variational approach to the zakharov shabat scattering problem it is shown that the zakharov shabat scattering problem associated with the inverse scattering transform technique for solving the nonlinear schr odinger equation can be reformulated as a variational problem this reformulation makes it possible to use direct variational methods for finding the eigenvalues of the scattering problem which determines the speed and amplitude of the solitons emerging from a given initial condition the approach is illustrated by an application to sech shaped initial conditions

multiplicative perturbation theory of the moore penrose inverse and the least squares problem abstract bounds for the variation of the moore penrose inverse of general matrices under multiplicative perturbations are presented their advantages with respect to classical bounds under additive perturbations and with respect to other bounds under multiplicative perturbations available in the literature are carefully studied and established closely connected to these developments a complete multiplicative perturbation theory for least squares problems valid for perturbations of any size is also presented improving in this way recent multiplicative perturbation bounds which are valid only to first order in the size of the perturbations the results in this paper are mainly based on exact expressions of the perturbed moore penrose inverse in terms of the unperturbed one the perturbation matrices and certain orthogonal projectors such feature makes the new results amenable to be generalized in the future to linear operators in infinite dimensional spaces

a b wavelet based noise reduction algorithm a wavelet based method is introduced for removing structured noise e g impulsive spikes or unwanted harmonic components from data for this type of noise the time and frequency localization capabilities of wavelets provide better noise detection and less signal distortion than direct filtering of data the procedure is applied to time series data with impulsive noise and transfer function data with multipath interference the authors use a single set of scaling and wavelet bases that can effectively represent a wide range of signals and then they perform noise reducing operations on the scaling and wavelet coefficients multiresolution analysis is introduced in order to describe the noise reduction algorithm

wavelet based schemes for linear advection dispersion equation abstract in this paper two wavelet based adaptive solvers are developed for linear advection dispersion equation the localization properties and multilevel structure of the wavelets in the physical space are used for adaptive computational methods for solution of equation which exhibit both smooth and shock like behaviour the first framework is based on wavelet galerkin and the second is based on multiscale decomposition of finite element method coiflet wavelet filter is incorporated in both the methods the main advantage of both the adaptive methods is the elimination of spurious oscillations at very high peclet number

fast surface interpolation using multiresolution wavelet transform discrete formulation of the surface interpolation problem usually leads to a large sparse linear equation system due to the poor convergence condition of the equation system the convergence rate of solving this problem with iterative method is very slow to improve this condition a multiresolution basis transfer scheme based on the wavelet transform is proposed by applying the wavelet transform the original interpolation basis is transformed into two sets of bases with larger supports while the admissible solution space remains unchanged with this basis transfer a new set of nodal variables results and an equivalent equation system with better convergence condition can be solved the basis transfer can be easily implemented by using an qmf matrix pair associated with the chosen interpolation basis the consequence of the basis transfer scheme can be regarded as a preconditioner to the subsequent iterative computation method the effect of the transfer is that the interpolated surface is decomposed into its low frequency and high frequency portions in the frequency domain it has been indicated that the convergence rate of the interpolated surface is dominated by the low frequency portion with this frequency domain decomposition the low frequency portion of the interpolated surface can be emphasized as compared with other acceleration methods this basis transfer scheme provides a more systematical approach for fast surface interpolation the easy implementation and high flexibility of the proposed algorithm also make it applicable to various regularization problems

enhanced b wavelets via mixed composite packets a modified b wavelet construction with enhanced filter characteristics is considered the design comprises a superposition of tessellated integer dilated sister wavelet functions we here propose a cascaded filter bank realization of this wavelet family together with some notable extensions we prove that modifications of low order members exist in the multiresolution subspace spanned by the half translates of the original wavelets and hence that the resulting modified wavelet coefficients can be computed as convolutions of the undecimated original wavelet coefficients finite impulse response filters are thus designed and incorporated into a b wavelet packet architecture such that the mainlobe to sidelobe ratio of the resulting wavelet filter characteristic is improved this is achieved by designing the filters so that zeros are introduced near to the maxima of the harmonics it is shown that the numbers of zeros can be balanced with the length of the corresponding filters by controlling the modification order several constructions are presented we prove that two such constructions satisfy the perfect reconstruction property for all orders the resulting modified wavelets preserve many of the properties of the original b wavelets such as differentiability number of vanishing moments symmetry anti symmetry finite support and the existence of a closed form expression

two direction refinable functions and two direction wavelets with dilation factor m abstract motivated by cardinal b spline scaling functions we introduce the concept of two direction refinable functions and two direction wavelets with dilation factor m we investigate the following two direction refinement equation x k p k mx k k p k k mx where m 2 is an integer here two sequences p k k and p k k are called the positive mask and negative mask respectively based on the positive mask and negative mask the conditions that the above equation has compactly distributional solutions are established the conditions that the above equation has l2 stable solutions are also presented the support of x is discussed amply an algorithm for constructing orthogonal two direction refinable functions and their two direction wavelets is presented based on orthogonal two direction scaling functions and two direction wavelets orthogonal and symmetric multiscaling functions and their multiwavelets can be constructed easily in the end a construction example is given

robust l estimation based forms of signal transforms and time frequency representations the l estimation based signal transforms and time frequency tf representations are introduced by considering the corresponding minimization problems in the huber 1981 1998 estimation theory the standard signal transforms follow as the maximum likelihood solutions for the gaussian additive noise environment for signals corrupted by an impulse noise the median based transforms produce robust estimates of the non noisy signal transforms when the input noise is a mixture of gaussian and impulse noise the l estimation based signal transforms can outperform other estimates in quadratic and higher order tf analysis the resulting noise is inherently a mixture of the gaussian input noise and an impulse noise component in this case the l estimation based signal representations can produce the best results these transforms and tf representations give the standard and the median based forms as special cases a procedure for parameter selection in the l estimation is proposed the theory is illustrated and checked numerically

