loop star decomposition of basis functions in the discretization of the efie a general and readily applicable scheme is presented for the determination of the basis functions that allow the decomposition of the surface current into a solenoidal part and a nonsolenoidal remainder the proposed approach brings into correspondence these two parts with two scalar functions and generates the known loop and star basis functions the completeness of the loop star basis is discussed employing the presented scheme the issue of the irrotational property of the nonsolenoidal functions is addressed

the numerical solution of integro differential equation by means of the sinc method we develop a numerical procedure for solving a class of linear integro differential equation of fredholm type using the globally defined sinc basis functions properties of the sinc procedure are utilized to reduce the computation of the linear integro differential equation to some algebraic equations we used three numerical examples to illustrate the accuracy and to show the implementation of our method

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

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

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

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

stratification and weighting via the propensity score in estimation of causal treatment effects a comparative study estimation of treatment effects with causal interpretation from observational data is complicated because exposure to treatment may be confounded with subject characteristics the propensity score the probability of treatment exposure conditional on covariates is the basis for two approaches to adjusting for confounding methods based on stratification of observations by quantiles of estimated propensity scores and methods based on weighting observations by the inverse of estimated propensity scores we review popular versions of these approaches and related methods offering improved precision describe theoretical properties and highlight their implications for practice and present extensive comparisons of performance that provide guidance for practical use

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

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

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

an impedance boundary condition efie that is low frequency and refinement stable a discretization of the impedance boundary condition electric field integral equation ibc efie is introduced that 1 yields the correct solution at arbitrarily small frequencies and 2 requires for its solution a number of matrix vector products bounded as the frequency tends to zero and as the mesh density increases the low frequency stabilization is based on a projector based discrete helmholtz splitting rescaling and recombination that depends on the low frequency behavior of both the efie operator and the surface impedance condition the dense mesh stabilization is a modification of the perfect electric conductor operator preconditioning approach taking into account the effect on the singular value spectrum of the ibc term

efficient estimation of average treatment effects using the estimated propensity score we are interested in estimating the average effect of a binary treatment on a scalar outcome if assignment to the treatment is exogenous or unconfounded that is independent of the potential outcomes given covariates biases associated with simple treatmentcontrol average comparisons can be removed by adjusting for differences in the covariates rosenbaum and rubin 1983 show that adjusting solely for differences between treated and control units in the propensity score removes all biases associated with differences in covariates although adjusting for differences in the propensity score removes all the bias this can come at the expense of efficiency as shown by hahn 1998 heckman ichimura and todd 1998 and robins mark and newey 1992 we show that weighting by the inverse of a nonparametric estimate of the propensity score rather than the true propensity score leads to an efficient estimate of the average treatment effect we provide intuition for this result by showing that this estimator can be interpreted as an empirical likelihood estimator that efficiently incorporates the information about the propensity score

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

uniform surface integral equation formulation for mixed impedance boundary conditions a numerical approach based on the surface integral equation sie method is developed for the analysis of time harmonic electromagnetic scattering by impenetrable objects equipped with mixed impedance boundary conditions mibcs mibc is an axially anisotropic generalization of the conventional isotropic leontovich impedance boundary condition the developed sie formulation utilizes surface helmholtz decomposition the self dual formulation of yan and jin and loop and star rwg functions a combined field formulation is applied to avoid internal resonances

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 numerical solution of the non linear integro differential equations based on the meshless method this article investigates the numerical solution of the nonlinear integro differential equations the numerical scheme developed in the current paper is based on the moving least square method the moving least square methodology is an effective technique for the approximation of an unknown function by using a set of disordered data it consists of a local weighted least square fitting valid on a small neighborhood of a point and only based on the information provided by its n closet points hence the method is a meshless method and does not need any background mesh or cell structures the error analysis of the proposed method is provided the validity and efficiency of the new method are demonstrated through several tests

the central role of the propensity score in observational studies for causal effects abstract the results of observational studies are often disputed because of nonrandom treatment assignment for example patients at greater risk may be overrepresented in some treatment group this paper discusses the central role of propensity scores and balancing scores in the analysis of observational studies the propensity score is the estimated conditional probability of assignment to a particular treatment given a vector of observed covariates both large and small sample theory show that adjustment for the scalar propensity score is sufficient to remove bias due to all observed covariates applications include matched sampling on the univariate propensity score which is equal percent bias reducing under more general conditions than required for discriminant matching multivariate adjustment by subclassification on balancing scores where the same subclasses are used to estimate treatment effects for all outcome variables and in all subpopulations and visual representation of multivariate adjustment by a two dimensional plot author

