parametrization of compactly supported orthonormal wavelets the parametrization of two channel reconstruction quadrature mirror filter banks is extended to obtain a single characterization of discrete orthonormal wavelets in particular it is shown that discrete orthonormal wavelets of support less than 2n 1 where n is an even integer are parametrized by n 1 real parameters each taking values in 0 2 pi a parametrization of discrete orthonormal wavelets with a number of vanishing moments greater than one is also given

wavelets with convolution type orthogonality conditions wavelets with free parameters are constructed using a convolution type orthogonality condition first finer and coarser scaling function spaces are introduced with the help of a two scale relation for scaling functions an inner product and a norm having convolution parameters are defined in the finer scaling function space which becomes a hilbert space as a result the finer scaling function space can be decomposed into the coarser one and its orthogonal complement a wavelet function is constructed as a mother function whose shifted functions form an orthonormal basis in the complement space such wavelet functions contain the daubechies compactly supported wavelets as a special case in some restricted cases several symmetric and almost compactly supported wavelets are constructed analytically by tuning free convolution parameters contained in the wavelet functions

minimax entropy principle and its application to texture modeling this article proposes a general theory and methodology called the minimax entropy principle for building statistical models for images or signals in a variety of applications this principle consists of two parts the first is the maximum entropy principle for feature binding or fusion for a given set of observed feature statistics a distribution can be built to bind these feature statistics together by maximizing the entropy over all distributions that reproduce them the second part is the minimum entropy principle for feature selection among all plausible sets of feature statistics we choose the set whose maximum entropy distribution has the minimum entropy computational and inferential issues in both parts are addressed in particular a feature pursuit procedure is proposed for approximately selecting the optimal set of features the minimax entropy principle is then corrected by considering the sample variation in the observed feature statistics and an information criterion for feature pursuit is derived the minimax entropy principle is applied to texture modeling where a novel markov random field mrf model called frame filter random field and minimax entropy is derived and encouraging results are obtained in experiments on a variety of texture images the relationship between our theory and the mechanisms of neural computation is also discussed

construction of parseval wavelets from redundant filter systems we consider wavelets in l2 rd which have generalized multiresolutions this means that the initial resolution subspace v0 in l2 rd is not singly generated as a result the representation of the integer lattice zd restricted to v0 has a nontrivial multiplicity function we show how the corresponding analysis and synthesis for these wavelets can be understood in terms of unitary matrix valued functions on a torus acting on a certain vector bundle specifically we show how the wavelet functions on rd can be constructed directly from the generalized wavelet filters

exponential bases paley wiener spaces and applications abstract we investigate the connection between translation bases for paley wiener spaces and exponential fourier bases for a domain we apply these results to the characterization of vector valued time frequency translations of a paley wiener window signal

a wavelet or lifting scheme based imputation method summary the paper proposes a new approach to imputation using the expected sparse representation of a surface in a wavelet or lifting scheme basis our method incorporates a bayesian mixture prior for these wavelet coefficients into a gibbs sampler to generate a complete posterior distribution for the variable of interest intuitively the estimator operates by borrowing strength from those observed neighbouring values to impute at the unobserved sites we demonstrate the strong performance of our estimator in both one and two dimensional imputation problems where we also compare its application with the standard imputation techniques of kriging and thin plate splines

saturation theorems for interpolation and the bernstein schnabl operator we shall study properties of box spline operators cardinal interpolation convolution and the bernstein schnabl operator we prove the saturation theorem

biorthogonal spline type wavelets let f be an orthonormal scaling function with approximation degree p 1 and let b nbe the b spline of order n then spline type scaling functions defined by n b n n 1 2 possess higher approximation order p n 1 and compact support the corresponding biorthogonal wavelet functions are also constructed this technique is extended to the case of biorthogonal scaling function system as an application of the method supplied in this paper one can easily construct a sequence of pairs of biorthogonal spline type scaling functions from one pair of biorthogonal scaling functions or an orthonormal scaling function in particular if both the method and the lifting scheme of sweldens see 1 are applied then all pairs of biorthogonal spline type scaling functions shown in references 2 and 3 can be constructed from the haar scaling function

