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The Lifting Scheme: A Construction Of Second Generation Wavelets
, 1997
"... . We present the lifting scheme, a simple construction of second generation wavelets, wavelets that are not necessarily translates and dilates of one fixed function. Such wavelets can be adapted to intervals, domains, surfaces, weights, and irregular samples. We show how the lifting scheme leads to ..."
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Cited by 541 (16 self)
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. We present the lifting scheme, a simple construction of second generation wavelets, wavelets that are not necessarily translates and dilates of one fixed function. Such wavelets can be adapted to intervals, domains, surfaces, weights, and irregular samples. We show how the lifting scheme leads to a faster, inplace calculation of the wavelet transform. Several examples are included. Key words. wavelet, multiresolution, second generation wavelet, lifting scheme AMS subject classifications. 42C15 1. Introduction. Wavelets form a versatile tool for representing general functions or data sets. Essentially we can think of them as data building blocks. Their fundamental property is that they allow for representations which are efficient and which can be computed fast. In other words, wavelets are capable of quickly capturing the essence of a data set with only a small set of coefficients. This is based on the fact that most data sets have correlation both in time (or space) and frequenc...
Using the refinement equation for the construction of prewavelets II
, 1991
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On the construction of multivariate (pre)wavelets
, 1992
"... A new approach for the construction of wavelets and prewavelets on IR d from multiresolution is presented. The method uses only properties of shiftinvariant spaces and orthogonal projectors from L2(IR d) onto these spaces, and requires neither decay nor stability of the scaling function. Furthermo ..."
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Cited by 108 (14 self)
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A new approach for the construction of wavelets and prewavelets on IR d from multiresolution is presented. The method uses only properties of shiftinvariant spaces and orthogonal projectors from L2(IR d) onto these spaces, and requires neither decay nor stability of the scaling function. Furthermore, this approach allows a simple derivation of previous, as well as new, constructions of wavelets, and leads to a complete resolution of questions concerning the nature of the intersection and the union of a scale of spaces to be used in a multiresolution.
Multiresolution and wavelets
 Proc. Edinburgh Math. Soc
, 1994
"... Multiresolution is investigated on the basis of shiftinvariant spaces. Given a finitely generated shiftinvariant subspace S of L2(IR d), let Sk be the 2 kdilate of S (k ∈ Z). A necessary and sufficient condition is given for the sequence {Sk}k ∈ Z to form a multiresolution of L2(IR d). A general ..."
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Cited by 69 (32 self)
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Multiresolution is investigated on the basis of shiftinvariant spaces. Given a finitely generated shiftinvariant subspace S of L2(IR d), let Sk be the 2 kdilate of S (k ∈ Z). A necessary and sufficient condition is given for the sequence {Sk}k ∈ Z to form a multiresolution of L2(IR d). A general construction of orthogonal wavelets is given, but such wavelets might not have certain desirable properties. With the aid of the general theory of vector fields on spheres, it is demonstrated that the intrinsic properties of the scaling function must be used in constructing orthogonal wavelets with a certain decay rate. When the scaling function is skewsymmetric about some point, orthogonal wavelets and prewavelets are constructed in such a way that they possess certain attractive properties. Several examples are provided to illustrate the general theory.
Approximation error for quasiinterpolators and (multi)wavelet expansions
 APPL. COMPUT. HARMON. ANAL
, 1999
"... We investigate the approximation properties of general polynomial preserving operators that approximate a function into some scaled subspace of L² via an appropriate sequence of inner products. In particular, we consider integer shiftinvariant approximations such as those provided by splines and wa ..."
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Cited by 64 (22 self)
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We investigate the approximation properties of general polynomial preserving operators that approximate a function into some scaled subspace of L² via an appropriate sequence of inner products. In particular, we consider integer shiftinvariant approximations such as those provided by splines and wavelets, as well as finite elements and multiwavelets which use multiple generators. We estimate the approximation error as a function of the scale parameter T when the function to approximate is sufficiently regular. We then present a generalized sampling theorem, a result that is rich enough to provide tight bounds as well as asymptotic expansions of the approximation error as a function of the sampling step T. Another more theoretical consequence is the proof of a conjecture by Strang and Fix, which states the equivalence between the order of a multiwavelet space and the order of a particular subspace generated by a single function. Finally, we consider refinable generating functions and use the twoscale relation to obtain explicit formulae for the coefficients of the asymptotic development of the error. The leading constants are easily computable and can be the basis for the comparison of the approximation power of wavelet and multiwavelet expansions of a given order L.
