scaling functions and wavelets with vanishing moments abstract the authors investigate conditions equivalent to the vanishing of moments of wavelets in a multiresolution analysis

one point quadrature rule with cardinal b spline to compute approximately an integral 1 0m m x f x dx where m is cardinal b spline we used composite rectangular rule we proved that on the quasi uniform mesh the used formula has conditionally speaking algebraic degree of exactness m 1 under additional assumptions algebraic degree of exactness is m

gauss quadrature rules for partial support and logarithmic singular integrals with compactly supported wavelets abstract wavelet gauss quadrature methods are effective approaches to evaluate the inner products with wavelets or with associated scaling functions by a lifting trick a barinka t barsch s dahlke et al some remarks on quadrature formulas for refinable functions and wavelets z angew math mech 81 12 2001 839 885 the compactly supported wavelets and the associated scaling functions can be used as weighted functions gauss quadrature rules for wavelets on partial support are obtained in this paper based on the wavelet moments and the partial moments formulas for calculating the logarithmic wavelet moments are introduced finally gauss quadrature rules with the product of scaling functions and logarithmic singular functions as the weight function are constructed the accuracy of the rules is illustrated by two numerical experiments

direct and inverse computation of jacobi matrices of infinite iterated function systems we introduce a new set of algorithms to compute the jacobi matrices associated with invariant measures of infinite iterated function systems composed of one dimensional homogeneous affine maps we demonstrate their utility in the study of theoretical problems like the conjectured almost periodicity of such jacobi matrices the singularity of the measures and the logarithmic capacity of their support since our technique is based on a reversible transformation between pairs of jacobi matrices it can also be applied to solve an inverse approximation problem the proposed algorithms are tested in significant highly sensitive cases they perform in a stable fashion and can reliably compute jacobi matrices of large order

on the approximation power of convolution based least squares versus interpolation there are many signal processing tasks for which convolution based continuous signal representations such as splines and wavelets provide an interesting and practical alternative to the more traditional sine based methods the coefficients of the corresponding signal approximations are typically obtained by direct sampling interpolation or quasi interpolation or by using least squares techniques that apply a prefilter prior to sampling we compare the performance of these approaches and provide quantitative error estimates that can be used for the appropriate selection of the sampling step h specifically we review several results in approximation theory with a special emphasis on the strang fix 1971 conditions which relate the general o h sup l behavior of the error to the ability of the representation to reproduce polynomials of degree n l 1 we use this theory to derive pointwise error estimates for the various algorithms and to obtain the asymptotic limit of the l sub 2 error as h tends to zero we also propose a new improved l sub 2 error bound for the least squares case in the process we provide all the relevant bound constants for polynomial splines some of our results suggest the existence of an intermediate range of sampling steps where the least squares method is roughly equivalent to an interpolator with twice the order we present experimental examples that illustrate the theory and confirm the adequacy of our various bound and limit determinations

quadrature formulae and asymptotic error expansions for wavelet approximations of smooth functions 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 ill conditioned also a modified well conditioned 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 derive asymptotic interpolating properties of the wavelet approximation and to construct a convergence acceleration algorithm this is illustrated with numerical examples

integration of products of gaussians and wavelets with applications to electronic structure calculations in this work we consider a scheme for the computation of integrals of gaussian functions with products of wavelets which is applicable to wavelet bases given only implicitly in terms of refinement relations the integrals under consideration arise in wavelet discretizations of operator equations involving separable approximations of coulomb potentials motivated by this application we study in particular the case of large exponents in the gaussian terms the numerical tests demonstrate the practical efficiency of the method

a statistical inter cell interference model for downlink cellular ofdma networks under log normal shadowing and multipath rayleigh fading the probability density function pdf of the intercell interference for downlink cellular orthogonal frequency division multiple access networks under log normal shadowing and multipath rayleigh fading is derived the derived pdf is observed to highly deviate from the gaussian distribution especially at moderate to small network loads based on the derived inter cell interference statistics the decoder performance can be significantly improved compared to the case when the traditional decoder optimized for the awgn channel is employed a simple suboptimum decoding algorithm that closely approximates the derived optimum decoder performance is also proposed

nonuniform sampling and reconstruction in shift invariant spaces this article discusses modern techniques for nonuniform sampling and reconstruction of functions in shift invariant spaces it is a survey as well as a research paper and provides a unified framework for uniform and nonuniform sampling and reconstruction in shift invariant subspaces by bringing together wavelet theory frame theory reproducing kernel hilbert spaces approximation theory amalgam spaces and sampling inspired by applications taken from communication astronomy and medicine the following aspects will be emphasized a the sampling problem is well defined within the setting of shift invariant spaces b the general theory works in arbitrary dimension and for a broad class of generators c the reconstruction of a function from any sufficiently dense nonuniform sampling set is obtained by efficient iterative algorithms these algorithms converge geometrically and are robust in the presence of noise d to model the natural decay conditions of real signals and images the sampling theory is developed in weighted l p spaces