wavelets generated by using discrete singular convolution kernels this paper explores the connection between wavelet methods and an efficient computational algorithm the discrete singular convolution dsc many new dsc kernels are constructed and they are identified as wavelet scaling functions two approaches are proposed to generate wavelets from dsc kernels two well known examples the canny filter and the mexican hat wavelet are found to be special cases of the present dsc kernel generated wavelets a family of wavelet generators proposed in this paper are found to form an infinite dimensional lie group which has an invariant subgroup of translation and dilation if dsc kernels form an orthogonal system they are found to span a wavelet subspace in a multiresolution analysis

computational approach to solvability of refinement equations the solvability and fredholm properties of refinement equations in spaces of square integrable functions are studied necessary and jointly necessary and sufficient conditions for the solvability of homogeneous and non homogeneous refinement equations are established it is shown that in the space l 2 r the kernel space of any homogeneous equation with a non trivial solution is infinite dimensional moreover the solvability problem is reduced to the study of singular values of certain matrix sequences these sequences arise from galerkin approximations of auxiliary linear operators the corresponding constructions use only the coefficients of refinement equations that generate multiresolution analysis and the coefficients of the refinement equation studied for the equations with polynomial symbols the most complete results are obtained if the corresponding operator is considered on an appropriate subspace of the space l 2

nonuniform multiresolution analyses and spectral pairs abstract a generalization of the notion of multiresolution analysis based on the theory of spectral pairs is considered in contrast to the standard setting the associated subspacev0ofl2 r has as an orthonormal basis a collection of translates of the scaling function of the form x where 0 r n 2z n 1 is an integer andris an odd integer with 1 r 2n 1 such thatrandnare relatively prime andzis the set of all integers furthermore the corresponding dilation factor is 2n the case wheren 1 corresponding to the usual definition of a multiresolution analysis with dilation factor 2 a necessary and sufficient condition for the existence of associated wavelets which is always satisfied whenn 1 or 2 is obtained and is shown to always hold if the fourier transform of is a constant multiple of the characteristic function of a set

information theoretic wavelet noise removal for inverse elastic wave scattering theory a discussion of noise removal in ultrasound elastic wave scattering for nondestructive evaluation is given the methods used in this paper include a useful suboptimal wiener filter information theory and orthonormal wavelets the multiresolution analysis mra due to mallat is the key wavelet feature used here whereas fourier transforms have a translational symmetry wavelets have a dilation or affine symmetry which consists of the semi direct product of a translation with a change of scale of the variable the mra describes the scale change features of orthonormal wavelet families first an empirical method of noise removal from scattered elastic waves using wavelets is shown to markedly improve the l 1 and l 2 error norms this suggests that the wavelet scale can act as dial to tune out noise maximization of the kullback liebler information is also shown to provide a scale dependent noise removal technique that supports but does not prove the intuition that certain small energy coefficients that are retained contain large information content the wavelet mra thereby locates islands of information in the phase space of the signal it is conjectured that this method holds more generally

families of biorthogonal wavelets several families of biorthogonal wavelet bases are constructed with various properties in particular for a given filter fn of finite length 2n 1 a parametric family of dual filters fvn of length 2n 1 is constructed the parametric nature of the dual filters makes it possible to design the optimum dual filter fv0n corresponding to a fixed filter fn

development of cfl free explicit schemes for multidimensional advection reaction equations we combine an eulerian lagrangian approach and multiresolution analysis to develop unconditionally stable explicit multilevel methods for multidimensional linear hyperbolic equations the derived schemes generate accurate numerical solutions even if large time steps are used furthermore these schemes have the capability of carrying out adaptive compression without introducing mass balance error computational results are presented to show the strong potential of the numerical methods developed