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

estimation of regression coefficients when some regressors are not always observed abstract in applied problems it is common to specify a model for the conditional mean of a response given a set of regressors a subset of the regressors may be missing for some study subjects either by design or happenstance in this article we propose a new class of semiparametric estimators based on inverse probability weighted estimating equations that are consistent for parameter vector 0 of the conditional mean model when the data are missing at random in the sense of rubin and the missingness probabilities are either known or can be parametrically modeled we show that the asymptotic variance of the optimal estimator in our class attains the semiparametric variance bound for the model by first showing that our estimation problem is a special case of the general problem of parameter estimation in an arbitrary semiparametric model in which the data are missing at random and the probability of observing complete data is bounded away from 0 and then deriving a representation for the efficient score

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

a semi analytical approach to solve integro differential equations in this work we present a semi analytical method for solving integro differential equations under multi point or mixed boundary conditions the proposed method solves linear and nonlinear fredholmvolterra integro differential equations a convergence analysis of the proposed method is directly examined numerical examples are worked out to demonstrate the main results moreover proper graphs are provided to confirm the efficiency and the accuracy of the proposed scheme we show that with a few number of obtained approximating terms we achieve a high accuracy level of the obtained results however increasing the number of approximating terms yields a significant decrease of the error of the approximation the proposed method is very useful reliable and flexible for solving different kinds of integro differential equations

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

models for the propensity score that contemplate the positivity assumption and their application to missing data and causality generalized linear models are often assumed to fit propensity scores which are used to compute inverse probability weighted ipw estimators to derive the asymptotic properties of ipw estimators the propensity score is supposed to be bounded away from zero this condition is known in the literature as strict positivity or positivity assumption and in practice when it does not hold ipw estimators are very unstable and have a large variability although strict positivity is often assumed it is not upheld when some of the covariates are unbounded in real data sets a data generating process that violates the positivity assumption may lead to wrong inference because of the inaccuracy in the estimations in this work we attempt to conciliate between the strict positivity condition and the theory of generalized linear models by incorporating an extra parameter which results in an explicit lower bound for the propensity score an additional parameter is added to fulfil the overlap assumption in the causal framework

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

bessel polynomial solutions of high order linear volterra integro differential equations abstract in this study a practical matrix method which is based on collocation points is presented to find approximate solutions of high order linear volterra integro differential equations vides under the mixed conditions in terms of bessel polynomials numerical examples are included to demonstrate the validity and applicability of the technique and comparisons are made with the existing results the results show the efficiency and accuracy of the present work all of the numerical computations have been performed on the computer using a program written in matlab v7 6 0 r2008a

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

comparison between the homotopy perturbation method and the sine cosine wavelet method for solving linear integro differential equations this paper compares the homotopy perturbation method with the sine cosine wavelet method for solving linear integro differential equations from the computational viewpoint the homotopy perturbation method is more efficient and easier than the sine cosine wavelet method

a new collocation method for solution of mixed linear integro differential difference equations numerical solution of mixed linear integro differential difference equation is presented using chebyshev collocation method the aim of this article is to present an efficient numerical procedure for solving mixed linear integro differential difference equations our method depends mainly on a chebyshev expansion approach this method transforms mixed linear integro differential difference equations and the given conditions into matrix equation which corresponds to a system of linear algebraic equation the reliability and efficiency of the proposed scheme are demonstrated by some numerical experiments and performed on the computer algebraic system maple10

dealing with limited overlap in estimation of average treatment effects summary estimation of average treatment effects under unconfounded or ignorable treatment assignment is often hampered by lack of overlap in the covariate distributions between treatment groups this lack of overlap can lead to imprecise estimates and can make commonly used estimators sensitive to the choice of specification in such cases researchers have often used ad hoc methods for trimming the sample we develop a systematic approach to addressing lack of overlap we characterize optimal subsamples for which the average treatment effect can be estimated most precisely under some conditions the optimal selection rules depend solely on the propensity score for a wide range of distributions a good approximation to the optimal rule is provided by the simple rule of thumb to discard all units with estimated propensity scores outside the range 0 1 0 9