biorthogonal two direction refinable function and two direction wavelet the concept of two direction refinable function and two direction wavelet is generalized to the biorthogonal setting the definition of biorthogonal two direction refinable function is introduced and the biorthogonality criteria for two direction refinable function is established we present an algorithm for constructing a pair of biorthogonal two direction refinable function and the corresponding biorthogonal two direction wavelet finally we give two examples illustrating how to use our method to construct a pair of biorthogonal refinable function and its corresponding biorthogonal two direction wavelet pair

wavelet localization of the radon transform the authors develop an algorithm which significantly reduces radiation exposure in x ray tomography when a local region of the body is to be imaged the algorithm uses the properties of wavelets to essentially localize the radon transform this algorithm differs from previous algorithms for doing local tomography because it recovers an approximation to the original image not the image module the nullspace of the local tomography operator or the lambda transform of the image this is possible because the authors do not truly invert the interior radon transform but rather sample the radon transform sparsely away from the local region of interest much attention in the field has been directed towards localized tomography the authors believe that this technique represents a significant contribution towards this effort

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

pointwise error bounds for orthogonal cardinal spline approximation for orthogonal cardinal spline approximation closed form expressions of the reproducing kernel and the peano kernels in terms of exponential splines are proved concrete and sharp pointwise error bounds are deduced for low degree splines

hybrid spline frames using their unitary extension principle ron and shen have constructed a normalized tight frame for l 2 consisting of spline functions with uniform knots this paper constructs a normalized tight frame for l 2 0 oo comprising spline functions with knots on a hybrid of uniform and geometric mesh the construction is motivated by applications in adaptive approximation using spline functions on a hybrid mesh that admits a natural dyadic multiresolution approximation of l 2 0 based on dilation and translation

semi orthogonal wavelet packet bases for parallel least squares approximation a generalization to the wavelet packet case of the semi orthogonal wavelets given by c k chui is presented and a simple numerical algorithm for their practical evaluation is developed to provide a motivation it is shown how using these new bases the least squares approximation problem in a multiresolution space can easily be solved by means of an efficient parallel algorithm

iterative method for computing the moore penrose inverse based on penrose equations an iterative algorithm for estimating the moore penrose generalized inverse is developed the main motive for the construction of the algorithm is simultaneous usage of penrose equations 2 and 4 convergence properties of the introduced method as well as their first order and second order error terms are considered numerical experiment is also presented

an algorithm for matrix extension and wavelet construction this paper gives a practical method of extending an n x r matrix p z r n with laurent polynomial entries in one complex variable z to a square matrix also with laurent polynomial entries if p z has orthonormal columns when z is restricted to the torus t it can be extended to a paraunitary matrix if p z has rank r for each z t it can be extended to a matrix with nonvanishing determinant on t the method is easily implemented in the computer it is applied to the construction of compactly supported wavelets and prewavelets from multiresolutions generated by several univariate scaling functions with an arbitrary dilation parameter

a class of numerical algorithms for computing outer inverses the aim of this study is to present a class of numerical algorithms for finding outer inverses with prescribed range and null space general convergence theorems are proven some particular cases of the general algorithm are introduced and demonstrated theoretical convergence speed and computational efficiency will further be supported by employing some numerical tests

a fast and accurate algorithm for solving bernstein vandermonde linear systems abstract a fast and accurate algorithm for solving a bernstein vandermonde linear system is presented the algorithm is derived by using results related to the bidiagonal decomposition of the inverse of a totally positive matrix by means of neville elimination the use of explicit expressions for the determinants involved in the process serves to make the algorithm both fast and accurate

bidiagonal decomposition of rectangular totally positive said ball vandermonde matrices error analysis perturbation theory and applications abstract an algorithm for computing the bidiagonal decomposition of said ball vandermonde matrices in the rectangular case is presented which allows one to use known algorithms for totally positive matrices represented by their bidiagonal decomposition the algorithm is fast and possesses high relative accuracy the error analysis of the algorithm is also addressed along with the perturbation theory for the bidiagonal factorization of totally positive said ball vandermonde matrices some numerical experiments showing the good behavior of the algorithm are included

a theory for multiresolution signal decomposition the wavelet representation multiresolution representations are effective for analyzing the information content of images the properties of the operator which approximates a signal at a given resolution were studied it is shown that the difference of information between the approximation of a signal at the resolutions 2 sup j 1 and 2 sup j where j is an integer can be extracted by decomposing this signal on a wavelet orthonormal basis of l sup 2 r sup n the vector space of measurable square integrable n dimensional functions in l sup 2 r a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function psi x this decomposition defines an orthogonal multiresolution representation called a wavelet representation it is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters wavelet representation lies between the spatial and fourier domains for images the wavelet representation differentiates several spatial orientations the application of this representation to data compression in image coding texture discrimination and fractal analysis is discussed

time varying system identification using an ultra orthogonal forward regression and multiwavelet basis functions with applications to eeg a new parametric approach is proposed for nonlinear and nonstationary system identification based on a time varying nonlinear autoregressive with exogenous input tv narx model the tv coefficients of the tv narx model are expanded using multiwavelet basis functions and the model is thus transformed into a time invariant regression problem an ultra orthogonal forward regression uofr algorithm aided by mutual information mi is designed to identify a parsimonious model structure and estimate the associated model parameters the uofr mi algorithm which uses not only the observed data themselves but also weak derivatives of the signals is more powerful in model structure detection the proposed approach combining the advantages of both the basis function expansion method and the uofr mi algorithm is proved to be capable of tracking the change of tv parameters effectively in both numerical simulations and the real eeg data