multiresolution representation in unstructured meshes in this paper we describe techniques to represent data which originate from discretization of functions in unstructured meshes in terms of their local scale components to do so we consider a nested sequence of discretization which corresponds to increasing levels of resolution and we define the scales as the difference in information between any two successive levels we obtain data compression by eliminating scale coefficients which are sufficiently small this capability for data compression can be used to reduce the cost of numerical schemes by solving for the more compact representation of the numerical solution in terms of its significant scale coefficients

a random laplace transform method for solving random mixed parabolic differential problems this paper deals with the explicit solution of random mixed parabolic equations in unbounded domains by using the random laplace transform to second order stochastic processes the mean square random laplace operational calculus is stated and its application to the random parabolic equation together with previous results of the underlying random ordinary differential equations allow us to obtain an explicit solution of the problem a numerical example which includes simulations illustrates the developed method

wavelet methods for pdes some recent developments abstract this paper is concerned with recent developments of wavelet schemes for the numerical treatment of operator equations with special emphasis on two issues adaptive solution concepts and nontrivial domain geometries after describing a general multiresolution framework the key features of wavelet bases are highlighted namely locality norm equivalences and cancellation properties assuming first that wavelet bases with these properties are available on the relevant problem domains the relevance of these features for a wide class of stationary problems is explained in subsequent sections the main issues are preconditioning and the efficient adaptive application of wavelet representations of the involved operators we indicate then how these ingredients combined with concepts from nonlinear or best n term approximation culminate in an adaptive wavelet scheme for elliptic selfadjoint problems covering boundary value problems as well as boundary integral equations these schemes can be shown to exhibit convergence rates that are in a certain sense asymptotically optimal we conclude this section with some brief remarks on data structures and implementation interrelations with regularity in a certain scale of besov spaces and strategies of extending such schemes to unsymmetric or indefinite problems we address then the adaptive evaluation of nonlinear functionals of wavelet expansions as a central task arising in connection with nonlinear problems wavelet bases on nontrivial domains are discussed next the main issues are the development of fourier free construction principles and criteria for the validity of norm equivalences finally we indicate possible combinations of wavelet concepts with conventional discretizations such as finite element or finite volume schemes in connection with convection dominated and hyperbolic problems

the wavelet transform of stochastic processes with stationary increments and its application to fractional brownian motion the wavelet transform of random processes with wide sense stationary increments is shown to be a wide sense stationary process whose correlation function and spectral distribution are determined the second order properties of the coefficients in the wavelet orthonormal series expansion of such processes is obtained applications to the spectral analysis and to the synthesis of fractional brownian motion are given

on the adaptive numerical solution of nonlinear partial differential equations in wavelet bases this work develops fast and adaptive algorithms for numerically solving nonlinear partial differential equations of the formut lu nf u where l and n are linear differential operators andf u is a nonlinear function these equations are adaptively solved by projecting the solutionuand the operators l and n into a wavelet basis vanishing moments of the basis functions permit a sparse representation of the solution and operators using these sparse representations fast and adaptive algorithms that apply operators to functions and evaluate nonlinear functions are developed for solving evolution equations for a wavelet representation of the solutionuthat containsnssignificant coefficients the algorithms update the solution usingo ns operations the approach is applied to a number of examples and numerical results are given

