / L g; e \Psi = f ~ / 1 ; : : : ; ~ / L g; in L 2 (R n ), to arise from a pair of biorthogonal MRA's. The condition is given in terms of simple equations involving the function ..."
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Cited by 2 (1 self)
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In this paper we give a necessary and sufficient condition for a pair of wavelet families \Psi = f/ 1 ; : : : ; / L g; e \Psi = f ~ / 1 ; : : : ; ~ / L g; in L 2 (R n ), to arise from a pair of biorthogonal MRA's. The condition is given in terms of simple equations involving the functions / ` and ~ / ` . To work in greater generality, we allow multiresolution analyses of arbitrary multiplicity, based on lattice translations and matrix dilations. Our result extends the characterization theorem of G. Gripenberg and X. Wang for dyadic orthonormal wavelets in L 2 (R), and includes, as particular cases, the sufficient conditions of P. Auscher and P.G. Lemari'e in the biorthogonal situation. 1
STABILITY OF WAVELET FRAMES WITH MATRIX DILATIONS
, 2005
"... Under certain assumptions we show that a wavelet frame {τ(Aj,bj,k)ψ}j,k∈Z: = {det Aj  −1/2 ψ(A −1 j (x − bj,k))}j,k∈Z in L 2 (R d) remains a frame when the dilation matrices Aj and the translation parameters bj,k are perturbed. As a special case of our result, we obtain that if {τ(A j,A j Bn)ψ} j ..."
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Cited by 2 (0 self)
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Under certain assumptions we show that a wavelet frame {τ(Aj,bj,k)ψ}j,k∈Z: = {det Aj  −1/2 ψ(A −1 j (x − bj,k))}j,k∈Z in L 2 (R d) remains a frame when the dilation matrices Aj and the translation parameters bj,k are perturbed. As a special case of our result, we obtain that if {τ(A j,A j Bn)ψ} j∈Z,n∈Z d is a frame for an expansive matrix A andaninvertiblematrixB, then{τ(A ′ j,Aj Bλn)ψ} j∈Z,n∈Z d is a frame if �A −j A ′ j − I�2 ≤ ε and �λn − n� ∞ ≤ η for sufficiently small ε, η> 0.