stable edge adaptive multiscale decompositions using updated normal offsets the normal offset decomposition is an adaptive nonlinear multiscale data transform constructed in a way similar to the lifting scheme for wavelet transforms the main difference lies in the fact that the detail coefficients are not wavelet coefficients corresponding to a wavelet basis function wavelet coefficients carry detail information situated at the location of the basis function besides detail information normal offset coefficients also contain directional geometrical information i e information on where to locate the details the normal offset decomposition is a jump adaptive transform leading to a sparse representation of edges in images and sharp reconstructions of these edges a major issue in this paper is the design of a new class of wavelet transforms that can be extended to normal offset decompositions the combination of normal offsets and a stable lifting scheme is nontrivial a necessary condition for an stable lifted wavelet transform is that the wavelet basis functions have a vanishing integral i e at least one vanishing moment although the normal offset decomposition is not a basis transform in the strict sense a similar criterion for stability can be established and instability is equally a problem in a naive implementation this paper constructs an ldquoupdatedrdquo normal offset decomposition that is stable and therefore appropriate for use in applications that sensitive to the numerical condition such as applications involving heavy noise the new scheme combines numerical stability fast computation and sharp edge representations

riesz bases of wavelets and applications to numerical solutions of elliptic equations we investigate riesz bases of wavelets in sobolev spaces and their applications to numerical solutions of the biharmonic equation and general elliptic equations of fourth order first we study bi cubic splines on the unit square with homogeneous boundary conditions the approximation properties of these cubic splines are established and applied to convergence analysis of the finite element method for the biharmonic equation second we develop a fairly general theory for riesz bases of hilbert spaces equipped with induced norms under the guidance of the general theory we are able to construct wavelet bases for sobolev spaces on the unit square the condition numbers of the stiffness matrices associated with the wavelet bases are relatively small and uniformly bounded third we provide several numerical examples to show that the numerical schemes based on our wavelet bases are very efficient finally we extend our study to general elliptic equations of fourth order and demonstrate that our numerical schemes also have superb performance in the general case

two direction poly scale refinability the concept of two direction poly scale refinable functions with dilation factor a and shift invariant spaces is introduced the existence of solutions of two direction poly scale refinable equations is investigated in particular a necessary and sufficient condition for the orthonormality of the solutions is established some properties of two direction poly scale refinable functions are discussed finally two examples of two direction poly scale refinable functions are given

sub band operators and saddle point wavelets in this paper a concept of sub band operator is defined a 2 circular matrix method is developed and the exact bounds of the sub band operators are obtained the method to calculate the bounds of the sub band operators is described by computing the maximum of a function in addition it is found that the size of bound of a sub band operator is very sensitive to the performance of wavelet transforms so a model to minimize the bounds is built and then a class of wavelets namely saddle point wavelets is designed experiments show that some saddle point wavelets with the filters of even lengths perform better than the well known 9 7 tap wavelet in terms of image compression

a multiresolution analysis for tensor product splines using weighted spline wavelets we construct biorthogonal spline wavelets for periodic splines which extend the notion of lazy wavelets for linear functions where the wavelets are simply a subset of the scaling functions to splines of higher degree we then use the lifting scheme in order to improve the approximation properties with respect to a norm induced by a weighted inner product with a piecewise constant weight function using the lifted wavelets we define a multiresolution analysis of tensor product spline functions and apply it to image compression of black and white images by performing as a model problem image compression with black and white images we demonstrate that the use of a weight function allows to adapt the norm to the specific problem

total positivity and neville elimination abstract neville elimination is described in terms of schur complements of matrices and used to improve some well known characterizations of totally positive and strictly totally positive matrices

two dimensional mesh embedding for b spline methods many factors motivate consideration of b splines as basis functions for solving partial differential equations these are arbitrary orders of accuracy and high resolving powers similar to those of compact schemes furthermore if one uses a galerkin scheme one gets in addition to conservation of the discretized quantities conservation of quadratic invariants such as energy this work develops another property namely the ability to treat semi structured embedded or zonal meshes for two dimensional geometries this can drastically reduce the number of grid points in many applications an algorithm is presented for constructing a global spline basis that automatically hasd 1 continuous derivatives at mesh block boundaries as everywhere else heredis the polynomial degree the basis functions are simply suitable products of one dimensional b splines both integer and noninteger refinement ratios are allowed across mesh blocks finally test cases for linear scalar equations such as the poisson and advection equation are presented

two improvements of the iterative method for computing moore penrose inverse based on penrose equations abstract two improvements of the iterative method for computing the moore penrose inverse introduced in petkovic and stanimirovic 2011 are introduced the first improvement defines new choice for the initial approximation and ensures better stability of the method the second one defines its extension to the set of outer inverses with prescribed range and null space numerical examples for both of these improvements are presented

neural networks for localized approximation we prove that feedforward artificial neural networks with a single hidden layer and an ideal sigmoidal response function cannot provide localized approximation in a euclidean space of dimension higher than one we also show that networks with two hidden layers can be designed to provide localized approximation since wavelet bases are most effective for local approximation we give a discussion of the implementation of spline wavelets using multilayered networks where the response function is a sigmoidal function of order at least two