cardinal interpolation by multivariate splines the purpose of this paper is to investigate cardinal interpolation using locally supported piecewise polynomials in particular the notion of a commutator is introduced and its connection with the marsden identity is observed the order of a commutator is shown to be equivalent to the strang and fix conditions that arise in the study of the local approxima tion orders using quasi interpolants we also prove that scaled cardinal interpolants give these local approximation orders introduction cardinal interpolation by bivariate box splines was first studied by de boor hollig and riemenschneider 3 the purpose of our paper is to investigate the cardinal interpolation problem from a different point of view in particular the notion of a commutator is introduced it will be shown that this notion generalizes the marsden identity for univariate splines to the multivariate setting the order of a commutator will be shown to be equivalent to the strang and fix conditions used in the study of the order of controlled approximation by dahmen and micchelli 8 or local approximation by de boor and jia 4 an application to obtain approximation orders through the constructive method of scaled cardinal interpolation will also be studied in this paper 1 preliminaries this section consists of preliminary material for multivariate cardinal interpolation our approach is motivated by the work in 5 and 13 where cardinal interpolation in l2 was connected with certain convolution operators lr let z denote the set of integers and z the nonnegative ones for any given complex sequence by

periodic quasi orthogonal spline bases and applications to least squares curve fitting of digital images presents a new covariant basis dubbed the quasi orthogonal q spline basis for the space of n degree periodic uniform splines with k knots this basis is obtained analogously to the b spline basis by scaling and periodically translating a single spline function of bounded support the construction hinges on an important theorem involving the asymptotic behavior in the dimension of the inverse of banded toeplitz matrices the authors show that the gram matrix for this basis is nearly diagonal hence the name quasi orthogonal the new basis is applied to the problem of approximating closed digital curves in 2d images by least squares fitting since the new spline basis is almost orthogonal the least squares solution can be approximated by decimating a convolution between a resolution dependent kernel and the given data the approximating curve is expressed as a linear combination of the new spline functions and new control points another convolution maps these control points to the classical b spline control points a generalization of the result has relevance to the solution of regularized fitting problems

time frequency featured co movement between the stock and prices of crude oil and gold abstract the nonlinear relationships among variables caused by the hidden frequency information complicate the time series analysis to shed more light on this nonlinear issue we examine their relationships in joint time frequency domain with multivariate framework and the analyses in the time domain and frequency domain serve as comparisons the daily brent oil prices london gold fixing price and shanghai composite index from january 1991 to september 2014 are adopted as example first they have long term cointegration relationship in time domain from holistic perspective second the granger causality tests in different frequency bands are heterogeneous finally the comparison between results from wavelet coherence and multiple wavelet coherence in the joint time frequency domain indicates that in the high 1 14 days and medium frequency 14 128 days bands the combination of brent and gold prices has stronger correlation with the stock in the low frequency band 256 512 days year 2003 is the structure broken point before which brent and oil are ideal choice for hedging the risk of the stock market thus this paper offers more details between the chinese stock market and the commodities markets of crude oil and gold which suggests that the decisions for different time and frequencies should consider the corresponding benchmark information

multiresolution parameterization for geophysical inverse problems a model parameterization based on a multiresolution wavelet representation is proposed for geophysical inverse problems in this study the matrix equation for the wavelet representation of the model is constructed by dual wavelet transforms of a readily built matrix based on the pixel parameterization we demonstrate the merits of the proposed parameterization using the simple examples of geophysical data gridding and 3d seismic tomography and comparing the results with those obtained by the conventional pixel parameterization it is shown that for the conventional damped least squares solutions of pixel parameterized models regularization by norm damping deteriorates the underlying model correlation due to sparse and heterogeneous sampling whereas roughness damping schemes impose a priori arbitrary correlation length irrespective of the heterogeneous sampling the scale hierarchy embedded in the proposed multiresolution parameterization facilitates the compromise between the dual spatial and scale resolution depending on the local richness of the data constraints across different scales at a site model variation of various scale spectra can be robustly resolved through the scale hierarchy there is thus no need to invoke additional smoothness regularization

joint similarities and parameterizations for naimark complementary frames abstract it is known that the naimark complementary frames for a given frame are not necessarily unique up to the similarity in this paper we introduce the concept of joint complementary frame pairs for a given dual frame pair and prove that they are unique up to the joint similarity as an application we give a necessary and sufficient condition under which two naimark complementary frames are similar for different pairs of dual frames we present an operator parameterization for their joint complementary frame pairs