an overview of wavelet based multiresolution analyses in this paper an overview of wavelet based multiresolution analyses is presented first the continuous wavelet transform in its simplest form is discussed then the definition of a multiresolution analysis is given and how wavelets fit into it is shown the authors take a closer look at orthogonal biorthogonal and semiorthogonal wavelets the fast wavelet transform wavelets on an interval multidimensional wavelets and wavelet packets are discussed several examples of wavelet families are introduced and compared finally the essentials of two major applications are outlined data compression and compression of linear operators

a dynamically adaptive multilevel wavelet collocation method for solving partial differential equations in a finite domain a dynamically adaptive multilevel wavelet collocation method is developed for the solution of partial differential equations the multilevel structure of the algorithm provides a simple way to adapt computational refinements to local demands of the solution high resolution computations are performed only in regions where sharp transitions occur the scheme handles general boundary conditions the method is applied to the solution of the one dimensional burgers equation with small viscosity a moving shock problem and a nonlinear thermoacoustic wave problem the results indicate that the method is very accurate and efficient

asymptotic error expansion of wavelet approximations of smooth functions ii we generalize earlier results concerning an asymptotic error expansion of wavelet approximations the properties of the monowavelets which are the building blocks for the error expansion are studied in more detail and connections between spline wavelets and euler and bernoulli polynomials are pointed out the expansion is used to compare the error for different wavelet families we prove that the leading terms of the expansion only depend on the multiresolution subspaces v j and not on how the complementary subspaces w j are chosen consequently for a fixed set of subspaces v j the leading terms do not depend on the fact whether the wavelets are orthogonal or not we also show that daubechies orthogonal wavelets need in general one level more than spline wavelets to obtain an approximation with a prescribed accuracy these results are illustrated with numerical examples

computational methods for integrals involving functions and daubechies wavelets abstract when wavelets are used as basis functions in galerkin approach to solve the integral equations integrals of the form supp j k f x j k x d x occur by a change of variable these integrals can be translated into integrals involving only in this paper we find quadrature rule on the supp for the integrals of the form supp f x x d x wavelets in this article are those discovered by daubechies i daubechies orthonormal bases of compactly supported wavelets comm pure appl math 41 1988 909 996 where is the scaling function and is the wavelet function

multiwavelet moments and projection prefilters an efficient projection procedure is derived for use of orthogonal multiwavelets in the analysis of discrete data sequences a family of simple prefilters corresponding to numerical quadrature evaluation of the projection integrals provides exact results for locally polynomial data the full approximation order of the multiwavelet basis can thus always be enabled for nonpolynomial signals the prefilters provide approximations to the coefficients of the multiwavelet series whose convergence accelerates quickly with increase in sampling rate comparison is also made with previous time invariant multiwavelet prefilters

performance bounds for ds ssma communications with complex signature sequences a unified performance analysis of direct sequence spread spectrum multiple access ds ssma communications with deterministic complex signature sequences is presented the probability density function pdf of the multiple user interference is determined using a round down and round up procedure on the pdf arbitrarily tight lower and upper bounds on the probability of bit error pbe are obtained results based on the gaussian approximation method are also presented it is shown that complex sequences can yield better pbe performance than binary sequences using complex sequences the number of signature sequences that have good auto and cross correlation properties is greatly enlarged new users employing complex or binary signature sequences can be added to existing systems with graceful performance degradation

bit error rate evaluation for spread spectrum multiple access systems a unified method for approximating and bounding the average bit error probability for spread spectrum multiple access communication systems is presented various forms of direct sequence spreadspectrum modulation are considered including binary phase shift keying quadriphase shift keying and minimum shift keying the analysis of the multiple access interference makes use of a number of moments sufficent to evaluate the error probability with a high degree of accuracy a computationally efficient algorithm for computing the moments is also given the subsequent transformation from the moments to the average bit error probability is carried out by means of gauss type numerical integration formulas it is shown that the same approach can be exploited for evaluating two classes of upper and lower bounds on the bit error rate finally some results and comparisons are reported

an efficient numerical quadrature for the calculation of the potential energy of wavefunctions expressed in the daubechies wavelet basis an efficient numerical quadrature is proposed for the approximate calculation of the potential energy in the context of pseudo potential electronic structure calculations with daubechies wavelet and scaling function basis sets our quadrature is also applicable in the case of adaptive spatial resolution our theoretical error estimates are confirmed by numerical test calculations of the ground state energy and wavefunction of the harmonic oscillator in one dimension with and without adaptive resolution as a byproduct we derive a filter which upon application on the scaling function coefficients of a smooth function renders the approximate grid values of this function this also allows for a fast calculation of the charge density from the wavefunction