fractional splines and wavelets we extend schoenberg s family of polynomial splines with uniform knots to all fractional degrees alpha 1 these splines which involve linear combinations of the one sided power functions x alpha max 0 x alpha are alpha holder continuous for alpha 0 we construct the corresponding b splines by taking fractional finite differences and provide an explicit characterization in both time and frequency domains we show that these functions satisfy most of the properties of the traditional b splines including the convolution property and a generalized fractional differentiation rule that involves finite differences only we characterize the decay of the b splines that are not compactly supported for nonintegral alpha s their most astonishing feature in reference to the strang fix theory is that they have a fractional order of approximation alpha 1 while they reproduce the polynomials of degree lceil alpha rceil for alpha frac 1 2 they satisfy all the requirements for a multiresolution analysis of ll 2 riesz bounds two scale relation and may therefore be used to build new families of wavelet bases with a continuously varying order parameter our construction also yields symmetrized fractional b splines which provide the connection with duchon s general theory of radial m s splines including thin plate splines in particular we show that the symmetric version of our splines can be obtained as the solution of a variational problem involving the norm of a fractional derivative

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

efficient wavelets based valuation of synthetic cdo tranches we present new formulae for the valuation of synthetic collateralized debt obligation cdo tranches under a one factor model these formulae are based on the wavelet theory and the method used is called wa a b we approximate the cumulative distribution function cdf of the underlying pool by a finite combination of j th order b spline basis where the b spline basis of order zero is typically called a haar basis we provide an error analysis and we show that for this type of distributions the rate of convergence in the approximation is similar regardless of the order of the b spline basis employed the resulting formula for the haar basis case is much easier to implement and performs better than the formula for the b spline basis of order one in terms of computational time the numerical experiments confirm the impressive speed and accuracy of the wa a b method equipped with a haar basis independently of the inhomogeneity features of the underlying pool the method appears to be particularly fast for multiple tranche valuation

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

accurate computations with collocation matrices of rational bases the rational bernstein basis plays a crucial role in computer aided geometric design the collocation matrices of this basis are called rational bernstein vandermonde matrices in this paper we provide algorithms for computing the bidiagonal decomposition of rational bernstein vandermonde matrices and their inverses with high relative accuracy similar results are obtained for the collocation matrices of another important rational basis the rational said ball basis it is also shown that these algorithms can be used to perform accurately some computations with these matrices such us the calculation of their inverses their eigenvalues or their singular values numerical experiments illustrate the results

accurate bidiagonal decomposition of totally positive h bernstein vandermonde matrices and applications abstract an algorithm for the computation of the bidiagonal decomposition of strictly totally positive h bernstein vandermonde matrices is presented these matrices are collocation matrices of h bernstein bases a generalization of the bernstein basis of the space of polynomials of degree less than or equal to n the algorithm is fast its computational complexity is o n 2 and has high relative accuracy the computation of this bidiagonal decomposition is used as the initial stage for the accurate and efficient solution of several linear algebra problems with these matrices linear system solving eigenvalue computation and inverse computation numerical experiments are also included

c 1 spline wavelets on triangulations in this paper we investigate spline wavelets on general triangulations in particular we are interested in c 1 wavelets generated from piecewise quadratic polynomials by using the powell sabin elements we set up a nested family of spaces of c 1 quadratic splines which are suitable for multiresolution analysis of besov spaces consequently we construct c 1 wavelet bases on general triangulations and give explicit expressions for the wavelets on the three direction mesh a general theory is developed so as to verify the global stability of these wavelets in besov spaces the wavelet bases constructed in this paper will be useful for numerical solutions of partial differential equations

scale space derived from b splines this paper proposes a scale space theory based on b spline kernels our aim is twofold 1 present a general framework and show how b splines provide a flexible tool to design various scale space representations in particular we focus on the design of continuous scale space and dyadic scale space frame representations a general algorithm is presented for fast implementation of continuous scale space at rational scales in the dyadic case efficient frame algorithms are derived using b spline techniques to analyze the geometry of an image the relationship between several scale space approaches is explored the behavior of edge models the properties of completeness causality and other properties in such a scale space representation are examined in the framework of b splines it is shown that besides the good properties inherited from the gaussian kernel the b spline derived scale space exhibits many advantages for modeling visual mechanism including the efficiency compactness orientation feature and parallel structure

biorthogonal butterworth wavelets derived from discrete interpolatory splines we present a new family of biorthogonal wavelet transforms and a related library of biorthogonal periodic symmetric waveforms for the construction we used the interpolatory discrete splines which enabled us to design a library of perfect reconstruction filterbanks these filterbanks are related to butterworth filters the construction is performed in a lifting manner the difference from the conventional lifting scheme is that all the transforms are implemented in the frequency domain with the use of the fast fourier transform fft two ways to choose the control filters are suggested the proposed scheme is based on interpolation and as such it involves only samples of signals and it does not require any use of quadrature formulas these filters have a linear phase property and the basic waveforms are symmetric in addition these filters yield refined frequency resolution

from cardinal spline wavelet bases to highly coherent dictionaries wavelet families arise by scaling and translations of a prototype function called the mother wavelet the construction of wavelet bases for cardinal spline spaces is generally carried out within the multi resolution analysis scheme thus the usual way of increasing the dimension of the multi resolution subspaces is by augmenting the scaling factor we show here that when working on a compact interval the identical effect can be achieved without changing the wavelet scale but reducing the translation parameter by such a procedure we generate a redundant frame called a dictionary spanning the same spaces as a wavelet basis but with wavelets of broader support we characterize the correlation of the dictionary elements by measuring their coherence and produce examples illustrating the relevance of highly coherent dictionaries to problems of sparse signal representation