truncated taylor series solutions to a generalized burgers equation a generalized burgers equation is solved in the frequency domain by deriving taylor series expansions in powers of the range variable the first five terms of the solution are derived for both plane and spherical deterministic waveforms for the boundary value problem of an arbitrary time history at the origin any frequency dependence of attenuation and dispersion is allowed particular solutions are found to agree with previous work on burgers equation the errors involved in truncating both plane and spherical wave series are investigated and a discussion of two special cases suggests a boundary to the domain in which such truncated solutions are useful in particular they are not well adapted to the prediction of high frequencies

multiresolution expansion approximation order and quasiasymptotic behavior of tempered distributions abstract multiresolution analysis of tempered distributions is studied through multiresolution analysis on the corresponding test function spaces s r r r n 0 for a function h which is smooth enough and of appropriate decay it is shown that the derivatives of its projections to the corresponding spaces v j j z in a regular multiresolution analysis of l 2 r denoted by h j multiplied by a polynomial weight converge in sup norm i e h j h in s r r as j analogous result for tempered distributions is obtained by duality arguments the analysis of the approximation order of the projection operator within the framework of the theory of shift invariant spaces gives a further refinement of the results the order of approximation is measured with respect to the corresponding space of test functions as an application we give abelian and tauberian type theorems concerning the quasiasymptotic behavior of a tempered distribution at infinity

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

a load identification method based on wavelet multi resolution analysis abstract load identification as a kind of indirect identification method uses system characteristic and responses to calculate loads a method based on wavelet multi resolution analysis is proposed in this paper by wavelet decomposition and transform at certain resolutions the proposed method transforms the convolution relation between responses and loads in time domain into the linear multiplicative relation between system responses and wavelet responses in the wavelet domain loads can be identified as long as the linear multiplicative relation is solved qualitative and quantitative rules are proposed for selecting parameters that affect the accuracy of the proposed method and are illustrated via numerical investigations the method is illustrated by a multi input multi output numerical simulation a multi input multi output laboratory experiment is performed to compare the proposed method with the frequency method on the identification ability

signal adapted multiresolution transform for image coding the authors consider the problem of designing multiresolution transforms that are adapted to the given image signal in the sense that they maximize the coding gain at each resolution level a simple alternating optimization algorithm is derived for solving this problem in the framework of the lattice realization of para unitary quadrature mirror filters the resulting large coding scheme is discussed in some detail and its performance is compared with that of the discrete cosine transform jpeg technique and with that of some nonadapted multiresolution transforms

multiresolution analysis haar bases and self similar tilings of r sup n orthonormal bases for l sup 2 r sup n are constructed that have properties that are similar to those enjoyed by the classical haar basis for l sup 2 r for example each basis consists of appropriate dilates and translates of a finite collection of piecewise constant functions the construction is based on the notion of multiresolution analysis and reveals an interesting connection between the theory of compactly supported wavelet bases and the theory of self similar tilings

translates of multivarlate splines abstract we study spaces generated by translations of a fixed function over lattice points we provide some new algebraic and approximation properties for these spaces which show their applicability for finite element analysis

linear dependence relations in wavelets and tilings abstract we give explicitly all linear dependence relations for integer translates of a tiling in r associated with z k with a prime k 2 as a tool we determine linear dependence relations for k refinable compactly supported distributions in terms of the mask sequence in the corresponding k refinable refinement equations