biorthogonal bases of compactly supported wavelets orthonormal bases of compactly supported wavelet bases correspond to subband coding schemes with exact reconstruction in which the analysis and synthesis filters coincide we show here that under fairly general conditions exact reconstruction schemes with synthesis filters different from the analysis filters give rise to two dual riesz bases of compactly supported wavelets we give necessary and sufficient conditions for biorthogonality of the corresponding scaling functions and we present a sufficient condition for the decay of their fourier transforms we study the regularity of these biorthogonal bases we provide several families of examples all symmetric corresponding to linear phase filters in particular we can construct symmetric biorthogonal wavelet bases with arbitrarily high preassigned regularity we also show how to construct symmetric biorthogonal wavelet bases close to a nonsymmetric orthonormal basis

non gaussian characterization of ds cdma noise in few user systems with complex signature sequences an higher order statistics hos based analysis of the non gaussian characteristics of the ds cdma global noise and a non gaussian evaluation of the expected ber in few user systems with complex signature sequences are performed different error rate performances provided by binary gold and four phase even odd equivalent gold eoe gold spreading sequences are also considered

solving second kind fredholm integral equations by periodic wavelet galerkin method in this paper we use the periodized daubechies wavelets based galerkin method pwgm to solve linear nonlinear and singular fredholm integral equations of the second kind a main advantage of the present pwgm over the existing wavelet galerkin methods lies in that the wavelet expansion coefficients are exactly obtained without calculating the wavelet integrations therefore the computational cost is low whereas the accuracy is high after discretization the linear and nonlinear integral equations is converted into a system of linear and nonlinear equations respectively and for the linear case the matrix can be converted into a sparse and symmetrical one by fast wavelet transform fwt numerical experiments show that the pwgm has a good degree of accuracy

generalized average sampling in shift invariant spaces average sampling is motivated by realistic needs e g physical limitation of acquisition devices or characteristics of sampling procedure as an extension of the average sampling we study generalized average sampling in shift invariant spaces with frame generators in which averages are taken from suitable channeled version of signals illustrative examples are given

error estimates for irregular sampling of band limited functions on a locally compact abelian group abstract band limited functions f can be recovered from their sampling values f x i by means of iterative methods if only the sampling density is high enough we present an error analysis for these methods treating the typical forms of errors i e jitter error truncation error aliasing error quantization error and their combinations the derived apply uniformly to whole families of spaces e g to weighted l p spaces over some locally compact abelian group with growth rate up to some given order in contrast to earlier papers we do not make use of any relative separation condition on the sampling sets furthermore we discard the assumption on polynomial growth of the weights that has been used over euclidean spaces consequently even for the case of regular sampling i e sampling along lattices in g the results are new in the given generality

a sampling theorem for wavelet subspaces the classical shannon sampling theorem is extended to the subspaces used in the multiresolution analysis in wavelet theory under weak hypotheses these subspaces are first shown to have a riesz basis formed from the reproducing kernels these in turn are used to construct the sampling sequences examples are given

initialization of orthogonal discrete wavelet transforms the asymptotic formulae of both the approximation error and the systematic error of a special prefilter projection and the quantitative estimates of the upper bounds of the errors are obtained in addition it is shown that for the daubechies orthogonal wavelet basis the estimated constant is optimal

a wavelet multilevel method for ill posed problems stabilized by tikhonov regularization an additive schwarz iteration is described for the fast resolution of linear ill posed problems which are stabilized by tikhonov regularization the algorithm and its analysis are presented in a general framework which applies to integral equations of the first kind discretized either by spline functions or daubechies wavelets numerical experiments are reported on to illustrate the theoretical results and to compare both discretization schemes

b spline signal processing ii efficiency design and applications for pt i see ibid vol 41 no 2 p 821 33 1993 a class of recursive filtering algorithms for the efficient implementation of b spline interpolation and approximation techniques is described in terms of simplicity of realization and reduction of computational complexity these algorithms compare favorably with conventional matrix approaches a filtering interpretation low pass filter followed by an exact polynomial spline interpolator of smoothing spline and least squares approximation methods is proposed these techniques are applied to the design of digital filters for cubic spline signal processing an efficient implementation of a smoothing spline edge detector is proposed it is also shown how to construct a cubic spline image pyramid that minimizes the loss of information in passage from one resolution level to the next in terms of common measures of fidelity this data structure appears to be superior to the gaussian laplacian pyramid

solving linear integro differential equation system by galerkin methods with hybrid functions in this paper we use the hybrid legendre and block pulse functions on interval 0 1 to solve the linear integro differential equation system and construct the quadrature formulae for the calculation of inner products of any functions which are required in the galerkin methods for integro differential equation system