on compactly supported spline wavelets and a duality principle let v 1 v 0 v 1 be a multiresolution analysis of l 2 generated by the mth order b spline n m x we exhibit a compactly supported basic wavelet m x that generates the corresponding orthogonal complementary wavelet subspaces w 1 w 0 w 1 consequently the two finite sequences that describe the two scale relations of n m x and m x in terms of n m 2x j j z yield an efficient reconstruction algorithm to give an efficient wavelet decomposition algorithm based on these two finite sequences we derive a duality principle which also happens to yield the dual bases n m x j and 59 m x j relative to n m x j and m x j respectively

an effective wavelet matrix transform approach for efficient solutions of electromagnetic integral equations an effective numerical method based on wavelet matrix transforms for efficient solution of electromagnetic em integral equations is proposed using the wavelet matrix transform produces highly sparse moment matrices which can be solved efficiently a fast construction method for various orthonormal or nonorthonormal wavelet basis matrices is also given it has been found that using nonsimilarity wavelet matrix transforms such as nonsimilarity nonorthonormal cardinal spline wavelet nsncsw transform one can obtain a much higher compression rate and much better accuracy of the approximate solutions than using similarity wavelet transforms such as daubechies 1992 orthonormal wavelet dow transform numerical examples are given to show the validity and effectiveness of the method

on asymptotic convergence of the dual filters associated with two families of biorthogonal wavelets the asymptotic behavior of the dual filters associated with biorthogonal spline wavelet bsw systems and general biorthogonal coifman wavelet gbcw systems are studied as the order of the wavelet systems approaches infinity the magnitude responses of the dual filters in the bsw systems either diverge or converge to some nonideal frequency responses however the synthesis filters in the gbcw systems converge to an ideal halfband lowpass filter without exhibiting any gibbs like phenomenon and a subclass of the analysis filters also converge to an ideal halfband lowpass filter but with a one sided gibbs like behavior the two approximations of the ideal lowpass filter by the filter associated with a daubechies orthonormal wavelet and by the synthesis filter in a gbcw system of the same order are compared such a study of the asymptotic behaviors of wavelet systems provides insightful characterization of these systems and systematic assessment and global comparison of different wavelet systems

construction of refinable function vector via gtst the concept of general two scale similarity transform gtst is introduced by applying gtst we present an algorithm of constructing a class of compactly supported refinable function vector in addition we prove that gtst can preserve the symmetry of the given a refinable function vector and give a sufficient condition that the corresponding transform matrices m z and n z satisfy when gtst can preserve symmetry some examples are given

accurate computations with lupa matrices lupa q analogues of the bernstein functions play an important role in approximation theory and computer aided geometric design their collocation matrices are called lupa matrices in this paper we provide algorithms for computing the bidiagonal decomposition of these matrices and their inverses to high relative accuracy it is also shown that these algorithms can be used to perform to high relative accuracy several algebraic calculations with these matrices such as the calculation of their inverses their eigenvalues or their singular values numerical experiments are included

a maximum likelihood digital receiver using coordinate ascent and the discrete wavelet transform in this paper a maximum likelihood ml method is presented for joint estimation of amplitude phase time delay and data symbols in a single user direct sequence spread spectrum communication system since maximization of the likelihood function is analytically intractable a novel coordinate ascent algorithm is used to obtain sequential updates of the data symbols and all unknown nuisance parameters the novelty of the algorithm is due to the use of a multiresolution expansion of the received signal and the use of polynomial rooting in the complex plane in place of a line search over the signal delay parameter the multiresolution structure of the algorithm is exploited to reduce sensitivity to impulsive noise via wavelet thresholding computer simulations of the single user system show that the algorithm has fast convergence and comparison with theoretical lower bounds establishes that the algorithm achieves nearly optimal error performance

accurate svds of polynomial vandermonde matrices involving orthonormal polynomials abstract we present a new o n3 algorithm for computing the svd of an n n polynomial vandermonde matrix vp pi 1 xj to high relative accuracy in o n3 time the pi are orthonormal polynomials deg pi i and xj are complex nodes the small singular values of vp can be arbitrarily smaller than the largest ones so that traditional algorithms typically compute them with no relative accuracy at all we show that the singular values even the tiniest ones are usually well conditioned functions of the data xj justifying this computation we also explain how this theory can be extended to other polynomial vandermonde matrices involving polynomials that are not orthonormal or even orthogonal

trigonometric wavelets for hermite interpolation the aim of this paper is to investigate a multiresolution analysis of nested subspaces of trigonometric polynomials the pair of scaling functions which span the sample spaces are fundamental functions for hermite interpolation on a dyadic partition of nodes on the interval 0 2 two wavelet functions that generate the corresponding orthogonal complementary subspaces are constructed so as to possess the same fundamental interpolatory properties as the scaling functions together with the corresponding dual functions these interpolatory properties of the scaling functions and wavelets are used to formulate the specific decomposition and reconstruction sequences consequently this trigonometric multiresolution analysis allows a completely explicit algorithmic treatment

spline prewavelets for non uniform knots in this paper we develop a framework suitable for performing a multiresolution analysis using univariate spline spaces of arbitrary degree and with non uniform knot sequences to this end we show among other things the existence of compactly supported prewavelets and of prewavelets that are globally supported but decay exponentially in each case we obtain a decomposition of a fine spline space as a sum of a coarse spline space plus a spline space spanned by prewavelets