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

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

on the spectrum of fractional brownian motions fractional brownian motions fbms provide useful models for a number of physical phenomena whose empirical spectra obey power laws of fractional order however due to the nonstationary nature of these processes the precise meaning of such spectra remains generally unclear two complementary approaches are proposed which are intended to clarify this point the first one based on a time frequency analysis takes into account the nonstationary nature of fbm and puts emphasis on time averaged measurements the second one based on a time scale analysis is matched to self similarity properties of fbm and reveals an underlying stationary structure relative to each time scaling

approximation algorithms for wavelet transform coding of data streams this paper addresses the problem of finding a b term wavelet representation of a given discrete function fepsirn whose distance from is minimized the problem is well understood when we seek to minimize the euclidean distance between f and its representation the first known algorithms for finding provably approximate representations minimizing general lp distances including linfin under a wide variety of compactly supported wavelet bases are presented in this paper for the haar basis a polynomial time approximation scheme is demonstrated these algorithms are applicable in the one pass sublinear space data stream model of computation they generalize naturally to multiple dimensions and weighted norms a universal representation that provides a provable approximation guarantee under all mu norms simultaneously and the first approximation algorithms for bit budget versions of the problem known as adaptive quantization are also presented further it is shown that the algorithms presented here can be used to select a basis from a tree structured dictionary of bases and find a b term representation of the given function that provably approximates its best dictionary basis representation

minimax estimation via wavelet shrinkage we attempt to recover an unknown function from noisy sampled data using orthonormal bases of compactly supported wavelets we develop a nonlinear method which works in the wavelet domain by simple nonlinear shrinkage of the empirical wavelet coefficients the shrinkage can be tuned to be nearly minimax over any member of a wide range of triebel and besov type smoothness constraints and asymptotically minimax over besov bodies with p q linear estimates cannot achieve even the minimax rates over triebel and besov classes with p 2 so the method can significantly outperform every linear method e g kernel smoothing spline sieve in a minimax sense variants of our method based on simple threshold nonlinear estimators are nearly minimax our method possesses the interpretation of spatial adaptivity it reconstructs using a kernel which may vary in shape and bandwidth from point to point depending on the data least favorable distributions for certain of the triebel and besov scales generate objects with sparse wavelet transforms many real objects have similarly sparse transforms which suggests that these minimax results are relevant for practical problems sequels to this paper which was first drafted in november 1990 discuss practical implementation spatial adaptation properties universal near minimaxity and applications to inverse problems

recovering edges in ill posed inverse problems optimality of curvelet frames we consider a model problem of recovering a function f x 1 x 2 from noisy radon data the function f to be recovered is assumed smooth apart from a discontinuity along a c 2 curve that is an edge we use the continuum white noise model with noise level varepsilon traditional linear methods for solving such inverse problems behave poorly in the presence of edges qualitatively the reconstructions are blurred near the edges quantitatively they give in our model mean squared errors mses that tend to zero with noise level varepsilon only as o varepsilon 1 2 as varepsilon to 0 a recent innovation nonlinear shrinkage in the wavelet domain visually improves edge sharpness and improves mse convergence to o varepsilon 2 3 however as we show here this rate is not optimal in fact essentially optimal performance is obtained by deploying the recently introduced tight frames of curvelets in this setting curvelets are smooth highly anisotropic elements ideally suited for detecting and synthesizing curved edges to deploy them in the radon setting we construct a curvelet based biorthogonal decomposition of the radon operator and build curvelet shrinkage estimators based on thresholding of the noisy curvelet coefficients in effect the estimator detects edges at certain locations and orientations in the radon domain and automatically synthesizes edges at corresponding locations and directions in the original domain we prove that the curvelet shrinkage can be tuned so that the estimator will attain within logarithmic factors the mse o varepsilon 4 5 as noise level varepsilon to 0 this rate of convergence holds uniformly over a class of functions which are c 2 except for discontinuities along c 2 curves and except for log terms is the minimax rate for that class our approach is an instance of a general strategy which should apply in other inverse problems we sketch a deconvolution example