on the generalized wavelet galerkin method abstract in the frame of the traditional wavelet galerkin method based on the compactly supported wavelets it is important to calculate the so called connection coefficients that are some integrals whose integrands involve products of wavelets their derivatives as well as some known coefficients in considered differential equations however even for linear differential equations with non constant coefficient the computation of connect coefficients becomes rather time consuming and often even impossible in this paper we propose a generalized wavelet galerkin method based on the compactly supported wavelets which is computationally very efficient even for differential equations with non constant coefficients no matter linear or nonlinear problems some related mathematical theorems are proved based on which the basic ideas of the generalized wavelet galerkin method are described in details in addition some examples are used to illustrate its validity and high efficiency a nonlinear example shows that the generalized wavelet galerkin method is not only valid to solve nonlinear problems but also possesses the ability to find new solutions of multi solution problems this method can be widely applied to various types of both linear and nonlinear differential equations in science and engineering

solving second kind integral equations by galerkin methods with continuous orthogonal wavelets abstract in this paper we use the continuous wavelets on the interval constructed by cohen et al appl comput harm anal 1 1993 54 81 to solve the second kind integral equations to this end we give the decomposition and reconstruction algorithm for these wavelets and construct the quadrature formulae for the calculation of inner products of any functions and the scaling functions which are required in the wavelet galerkin methods for integral equations in this method the integral kernels are represented in these wavelet bases as sparse matrices to high precision thus we present an efficient algorithm for numerical solution of second kind integral equations

on the existence of solutions for nonlinear first order implicit impulsive integro differential equations abstract by using the monch fixed point theorem the purpose of this paper is to obtain the new existence theorems of solutions for a new class of initial value problems of nonlinear first order implicit impulsive integro differential equations in banach spaces under some weaker conditions

monotone method for a system of nonlinear mixed type implicit impulsive integro differential equations in banach spaces in this paper by using a monotone iterative technique in the presence of lower and upper solutions we discuss the existence of solutions for a new system of nonlinear mixed type implicit impulsive integro differential equations in banach spaces under wide monotonicity conditions and the noncompactness measure conditions we also obtain the existence of extremal solutions and a unique solution between lower and upper solutions

a note on one point quadrature formula for daubechies scale function with partial support abstract this note is concerned with an efficient computation of integrals of products of a smooth function and daubechies scale function with partial support by using a one point quadrature rule the error estimate is obtained the rule is illustrated by considering an example from the literature

numerical solution of linear integro differential equation by using sine cosine wavelets abstract we use the continuous sine cosine wavelets on the interval 0 1 to solve the linear integro differential equation to do so we construct the quadrature formulae for the calculation of inner products of any functions which are required in the approximation for the integro differential equation then we reduced the integro differential equation to the solution of linear algebraic equations

filter bank methods for hyperbolic pdes we use biorthogonal filter banks to solve hyperbolic pdes adaptively with a sparse multilevel representation of the solution the methods described are of finite difference type and the filter banks are used to give a sparse representation of signals and to transform between grids on different scales we derive bounds for the error and number of coefficients in the sparse representation these bounds also apply for filter banks that are not associated with any wavelets we develop algorithms for fast differentiation and multiplication in detail the strength of the method is shown in various test problems

solving second kind integral equations with hybrid chebyshev and block pulse functions in this paper we use a combination of chebyshev and block pulse functions on the interval 0 1 to solve the linear integral equation of the second kind we convert the integral equation to a system of linear equations we show that our estimates have a good degree of accuracy

the discrete wavelet transform wedding the a trous and mallat algorithms two separately motivated implementations of the wavelet transform are brought together it is observed that these algorithms are both special cases of a single filter bank structure the discrete wavelet transform the behavior of which is governed by the choice of filters in fact the a trous algorithm is more properly viewed as a nonorthonormal multiresolution algorithm for which the discrete wavelet transform is exact moreover it is shown that the commonly used lagrange a trous filters are in one to one correspondence with the convolutional squares of the daubechies filters for orthonormal wavelets of compact support a systematic framework for the discrete wavelet transform is provided and conditions are derived under which it computes the continuous wavelet transform exactly suitable filter constraints for finite energy and boundedness of the discrete transform are also derived relevant signal processing parameters are examined and it is observed that orthonormality is balanced by restrictions on resolution

a fast adaptive wavelet collocation algorithm for multidimensional pdes a fast multilevel wavelet collocation method for the solution of partial differential equations in multiple dimensions is developed the computational cost of the algorithm is independent of the dimensionality of the problem and is o n where n is the total number of collocation points the method can handle general boundary conditions the multilevel structure of the algorithm provides a simple way to adapt computational refinements to local demands of the solution high resolution computations are performed only in regions where singularities or sharp transitions occur numerical results demonstrate the ability of the method to resolve localized structures such as shocks which change their location and steepness in space and time the present results indicate that the method has clear advantages in comparison with well established numerical algorithms