loop star and loop tree decompositions analysis and efficient algorithms a new analysis of the spectral properties of loop star and loop tree decompositions is presented in this work the analysis will shed light on the behavior of these decompositions when used with regular operators such as the magnetic field and the calderon preconditioned electric field integral operators this work will explain the ill conditioning problems reported in literature and will provide a family of efficient algorithms to solve the ill conditioning and regularizing several loop star tree decomposed equations of interest for applications the theory will be corroborated by numerical results that will show the practical impact of the theoretical developments

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

analysis of semiparametric regression models for repeated outcomes in the presence of missing data abstract we propose a class of inverse probability of censoring weighted estimators for the parameters of models for the dependence of the mean of a vector of correlated response variables on a vector of explanatory variables in the presence of missing response data the proposed estimators do not require full specification of the likelihood they can be viewed as an extension of generalized estimating equations estimators that allow for the data to be missing at random but not missing completely at random these estimators can be used to correct for dependent censoring and nonrandom noncompliance in randomized clinical trials studying the effect of a treatment on the evolution over time of the mean of a response variable the likelihood based parametric g computation algorithm estimator may also be used to attempt to correct for dependent censoring and nonrandom noncompliance but because of possible model misspecification the parametric g computation algorithm estimator in contrast with the proposed w

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

on a well conditioned electric field integral operator for multiply connected geometries all known integral equation techniques for simulating scattering and radiation from arbitrarily shaped perfect electrically conducting objects suffer from one or more of the following shortcomings i they give rise to ill conditioned systems when the frequency is low ii and or when the discretization density is high iii their applicability is limited to the quasi static regime iv they require a search for global topological loops v they suffer from numerical cancellations in the solution when the frequency is very low this work presents an equation that does not suffer from any of the above drawbacks when applied to smooth and closed objects the new formulation is obtained starting from a helmholtz decomposition of two discretizations of the electric field integral operator obtained by using rwgs and dual bases respectively the new decomposition does not leverage loop and star tree basis functions but projectors that derive from them following the decomposition the two discretizations are combined in a calderon like fashion resulting in a new overall equation that is shown to exhibit self regularizing properties without suffering from the limitations of existing formulations numerical results show the usefulness of the proposed method both for closed and open structures

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

solving ibc cfie with dual basis functions the surface integral equation method is applied to solve electromagnetic scattering by arbitrarily shaped three dimensional imperfectly conducting objects modeled with the leontovitch impedance boundary condition ibc dual basis functions are used to remove previously reported difficulties associated with the solution of the combined field integral equation for objects with the ibc by expanding the magnetic current with quasi curl conforming dual functions and the electric current with divergence conforming primary rwg functions leads to a stable discretization procedure for the rotation n operator required by the ibc

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

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

on the role of the propensity score in efficient semiparametric estimation of average treatment effects in this paper the role of the propensity score in the efficient estimation of average treatment effects is examined under the assumption that the treatment is ignorable given some observed characteristics it is shown that the propensity score is ancillary for estimation of the average treatment effects the propensity score is not ancillary for estimation of average treatment effects on the treated it is suggested that the marginal value of the propensity score lies entirely in the dimension reduction efficient semiparametric estimators of average treatment effects and average treatment effects on the treated are shown to take the form of relevant sample averages of the data completed by the nonparametric imputation method it is shown that the projection on the propensity score is not necessary for efficient semiparametric estimation of average treatment effects on the treated even if the propensity score is known an application to the experimental data reveals that conditioning on the propensity score may even result in a loss of efficiency

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

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

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

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

new construction of wavelets base on floor function in this paper the properties of the floor function has been used to find a function which is one on the interval 0 1 and is zero elsewhere the suitable dilation and translation parameters lead us to get similar function corresponding to the interval a b these functions and their combinations enable us to represent the stepwise functions as a function of floor function we have applied this method on haar wavelet sine cosine wavelet block pulse functions and hybrid fourier block pulse functions to get the new representations of these functions

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 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

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