Quadrature Formulae and Asymptotic Error Expansions for Wavelet Approximations of Smooth Functions

Sweldens, Wim; Piessens, Robert

doi:10.1137/0731065

Cited by 151 publications

(143 citation statements)

References 13 publications

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“…Let X c l2 denote the range of p(-, 4>). Because {<t>n}nez is a frame, the operator p(-, 4>) satisfies (14). The left inequality of (14) implies that p(-, <P) is injective and bounded below by the positive constant A.…”

Section: Mappings On L2 For the Construction Of Frames Of %Fmentioning

confidence: 97%

Portraits of Frames

Aldroubi¹

1995

Proceedings of the American Mathematical Society

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Abstract. We introduce two methods for generating frames of a Hubert space &. The first method uses bounded operators on ^ . The other method uses bounded linear operators on 1% to generate frames of %? . We characterize all the mappings that transform frames into other frames. We also show how to construct all frames of a given Hubert space &, starting from any given one. We illustrate the results by giving some examples from multiresolution and wavelet theory.

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Section: Mappings On L2 For the Construction Of Frames Of %Fmentioning

confidence: 97%

Portraits of Frames

Aldroubi¹

1995

Proceedings of the American Mathematical Society

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“…For general wavelet families this integration can be difficult and computationally expensive. 6 The interpolating wavelets discussed above are the glorious exception. Since both the dual scaling function and the dual wavelets are delta functions, 4 one or just a few data points suffice to do the integration exactly.…”

Section: Expanding Functions In a Wavelet Basismentioning

confidence: 99%

“…(6) Note that in both cases we need exactly 16 coefficients to represent the function. By doing a backward wavelet transform, we can go back to the original expansion of Eq.…”

Section: The Haar Wavelet Familymentioning

confidence: 99%

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Solution of Multiscale Partial Differential Equations Using Wavelets

Goedecker¹,

Иванов²

1998

Computers in Physics

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“…Approximation power. Given any function u ∈ L 2 (R), it can be approximated by the projection, this approximation has the following sharp estimate [17,18,19]:where Table 9 Compression ratio C 2 for matrix which converts the finite Chebyshev expansion to finite Legendre expansion. The matrix size is 1024 × 1024.…”

mentioning

confidence: 99%

Difference Wavelet - Theory and A Comparison Study

Chern¹,

Yen²

2002

Methods and Applications of Analysis

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Abstract. Wavelet methods with polynomial filters are usually favored in applications for their fast wavelet transforms and compact support. However, wavelet methods with rational filters have more freedom to achieve smaller condition numbers, more regularity and better efficiency. Such methods can be attractive if they also possess fast algorithms and have fast decay (as if the corresponding wavelets had compact support). In the first part of this paper, we propose a new wavelet method with rational filters which do have these properties. We call it the difference wavelet method. It is a generalization of Butterworth wavelets. The analysis part is simply averaging and finite differencing. The wavelet coefficients measure the finite differences of the averages of an input data sequence. Its synthesis part involves rational filters, which can be performed with linear computational complexity by the cyclic reduction method. Their Riesz basis property, biorthogonality, decay and regularity are investigated.In the second part of this paper, we perform comparison studies of the difference wavelet method (Diff) with three other popular wavelet methods: the Cohen-Daubechies-Feauveau biorthogonal wavelets (CDF), the Daubechies orthogonal wavelets (Daub) and the Chui-Wang semi-orthogonal wavelets (CW). Natural criteria in designing good wavelet methods for representing functions and operators are speed, stability and efficiency. Therefore, the items of our first comparison include (i) operation counts for performing transformations, (ii) condition numbers of the wavelet transformations, (iii) compression ratios, by some numerical experiments, for representing (smooth or non-smooth) data sequences and matrices (smooth or non-smooth kernels). The results show that (i) Diff, Daub and CDF have about the same operation counts, and CW has more; (ii) Diff has about the same condition numbers as those of CDF and CW; (iii) Diff has better compression ratio for both (smooth or non-smooth) data sequences and matrices (smooth or non-smooth kernels).The items of our second comparison include regularity, approximation power (the constant in the approximation estimate), approximation errors for non-smooth functions (where Gibbs phenomena appear) and the "essential supports." The results show that Diff has better regularity and better approximation ability with only slightly bigger essential supports. It is evident that the better efficiency of Diff for smooth functions is due to its regularity. It is surprising that, even for nonsmooth functions, Diff is comparable to, sometimes even superior to, other methods, despite its infinite-support property.This paper is organized as follows. Sec. 1 is preliminary. Sec. 2 provides the theory of the difference wavelet method. Sec. 3 contains the comparison studies. Experts are suggested to read Sec. 3 directly. Preliminary.A wavelet expansion method decomposes data (or functions) into fluctuations at various resolutions. It depends on four sets of filter coefficients: {h k } k∈Z , {g k } k∈Z , {...

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Quadrature Formulae and Asymptotic Error Expansions for Wavelet Approximations of Smooth Functions

Cited by 151 publications

References 13 publications

Portraits of Frames

Portraits of Frames

Solution of Multiscale Partial Differential Equations Using Wavelets

Difference Wavelet - Theory and A Comparison Study

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