quadrature prefilters for the discrete wavelet transform discrepancies between the discrete wavelet transform and the coefficients of the wavelet series are known to be reducible by initialization of input data prefilters based on lagrange interpolants are derived here for biorthogonal compact support wavelet systems providing exact subspace projection in cases of local polynomial smoothness the resulting divergence acceleration in a nonpolynomial test case is examined irregular sampling rates are also accommodated

error probabilities for binary direct sequence spread spectrum communications with random signature sequences binary direct sequence spread spectrum multiple access communications an additive white gaussian noise channel and a coherent correlation receiver are considered an expression for the output of the receiver is obtained for the case of random signature sequences and the corresponding characteristic function is determined the expression is used to study the density function of the multiple access interference and to determine arbitrarily tight upper and lower bounds on the average probability of error the bounds which are obtained without making a gaussian approximation are compared to results obtained using a gaussian approximation the effects of transmitter power the length of the signature sequences and the number of interfering transmitters are illustrated each transmitter is assumed to have the same power although the general approach can accommodate the case of transmitters with unequal powers

using the refinement equation for evaluating integrals of wavelets the wavelet galerkin method for solving partial differential equations leads to the problem of computing integrals of products of derivatives of wavelets this paper studies the problem from the point of view of stationary subdivision schemes one of the main results is to identify these integrals as components of the unique solution of a certain eigenvector moment problem associated with the coefficients of the refinement equation asymptotic expansions for the corresponding subdivision schemes form an important ingredient of our approach

wavelet simultaneous approximation from samples affected by noise the authors recently introduced a pth order wavelet regularization method together with the gcv criterion for approximating a function from a finite sample affected by noise convergence results of the method were proven for the l2 norm the present paper addresses the problem of simultaneous approximation which is of interest in applications where derivatives are useful for extracting several features from a signal it is proved that it is not needed to devise a special method to this purpose since the convergence of the method devised by the authors also works for the hq norm 0 q p

from the wavelet series to the discrete wavelet transform the initialization the discrete wavelet transform dwt is computed by subband filters bank and often used to approximate wavelet series ws and continuous wavelet transform cwt the approximation is often inaccurate because of improper initialized discretization of the continuous time signal in this correspondence the problem is analyzed and two simple algorithms for the initialization are introduced finally numerical examples are presented to show that our algorithms are more effective than others

wavelet families of increasing order in arbitrary dimensions we build discrete time compactly supported biorthogonal wavelets and perfect reconstruction filter banks for any lattice in any dimension with any number of primal and dual vanishing moments the associated scaling functions are interpolating our construction relies on the lifting scheme and inherits all of its advantages fast transform in place calculation and integer to integer transforms we show that two lifting steps suffice predict and update the predict step can be built using multivariate polynomial interpolation while update is a multiple of the adjoint of predict while we concentrate on the discrete time case some discussion of convergence and stability issues together with examples is given

efficient wavelet prefilters with optimal time shifts a wavelet prefilter maps sample values of an analyzed signal to the scaling function coefficient input of standard discrete wavelet transform dwt algorithms the prefilter is the inverse of a certain postfilter convolution matrix consisting of integer sample values of a noninteger shifted wavelet scaling function for the prefilter and the dwt algorithms to have similar computational complexity it is often necessary to use a short enough approximation of the prefilter in addition to well known quadrature formula and identity matrix prefilter approximations we propose a neumann series approximation which is a band matrix truncation of the optimal prefilter and derive simple formulas for the operator norm approximation error this error shows a dramatic dependence on how the postfilter noninteger shift is chosen we explain the meaning of this shift in practical applications describe how to choose it and plot optimally shifted prefilter approximation errors for 95 different daubechies symlet and b spline wavelets whereas the truncated inverse is overall superior the neumann filters are by far the easiest ones to compute and for some short support wavelets they also give the smallest approximation error for example for daubechies 1 5 wavelets the simplest neumann prefilter provide an approximation error reduction corresponding to 100 10 000 times oversampling in a nonprefiltered system

fast wavelet transforms and numerical algorithms i a class of algorithms is introduced for the rapid numerical application of a class of linear operators to arbitrary vectors previously published schemes of this type utilize detailed analytical information about the operators being applied and are specific to extremely narrow classes of matrices in contrast the methods presented here are based on the recently developed theory of wavelets and are applicable to all calderon zygmund and pseudo differential operators the algorithms of this paper require order o n or o n log n operations to apply an n n matrix to a vector depending on the particular operator and the version of the algorithm being used and our numerical experiments indicate that many previously intractable problems become manageable with the techniques presented here