variational principle of the zakharov shabat equations abstract we put forth a variational principle for the zakharov shabat zs equations which is the basis of the inverse scattering transform of a number of important nonlinear pde s using the variational representation of the zs equations we develop an approximate analytical technique for finding discrete eigenvalues of the complex spectral parameter in the zs equations for a given pulse shaped potential which is equivalent to the physically important problem of finding the soliton content of the given initial pulse we apply the technique to several particular shapes of the pulse and demonstrate that the simplest version of the variational approximation based on trial functions with one or two free parameters proves to be fully analytically tractable and it yields threshold conditions for the appearance of the first soliton or of the first soliton pair which are in a fairly good agreement with available numerical results however more free parameters are necessary to allow prediction of additional solitons produced by the given pulse

a block spin construction of ondelettes part i lemarie functions using block spin assignments we construct anl2 orthonormal basis consisting of dyadic scalings and translates of just a finite number of functions these functions have exponential localization and for even values of a construction parameterm one can make them classcm 1 with vanishing moments up to orderm inclusive such a basis has an important application to phase cell cluster expansions in quantum field theory

dispersion equation and eigenvalues for the zakharov shabat system using spectral parameter power series a general solution of the zakharov shabat system with a real valued potential is obtained in terms of spectral parameter power series it is used for deriving a dispersion characteristic equation for the zakharov shabat eigenvalue problem with a compactly supported potential due to its convenient form the dispersion equation lends itself to numerical solution providing a simple and accurate numerical method for solving the zakharov shabat eigenvalue problem

a unified wavelet based modelling framework for non linear system identification the wanarx model structure a new unified modelling framework based on the superposition of additive submodels functional components and wavelet decompositions is proposed for non linear system identification a non linear model which is often represented using a multivariate non linear function is initially decomposed into a number of functional components via the well known analysis of variance anova expression which can be viewed as a special form of the narx non linear autoregressive with exogenous inputs model for representing dynamic input output systems by expanding each functional component using wavelet decompositions including the regular lattice frame decomposition wavelet series and multiresolution wavelet decompositions the multivariate non linear model can then be converted into a linear in the parameters problem which can be solved using least squares type methods an efficient model structure determination approach based upon a forward orthogonal least squares ols algorithm which involves a stepwise o

full rank representations of outer inverses based on the qr decomposition abstract an efficient algorithm for computing a t s 2 inverses of a given constant matrix a based on the qr decomposition of an appropriate matrix w is presented correlations between the derived representation of outer inverses and corresponding general representation based on arbitrary full rank factorization are derived in particular cases we derive representations of 2 4 and 2 3 inverses numerical examples on different test matrices dense or sparse are presented as well as the comparison with several well known methods for computing the moore penrose inverse and the drazin inverse

componentwise backward error analysis of neville elimination abstract in this paper we perform a backward error analysis of neville elimination whenever there need row exchanges componentwise backward error bounds are presented for this elimination procedure applied to any nonsingular matrices consequently it is shown that neville elimination with two determinant pivoting proposed by cortes and pena 2007 5 is an excellent method for the triangularization of nonsingular sign regular matrices including totally nonnegative and totally nonpositive matrices which has a pleasantly small componentwise relative backward error in particular a small componentwise relative error bound is also provided for the bidiagonal factorization of totally nonnegative matrices

hyper power methods for the computation of outer inverses in this paper we propose new iterative schemes for the computation of outer inverse which reduce the total number of matrix multiplications per iteration in particular we consider how the hyper power method of orders 5 and 9 can be accelerated such that they require 4 and 5 matrix multiplications per iteration respectively these improvements are tested against quadratically convergent schultz method and fastest horner scheme hyper power method of order three numerical results show the superiority and practical applicability of the proposed methods finally it is shown that a possibly more efficient method should have the order at least r 14 making it useless for practical applications

spatiotemporal multi resolution approximation of the amari type neural field model neural fields are spatially continuous state variables described by integro differential equations which are well suited to describe the spatiotemporal evolution of cortical activations on multiple scales here we develop a multi resolution approximation mra framework for the integro difference equation ide neural field model based on semi orthogonal cardinal b spline wavelets in this way a flexible framework is created whereby both macroscopic and microscopic behavior of the system can be represented simultaneously state and parameter estimation is performed using the expectation maximization em algorithm a synthetic example is provided to demonstrate the framework

computation of the direct scattering transform for the nonlinear schroedinger equation abstract the cubic nonlinear schroedinger equation nls describes the space time evolution of narrow banded wave trains in one space and one time 1 1 dimensions the richness of nonlinear wave motions described by nls is exemplified by the fully nonlinear envelope soliton and breather solutions which are fully understood only in terms of the general solution of the equation as described by the inverse scattering transform 1st the method may be viewed as a nonlinear generalization of the linear fourier transform herein we develop a numerical algorithm for determining the scattering transform spectrum of a nonlinear wave train described by the nls equation the analysis of space or time series data obtained from computer simulations of nonlinear narrow band wave trains or from experimental measurements is thus a central point of discussion in particular we develop a numerical algorithm for computing the direct scattering transform dst which may be interpreted as the nonlinear fourier spectrum of the complex envelope function of a wave train the fact that the nonlinear fourier modes are constants of the motion for all time provides a physical basis for the analysis of data while the nonlinear fourier method is specifically applied to the nls equation the approach is easily generalized to include the class of spectral problems due to ablowitz kaup newell and segur akns this class includes several other nonlinear wave equations of physical interest in 1 1 including the kortewegde vries kdv modified kdv sine gordon and sinh gordon equations

