A non-MRA $$C^r$$ Frame Wavelet with Rapid Decay

Baggett, Larry; Jørgensen, Palle E. T.; Merrill, Kathy D.; Packer, Judith

doi:10.1007/s10440-005-9011-4

Cited by 24 publications

(24 citation statements)

References 26 publications

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“…This representation theoretic approach, adapted below to our present applications, fits best with how we use the Spectral Representation Theorem in understanding wavelets and generalized multiresolution analysis (GMRAs); see also [Bag00,BJMP05,BMM99]. The standard presentation of the Spectral Theorem in textbooks is typically different from the Spectral Representation Theorem, and here we spell out the connection; developing here a computational approach.…”

Section: Unitary Operatorsmentioning

confidence: 94%

Optimal Decompositions of Translations of L 2-Functions

Jørgensen

Song

2008

Complex anal.oper. theory

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Abstract. In this paper we offer a computational approach to the spectral function for a finite family of commuting operators, and give applications. Motivated by questions in wavelets and in signal processing, we study a problem about spectral concentration of integral translations of functions in the Hilbert space L 2 (R n ). Our approach applies more generally to families of n arbitrary commuting unitary operators in a complex Hilbert space H, or equivalent the spectral theory of a unitary representation U of the rank-n lattice Z n in R n . Starting with a non-zero vector ψ ∈ H, we look for relations among the vectors in the cyclic subspace in H generated by ψ. Since these vectors {U (k)ψ|k ∈ Z n } involve infinite "linear combinations," the problem arises of giving geometric characterizations of these non-trivial linear relations. A special case of the problem arose initially in work of Kolmogorov under the name L 2 -independence. This refers to infinite linear combinations of integral translates of a fixed function with l 2 -coefficients. While we were motivated by the study of translation operators arising in wavelet and frame theory, we stress that our present results are general; our theorems are about spectral densities for general unitary operators, and for stochastic integrals.Mathematics Subject Classification (2000). Primary 47B40, 47B06, 06D22, 62M15; Secondary 42C40, 62M20.

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Section: Unitary Operatorsmentioning

confidence: 94%

Optimal Decompositions of Translations of L 2-Functions

Jørgensen

Song

2008

Complex anal.oper. theory

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“…As such it is a natural problem to construct single wavelets with better temporal decay. Further, even on R, in order improve the temporal decay, one must consider systems of frames rather than orthonormal bases [4,10,22,23] or wavelets which have an MRA structure [25,26]. We shall address the problem of smoothing ψ by convolution, where ψ is derived by the so-called neighborhood mapping method, see Sect.…”

Section: Problemmentioning

confidence: 99%

“…A different smoothing idea is employed in [4]. The authors smooth the 1-d Journé wavelet using a Generalized Multiresolution Analysis.…”

Section: Baggett Jorgensen Merrill and Packer Smoothingmentioning

confidence: 99%

Smooth Functions Associated with Wavelet Sets on ℝ d , d≥1, and Frame Bound Gaps

Benedetto

King

2009

Acta Appl Math

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The theme is to smooth characteristic functions of Parseval frame wavelet sets by convolution in order to obtain implementable, computationally viable, smooth wavelet frames. We introduce the following: a new method to improve frame bound estimation; a shrinking technique to construct frames; and a nascent theory concerning frame bound gaps. The phenomenon of a frame bound gap occurs when certain sequences of functions, converging in L 2 to a Parseval frame wavelet, generate systems with frame bounds that are uniformly bounded away from 1. We prove that smoothing a Parseval frame wavelet set wavelet on the frequency domain by convolution with elements of an approximate identity produces a frame bound gap. Furthermore, the frame bound gap for such frame wavelets in L 2 (R d ) increases and converges as d increases.

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“…This fact does not hold for tight frame wavelets. In fact, Baggett et al [2] constructed a non-MRA C r tight frame wavelet with rapid decay for any r ∈ N. Moreover, their tight frame wavelet is associated with a GMRA having the same dimension/multiplicity function as the Journé wavelet. Once we allow non-tight frame wavelets we might lose even the GMRA property.…”

Section: Remarkmentioning

confidence: 99%

The canonical and alternate duals of a wavelet frame

Bownik

Lemvig

2007

Applied and Computational Harmonic Analysis

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We show that there exists a frame wavelet ψ with fast decay in the time domain and compact support in the frequency domain generating a wavelet system whose canonical dual frame cannot be generated by an arbitrary number of generators. On the other hand, there exists infinitely many alternate duals of ψ generated by a single function. Our argument closes a gap in the original proof of this fact by Daubechies and Han [The canonical dual frame of a wavelet frame, Appl. Comput. Harmon. Anal. 12 (3) (2002) 269-285].

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A non-MRA $$C^r$$ Frame Wavelet with Rapid Decay

Cited by 24 publications

References 26 publications

Optimal Decompositions of Translations of L 2-Functions

Optimal Decompositions of Translations of L 2-Functions

Smooth Functions Associated with Wavelet Sets on ℝ d , d≥1, and Frame Bound Gaps

The canonical and alternate duals of a wavelet frame

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