gauss type quadrature rule with complex nodes and weights for integrals involving daubechies scale functions and wavelets this paper deals with derivation of a gauss type quadrature rule named as gauss daubechies quadrature rule for numerical evaluation of integrals involving product of integrable function and daubechies scale functions wavelets some of the nodes and weights of the quadrature formula may be complex and appear with their conjugates this is in contrast with the standard gauss type quadrature rule for integrals involving products of integrable functions and non negative weight functions this quadrature rule has accuracy as good as the standard gauss type quadrature rule and is also found to be stable the efficacy of the quadrature rule derived here has been tested through some numerical examples

numerical approximation of 2d fredholm integral eigenvalue problems by orthogonal wavelets we investigate the numerical approximation of two dimensional second kind fredholm integral eigenvalue problems by the galerkin method with the cohen daubechies vial cdv wavelet family this choice provides us orthogonal bases for bounded domains avoiding the need of periodization or domain truncation the cdv family is indexed by the number of vanishing moments which drives the regularity of the basis we generate the galerkin basis from tensorized scaling functions and employ weighted gaussian quadratures derived from refinement equations numerical experiments address the relative computational cost of this approach with respect to the haar basis and the relationship between convergence rate and number of vanishing moments

composite quadrature formulae for the approximation of wavelet coefficients of piecewise smooth and singular functions the computation of wavelet coefficients of a function typically requires the computation of a large number of integrals these integrals represent the inner product of that function with a wavelet function on different scales or with the corresponding scaling function on a fine scale we develop quadrature rules for those integrals that converge fast for piecewise smooth and singular functions they do not require the evaluation of the scaling function and the convergence does not depend on the smoothness of that function the analysis and computation is based completely on the filter coefficients that define the scaling function an application is presented from the field of electromagnetics involving the inner product of a singular function with two dimensional tensor product wavelets

computation of wavelet coefficients from average samples there exist efficient methods to compute the wavelet coefficients of a function f t from its point samples f t email protected email protected n however in many applications the available samples are average samples of the type f t email protected u t dt where the averaging function u t reflects the characteristic of the acquisition device in this work methods to compute the coefficients in a biorthogonal wavelet system from average samples are studied error estimations are obtained and using them the optimal values for the parameters in the proposed approximation rules are calculated the obtained error estimations can also be applied to the rules that compute the coefficients from point samples and thus these estimations can be used to compare and to choose between the different methods proposed in the literature the methods proposed here also allow us to compute the biorthogonal wavelet coefficients from the coefficients in another biorthogonal wavelet system

wavelet coefficient computation with optimal prefiltering discrete wavelet transform dwt is often used to approximate wavelet series transform wst and continuous wavelet transform cwt since it can be computed numerically in this research we first study the accuracy of the computed dwt coefficients obtained from the shensa see ibid vol 40 no 10 p 2464 2482 1992 algorithm as an approximate of the wst coefficients based on the accuracy analysis we then propose a procedure to design optimal fir prefilters used in the shensa algorithm to reduce the approximation error finally numerical examples are presented to demonstrate the performance of the optimal fir prefilters

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

higher order wavelet reconstruction differentiation filters and gibbs phenomena an orthogonal wavelet basis is characterized by its approximation order which relates to the ability of the basis to represent general smooth functions on a given scale it is known though perhaps not widely known that there are ways of exceeding the approximation order i e achieving higher order error in the discretized wavelet transform and its inverse the focus here is on the development of a practical formulation to accomplish this first for 1d smooth functions then for 1d functions with discontinuities and then for multidimensional here 2d functions with discontinuities it is shown how to transcend both the wavelet approximation order and the 2d gibbs phenomenon in representing electromagnetic fields at discontinuous dielectric interfaces that do not simply follow the wavelet basis grid

wavelet packet transform initialization for piecewise polynomial and bandlimited inputs this paper considers the problem of initializing the wavelet packet transform when the analog input x t is either bandlimited or else is piecewise polynomial on the dyadic intervals this is accomplished by the construction of certain prefilter sequences that are applied to the samples x sub k x sub k x kt t is the sampling period in the piecewise polynomial case the prefilter sequences are computed with the aid of the interpolatory graphical display algorithm igda which is used to compute the so called incremental moment functions of the scaling function spl phi t on the dyadic points the moments of spl phi t are also employed

approximation power of biorthogonal wavelet expansions this paper looks at the effect of the number of vanishing moments on the approximation power of wavelet expansions the strang fix conditions imply that the error for an orthogonal wavelet approximation at scale a 2 sup i globally decays as a sup n where n is the order of the transform this is why for a given number of scales higher order wavelet transforms usually result in better signal approximations we prove that this result carries over for the general biorthogonal case and that the rate of decay of the error is determined by the order properties of the synthesis scaling function alone we also derive asymptotic error formulas and show that biorthogonal wavelet transforms are equivalent to their corresponding orthogonal projector as the scale goes to zero these results strengthen sweldens earlier analysis and confirm that the approximation power of biorthogonal and semi orthogonal wavelet expansions is essentially the same finally we compare the asymptotic performance of various wavelet transforms and briefly discuss the advantages of splines we also indicate how the smoothness of the basis functions is beneficial in reducing the approximation error