optimized general sparse grid approximation spaces for operator equations this paper is concerned with the construction of optimized sparse grid approximation spaces for elliptic pseudodifferential operators of arbitrary order based on the framework of tensor product biorthogonal wavelet bases and stable subspace splittings we construct operator adapted subspaces with a dimension smaller than that of the standard full grid spaces but which have the same approximation order as the standard full grid spaces provided that certain additional regularity assumptions on the solution are fulfilled specifically for operators of positive order their dimension is o 2 j independent of the dimension n of the problem compared to o 2 jn for the full grid space also for operators of negative order the overall cost is significantly in favor of the new approximation spaces we give cost estimates for the case of continuous linear information we show these results in a constructive manner by proposing a galerkin method together with optimal preconditioning the theory covers elliptic boundary value problems as well as boundary integral equations

an approximation procedure for the zakharov shabat scattering problem for real single humped potentials a procedure is presented for finding simple approximations of the discrete eigenvalues of the zakharov shabat scattering problem corresponding to the nonlinear schrodinger equation the approximation is in the form of an interpolation formula which combines results for small eigenvalues obtained by a direct variational approach and for large eigenvalues obtained by the bohr sommerfeld quantization rule

fast implementation of the continuous wavelet transform with integer scales we describe a fast noniterative algorithm for the evaluation of continuous spline wavelet transforms at any integer scale m in this approach the input signal and the analyzing wavelet are both represented by polynomial splines the algorithm uses a combination of moving sum and zero padded filters and its complexity per scale is o n where n is the signal length the computation is exact and the implementation is noniterative across scales we also present examples of spline wavelets exhibiting properties that are desirable for either singularity detection first and second derivative operators or gabor like time frequency signal analysis

new exact solutions for the kdv equation with higher order nonlinearity by using the variational method the korteweg de vries kdv equation with higher order nonlinearity models the wave propagation in one dimensional nonlinear lattice a higher order extension of the familiar kdv equation is produced for internal solitary waves in a density and current stratified shear flow with a free surface the variational approximation method is applied to obtain the solutions for the well known kdv equation explicit solutions are presented and compared with the exact solutions very good agreement is achieved demonstrating the high efficiency of variational approximation method the existence of a lagrangian and the invariant variational principle for the higher order kdv equation are discussed the simplest version of the variational approximation based on trial functions with two free parameters is demonstrated the jost functions by quadratic cubic and fourth order polynomials are approximated also we choose the trial jost functions in the form of exponential and sinh solutions all solutions are exact and stable and have applications in physics

on a partial order defined by the weighted moore penrose inverse the weighted moore penrose inverse of a matrix can be used to define a partial order on the set of mxn complex matrices and to introduce the concept of weighted ep matrices in this paper we study the weighted star partial order on the set of weighted ep matrices in addition some properties that relate the eigenprojection at zero with the weighted star partial order are obtained

accurate bidiagonal decomposition of totally positive cauchy vandermonde matrices and applications abstract cauchy vandermonde matrices play a fundamental role in rational interpolation theory and in other fields when all their corresponding nodes are different and positive and all poles are different and negative and follow adequate orderings these matrices are totally positive in this paper we provide fast algorithms for computing bidiagonal factorizations of these matrices and their inverses with high relative accuracy these algorithms can be used to solve with high relative accuracy other algebraic problems such as the computation of all singular values all eigenvalues or the solution of certain linear systems the error analysis of the algorithm for computing the bidiagonal factorization and the corresponding perturbation theory are also performed

fast integral wavelet transform on a dense set of the time scale domain the objective of this paper is to introduce a fast algorithm for computing the integral wavelet transform iwt on a dense set of points in the time scale domain by applying the duality principle and using a compactly supported spline wavelet as the analyzing wavelet this fast integral wavelet transform fiwt is realized by applying only fir moving average operations and can be implemented in parallel since this computational procedure is based on a local optimal order spline interpolation scheme and the fir filters are exact the iwt values so obtained are guaranteed to have zero moments up to the order of the cardinal spline functions the semi orthogonal s o spline wavelets used here cannot be replaced by any other biorthogonal wavelet spline or otherwise which is not s o since the duality principle must be applied to some subspace of the multiresolution analysis under consideration in contrast with the existing procedures based on direct numerical integration or an fft based multi voice per octave scheme the computational complexity of our fiwt algorithm does not increase with the increasing number of values of the scale parameter

biorthogonal multiwavelets on the interval for numerical solutions of burgers equation to numerically solve the burgers equation in this paper we propose a general method for constructing wavelet bases on the interval 0 1 derived from symmetric biorthogonal multiwavelets on the real line in particular we obtain wavelet bases with simple structures on the interval 0 1 from the hermite cubic splines in comparison with all other known constructed wavelets on the interval 0 1 our constructed wavelet bases on the interval 0 1 from the hermite cubic splines not only have good approximation and symmetry properties with extremely short supports but also employ a minimum number of boundary wavelets with a very simple structure these desirable properties make them to be of particular interest in numerical algorithms our constructed wavelet bases on the interval 0 1 are then used to solve the nonlinear burgers equation our method is based on the finite difference formula combined with the collocation method therefore our proposed numerical scheme in this paper is abbreviated as mfdcm mixed finite difference and collocation method some numerical examples are provided to demonstrate the validity and applicability of our proposed method which can be easily implemented to produce a desired accuracy

applications of optimally local interpolation to interpolatory approximants and compactly supported wavelets the objective of this paper is to introduce a general scheme for the construction of interpolatory approximation formulas and compactly supported wavelets by using spline functions with arbitrary nonuniform knots both construction procedures are based on certain optimally local interpolatory fundamental spline functions which are not required to possess any approximation property