Quadrature Formulae And Asymptotic Error Expansions For Wavelet Approximations Of Smooth Functions
 SIAM J. Numer. Anal
, 1994
"... . This paper deals with typical problems that arise when using wavelets in numerical analysis applications. The first part involves the construction of quadrature formulae for the calculation of inner products of smooth functions and scaling functions. Several types of quadratures are discussed and ..."
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Cited by 48 (7 self)
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. This paper deals with typical problems that arise when using wavelets in numerical analysis applications. The first part involves the construction of quadrature formulae for the calculation of inner products of smooth functions and scaling functions. Several types of quadratures are discussed and compared for different classes of wavelets. Since their construction using monomials is illconditioned, also a modified, wellconditioned construction using Chebyshev polynomials is presented. The second part of the paper deals with pointwise asymptotic error expansions of wavelet approximations of smooth functions. They are used to to derive asymptotic interpolating properties of the wavelet approximation and to construct a convergence acceleration algorithm. This is illustrated with numerical examples. Key words. wavelet, multiresolution analysis, quadrature formula, asymptotic error expansion, convergence acceleration, numerical extrapolation AMS subject classifications. 65D32, 42C05, ...
A Wavelet Based SpaceTime Adaptive Numerical Method for Partial Differential Equations
"... We describe a space and time adaptive numerical method based on wavelet orthonormal bases for solving partial differential equations. The multiresolution structure of wavelet orthonormal bases provides a simple way to adapt computational refinements to the local regularity of the solution [11] [16]. ..."
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Cited by 44 (0 self)
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We describe a space and time adaptive numerical method based on wavelet orthonormal bases for solving partial differential equations. The multiresolution structure of wavelet orthonormal bases provides a simple way to adapt computational refinements to the local regularity of the solution [11] [16]. High resolution computations are performed only in regions where singularities or sharp transitions occur. For many evolution equations it is necessary to adapt the time steps to the spatial resolution in order to maintain the stability and precision of the numerical scheme. We describe an algorithm that modifies the time discretization at each resolution, depending on the structure of the solution. The stability of this spacetime adaptive scheme is studied for the heat equation and the linear advection equation. We also explain how this algorithm can be used to solve the onedimensional Burgers equation with periodic boundary conditions. We present numerical results on the accuracy and complexity of the algorithm.
Compactly supported tight wavelet frames and orthonormal wavelets of exponential decay with a general dilation matrix
 J. Comput. Appl. Math
"... Tight wavelet frames and orthonormal wavelet bases with a general dilation matrix have applications in many areas. In this paper, for any d × d dilation matrix M, we demonstrate in a constructive way that we can construct compactly supported tight Mwavelet frames and orthonormal Mwavelet bases in ..."
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Cited by 33 (21 self)
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Tight wavelet frames and orthonormal wavelet bases with a general dilation matrix have applications in many areas. In this paper, for any d × d dilation matrix M, we demonstrate in a constructive way that we can construct compactly supported tight Mwavelet frames and orthonormal Mwavelet bases in L2(R d) of exponential decay, which are derived from compactly supported Mrefinable functions, such that they can have both arbitrarily high smoothness and any preassigned order of vanishing moments. This paper improves several
Construction of compactly supported biorthogonal wavelets II
, 1997
"... This paper deals with constructions of compactly supported biorthogonal wavelets from a pair of dual refinable functions in L 2 (R s ). In particular, an algorithmic method to construct wavelet systems and the corresponding dual systems from a given pair of dual refinable functions is given. ..."
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Cited by 32 (9 self)
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This paper deals with constructions of compactly supported biorthogonal wavelets from a pair of dual refinable functions in L 2 (R s ). In particular, an algorithmic method to construct wavelet systems and the corresponding dual systems from a given pair of dual refinable functions is given.