wavelets and dilation equations a brief introduction wavelets are new families of basis functions that yield the representation f x sum b jk w 2 j x k their construction begins with the solution phi x to a dilation equation with coefficients c k then w comes from phi and the basis comes by translation and dilation of w it is shown in part 1 how conditions on the c k lead to approximation properties and orthogonality properties of the wavelets part 2 describes the recursive algorithms also based on the c k that decompose and reconstruct f the object of wavelets is to localize as far as possible in both time and frequency with efficient algorithms

sampling approximation of smooth functions via generalized coiflets we present the sampling approximation power of a newly constructed class of compactly supported orthonormal wavelets called generalized coiflets we study the accuracy of generalized coiflets based sampling approximation of smooth functions by developing convergence rates for the pointwise approximation error as well as its l sup p norm we show i that the l sup 2 error due to the approximation of expansion coefficients by function samples is asymptotically negligible compared with that due to projection and ii that generalized coiflets can achieve asymptotically better approximation than the original coiflets

convergence of the red tower method for removing noise from data by coupling the wavelet transform with a particular nonlinear shrinking function the red telescopic optimal wavelet estimation of the risk tower method is introduced for removing noise from signals it is shown that the method yields convergence of the l sub 2 risk to the actual solution with optimal rate moreover the method is proved to be asymptotically efficient when the regularization parameter is selected by the generalized cross validation criterion gcv or the mallows criterion numerical experiments based on synthetic data are provided to compare the performance of the red tower method with hard thresholding soft thresholding and neigh coeff thresholding furthermore the numerical tests are also performed when the tower method is applied to hard thresholding soft thresholding and neigh coeff thresholding for which the full convergence results are still open

orthogonal polynomials for refinable linear functionals a refinable linear functional is one that can be expressed as a convex combination and defined by a finite number of mask coefficients of certain stretched and shifted replicas of itself the notion generalizes an integral weighted by a refinable function the key to calculating a gaussian quadrature formula for such a functional is to find the three term recursion coefficients for the polynomials orthogonal with respect to that functional we show how to obtain the recursion coefficients by using only the mask coefficients and without the aid of modified moments our result implies the existence of the corresponding refinable functional whenever the mask coefficients are nonnegative even when the same mask does not define a refinable function the algorithm requires o n 2 rational operations and thus can in principle deliver exact results numerical evidence suggests that it is also effective in floating point arithmetic

quantitative fourier analysis of approximation techniques i interpolators and projectors we present a general fourier based method that provides an accurate prediction of the approximation error as a function of the sampling step t our formalism applies to an extended class of convolution based signal approximation techniques which includes interpolation generalized sampling with prefiltering and the projectors encountered in wavelet theory we claim that we can predict the l sup 2 approximation error by integrating the spectrum of the function to approximate not necessarily bandlimited against a frequency kernel e spl omega that characterizes the approximation operator this prediction is easier yet more precise than was previously available our approach has the remarkable property of providing a global error estimate that is the average of the true approximation error over all possible shifts of the input function our error prediction is exact for stationary processes as well as for bandlimited signals we apply this method to the comparison of standard interpolation and approximation techniques our method has interesting implications for approximation theory in particular we use our results to obtain some new asymptotic expansions of the error as t spl rarr 0 as well as to derive improved upper bounds of the kind found in the strang fix 1971 theory we finally show how we can design quasi interpolators that are near optimal in the least squares sense

an adaptive wavelet vaguelette algorithm for the solution of pdes the paper first describes a fast algorithm for the discrete orthonormal wavelet transform and its inverse without using the scaling function this approach permits to compute the decomposition of a function into a lacunary wavelet basis i e a basis constituted of a subset of all basis functions up to a certain scale without modification the construction is then extended to operator adapted biorthogonal wavelets this is relevant for the solution of certain nonlinear evolutionary pdes where a priori information about the significant coefficients is available we pursue the approach described in j frohlich and k schneider europ j mech b fluids13 439 1994 which is based on the explicit computation of the scalewise contributions of the approximated function to the values at points of hierarchical grids here we present an improved construction employing the cardinal function of the multiresolution the new method is applied to the helmholtz equation and illustrated by comparative numerical results it is then extended for the solution of a nonlinear parabolic pde with semi implicit discretization in time and self adaptive wavelet discretization in space results with full adaptivity of the spatial wavelet discretization are presented for a one dimensional flame front as well as for a two dimensional problem