variational principle for zakharov shabat equations in two dimensions we study the corresponding scattering problem for zakharov and shabat compatible differential equations in two dimensions the representation for a solution of the nonlinear schrodinger equation is formulated as a variational problem in two dimensions we extend the derivation to the variational principle for the zakharov and shabat equations in one dimension we also developed an approximate analytical technique for finding discrete eigenvalues of the complex spectral parameters in zakharov and shabat equations for a given pulse shaped potential which is equivalent to the physically important problem of finding the soliton content of the given initial pulse using a trial function in a rectangular box we find the functional integral the general case for the two box potential can be obtained on the basis of a different ansatz where we approximate the jost function by polynomials of order n instead of a piecewise linear function we also demonstrated that the simplest version of the variational approximation based on trial functions with one two and n free parameters respectively and treated analytically

numerical solution of the integral equation of the second kind by using wavelet bases of hermite cubic splines in this paper we use the wavelet bases of hermite cubic splines to solve the second kind integral equationsx t 0 1k t s x s ds y t t 0 1 a pair of wavelets are constructed on the basis of hermite cubic splines this wavelets are in c 1 and supported on 0 2 moreover one wavelet is symmetric and the other is anti symmetric this spline wavelets are then adapted to the interval 0 1 the computational results demonstrate the advantage of the wavelet basis

uncertainty propagation using wiener linear b spline wavelet expansion abstract in this paper we have constructed a scheme combining generalized polynomial chaos gpc representation and b spline wavelets to begin with semi orthogonal compactly supported b spline wavelets are constructed for the bounded interval 0 1 which are used for pc expansion of possible stochastic processes to compute the deterministic coefficients of expansion we have applied galerkin projection on uncertain data and the solution variables then to ascertain the behavior of the random process the system of equations obtained from projection are integrated using fourth order runge kutta method to handle the nonlinearity we have compared galerkin projection with pseudo spectral projection the procedure is illustrated through three model problems of real life importance we conclude that galerkin approximation performs better in comparison to pseudo spectral approach which is numerically expected also it has been observed that the wavelet function based expansion shows superior results as compared to scaling function based expansion

semiclassical eigenvalue distribution of the zakharov shabat eigenvalue problem abstract in this paper we consider the semi classical limit of the non self adjoint zakharov shabat eigenvalue problem we conduct a series of careful numerical experiments which provide strong evidence that the number of eigenvalues scales like 1 just as in the self adjoint case and that the eigenvalues appear to approach a limiting curve one general choice of potential functions produces a y shaped spectrum we give an asymptotic argument which predicts a critical value for the phase for which the straight line spectra bifurcates to produce the y shaped spectra this asymptotic prediction agrees quite well with numerical experiments the asymptotic argument also predicts a symmetry breaking in the eigenfunctions which we are able to observe numerically we also show that the number of eigenvalues living away from the real axis for a restricted class of potentials is bounded by c 1 where c is an explicit constant a complete theory of the shape of the eigenvalue curve and a general bound on the number of eigenvalues is still lacking

an improved method for the computation of the moore penrose inverse matrix abstract in this article we provide a fast computational method in order to calculate the moore penrose inverse of singular square matrices and of rectangular matrices the proposed method proves to be much faster and has significantly better accuracy than the already proposed methods while works for full and sparse matrices

accurate computation of the moore penrose inverse of strictly totally positive matrices abstract the computation of the moore penrose inverse of structured strictly totally positive matrices is addressed since these matrices are usually very ill conditioned standard algorithms fail to provide accurate results an algorithm based on the q r factorization and which takes advantage of the special structure and the totally positive character of these matrices is presented the first stage of the algorithm consists of the accurate computation of the bidiagonal decomposition of the matrix numerical experiments illustrating the good behavior of our approach are included

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

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

a new family of spline based biorthogonal wavelet transforms and their application to image compression in this paper we design a new family of biorthogonal wavelet transforms and describe their applications to still image compression the wavelet transforms are constructed from various types of interpolatory and quasiinterpolatory splines the transforms use finite impulse response and infinite impulse response filters that are implemented in a fast lifting mode

a matricial description of neville elimination with applications to total positivity abstract the neville elimination process used by the authors in some previous papers in connection with totally positive matrices is studied in detail in the case of nonsingular matrices a wide class of matrices is found where neville elimination has a lower computational cost than gauss elimination finally some new characterizations are obtained for strictly totally positive and nonsingular totally positive matrices in terms of their neville elimination and that of their inverses

fast algorithms for periodic spline wavelets on sparse grids we consider boolean sums of univariate interpolation operators which define multivariate jth order blending interpolation operators on sparse grids sample spaces are defined as range of the blending operators sample and wavelet spaces have significantly lower dimension and good approximation order for certain function spaces fast decomposition and reconstruction algorithms for bivariate spline wavelets based on algorithms for univariate functions are described operation counts for the algorithms are given and it is shown that the complexity depends linearly on the dimension of sample spaces

wavelet theory demystified we revisit wavelet theory starting from the representation of a scaling function as the convolution of a b spline the regular part of it and a distribution the irregular or residual part this formulation leads to some new insights on wavelets and makes it possible to rederive the main results of the classical theory including some new extensions for fractional orders n a self contained accessible fashion in particular we prove that the b spline component is entirely responsible for five key wavelet properties order of approximation reproduction of polynomials vanishing moments multiscale differentiation property and smoothness regularity of the basis functions we also investigate the interaction of wavelets with differential operators giving explicit time domain formulas for the fractional derivatives of the basis functions this allows us to specify a corresponding dual wavelet basis and helps us understand why the wavelet transform provides a stable characterization of the derivatives of a signal additional results include a new peeling theory of smoothness leading to the extended notion of wavelet differentiability in the l sub p sense and a sharper theorem stating that smoothness implies order