quantitative fourier analysis of approximation techniques ii wavelets for pt i see ibid vol 47 no 10 p 2783 95 1999 in a previous paper we proposed a general fourier method that provides an accurate prediction of the approximation error irrespective of the scaling properties of the approximating functions here we apply our results when these functions satisfy the usual two scale relation encountered in dyadic multiresolution analysis as a consequence of this additional constraint the quantities introduced in our previous paper can be computed explicitly as a function of the refinement filter this is in particular true for the asymptotic expansion of the approximation error for biorthonormal wavelets as the scale tends to zero one of the contributions of this paper is the computation of sharp asymptotically optimal upper bounds for the least squares approximation error another contribution is the application of these results to b splines and daubechies 1988 1992 scaling functions which yields explicit asymptotic developments and upper bounds thanks to these explicit expressions we can quantify the improvement that can be obtained by using b splines instead of daubechies wavelets in other words we can use a coarser spline sampling and achieve the same reconstruction accuracy as daubechies specifically we show that this sampling gain converges to spl pi as the order tends to infinity

average sampling in shift invariant subspaces with symmetric averaging functions in this paper we study the reconstruction of functions in shift invariant subspaces from local averages with symmetric averaging functions we present an average sampling theorem for shift invariant subspaces and give quantitative results on the aliasing error and the truncation error we show that every square integrable function can be approximated by its average sampling series as special cases we also obtain new error bounds for regular sampling examples are given

gauss quadrature rules for cauchy principal value integrals with wavelets gauss quadrature rules for cauchy principal value integrals with wavelets are established by the construction of orthogonal polynomials of wavelets and scaling functions as weight functions the wavelet gauss quadrature formulae for cauchy principal value integrals are introduced numerical examples are also presented

on functional improper volterra integral equations and impulsive differential equations in ordered banach spaces in this paper we derive existence and comparison results for discontinuous improper functional integral equations of volterra type in an ordered banach space which has a regular order cone for this purpose we prove dominated and monotone convergence theorems for improper integrals the obtained results are then applied to first order impulsive differential equations concrete examples are also solved by using symbolic programming

a general approach for analysis and application of discrete multiwavelet transforms this paper proposes a general paradigm for the analysis and application of discrete multiwavelet transforms particularly to image compression first we establish the concept of an equivalent scalar wavelet filter bank system in which we present an equivalent and sufficient representation of a multiwavelet system of multiplicity r in terms of a set of r equivalent scalar filter banks this relationship motivates a new measure called the good multifilter properties gmps which define the desirable filter characteristics of the equivalent scalar filters we then relate the notion of gmps directly to the matrix filters as necessary eigenvector properties for the refinement masks of a given multiwavelet system second we propose a generalized efficient and nonredundant framework for multiwavelet initialization by designing appropriate preanalysis and post synthesis multirate filtering techniques finally our simulations verified that both orthogonal and biorthogonal multiwavelets that possess gmps and employ the proposed initialization technique can perform better than the popular scalar wavelets such as daubechies d8 wavelet and the d 9 7 wavelet and some of these multiwavelets achieved this with lower computational complexity

an evaluation of clenshaw curtis quadrature rule for integration w r t singular measures this work is devoted to the study of quadrature rules for integration with respect to w r t general probability measures with known moments automatic calculation of the clenshaw curtis rules is considered and analyzed it is shown that it is possible to construct these rules in a stable manner for quadrature w r t balanced measures in order to make a comparison gauss rules and their stable implementation for integration w r t balanced measures are recalled convergence rates are tested in the case of binomial measures

a general sampling theory for nonideal acquisition devices the authors first describe the general class of approximation spaces generated by translation of a function spl psi x and provide a full characterization of their basis functions they then present a general sampling theorem for computing the approximation of signals in these subspaces based on a simple consistency principle the theory puts no restrictions on the system input which can be an arbitrary finite energy signal bandlimitedness is not required in contrast to previous approaches this formulation allows for an independent specification of the sampling analysis and approximation synthesis spaces in particular when both spaces are identical the theorem provides a simple procedure for obtaining the least squares approximation of a signal they discuss the properties of this new sampling procedure and present some examples of applications involving bandlimited and polynomial spline signal representations they also define a spectral coherence function that measures the similarity between the sampling and approximation spaces and derive a relative performance bound for the comparison with the least squares solution

integral operators on sparse grids in this paper we are concerned with the construction and use of wavelet approximation spaces for the fast evaluation of integral expressions the spaces are based on biorthogonal anisotropic tensor product wavelets we introduce sparse grid hyperbolic cross approximation spaces which are adapted not only to the smoothness of the kernel but also to the norm in which the error is measured furthermore we introduce compression schemes for the corresponding discretizations numerical examples for the laplace equation with dirichlet boundary conditions and an additional integral term with a smooth kernel demonstrate the validity of our theoretical results