dynamic multiscale spatiotemporal models for gaussian areal data summary we introduce a new class of dynamic multiscale models for spatiotemporal processes arising from gaussian areal data specifically we use nested geographical structures to decompose the original process into multiscale coefficients which evolve through time following state space equations our approach naturally accommodates data that are observed on irregular grids as well as heteroscedasticity moreover we propose a multiscale spatiotemporal clustering algorithm that facilitates estimation of the nested geographical multiscale structure in addition we present a singular forward filter backward sampler for efficient bayesian estimation our multiscale spatiotemporal methodology decomposes large data analysis problems into many smaller components and thus leads to scalable and highly efficient computational procedures finally we illustrate the utility and flexibility of our dynamic multiscale framework through two spatiotemporal applications the first example considers mortality ratios in the state of missouri whereas the second example examines agricultural production in espirito santo state brazil

quantum field theory in ninety minutes on presente divers aspects de la theorie quantique des champs la theorie p 2 a 2 dimensions la theorie 3 4 a 3 dimensions la theorie de yang mills a 4 dimensions

multiresolution algorithms for the numerical solution of hyperbolic conservation laws given any scheme in conservation form and an appropriate uniform grid for the numerical solution of the initial value problem for one dimensional hyperbolic conservation laws we describe a multiresolution algorithm that approximates this numerical solution to a prescribed tolerance in an efficient manner to do so we consider the grid averages of the numerical solution for a hierarchy of nested diadic grids in which the given grid is the finest and introduce an equivalent multiresolution representation the multiresolution representation of the numerical solution consists of its grid averages for the coarsest grid and the set of errors in predicting the grid averages of each level of resolution in this hierarchy from those of the next coarser one once the numerical solution is resolved to our satisfaction in a certain locality of some grid then the prediction errors there are small for this particular grid and all finer ones this enables us to compress data by setting to zero small components of the representation which fall below a prescribed tolerance therefore instead of computing the time evolution of the numerical solution on the given grid we compute the time evolution of its compressed multiresolution representation algorithmically this amounts to computing the numerical fluxes of the given scheme at the points of the given grid by a hierarchical algorithm which starts with the computation of these numerical fluxes at the points of the coarsest grid and then proceeds through diadic refinements to the given grid at each step of refinement we add the values of the numerical flux at the center of the coarser cells the information in the multiresolution representation of the numerical solution is used to determine whether the solution is locally well resolved when this is the case we replace the costly exact value of the numerical flux with an accurate enough approximate value which is obtained by an inexpensive interpolation from the coarser grid the computational efficiency of this multiresolution algorithm is proportional to the rate of data compression for a prescribed level of tolerance that can be achieved for the numerical solution of the given scheme

coherent structures in random media and wavelets abstract we construct low dimensional dynamical models for the motion of coherent structures such as those frequently observed in extended turbulent flows these models are derived from the wavelet based galerkin projection of the pde describing the flow and account for the local interaction of a small number of coherent structures we show that under rather general assumptions the wavelet projection is close to the proper orthogonal decomposition in the average energy sense in the specific case of the 1d kuramoto sivashinsky equation we show numerically that the dynamics of our model agrees qualitatively with that of the original ks equation

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

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

eigenvalues and eigenvectors of the finite fourier transform the eigenvalues and eigenvectors of the n n unitary matrix of finite fourier transform whose j k element is 1 n 1 2 exp 2 i n jk i 1 1 2 is determined in doing so a multitude of identities some of which may be new are encountered a conjecture is advanced

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

refinement of operator valued reproducing kernels this paper studies the construction of a refinement kernel for a given operator valued reproducing kernel such that the vector valued reproducing kernel hilbert space of the refinement kernel contains that of the given kernel as a subspace the study is motivated from the need of updating the current operator valued reproducing kernel in multi task learning when underfitting or overfitting occurs numerical simulations confirm that the established refinement kernel method is able to meet this need various characterizations are provided based on feature maps and vector valued integral representations of operator valued reproducing kernels concrete examples of refining translation invariant and finite hilbert schmidt operator valued reproducing kernels are provided other examples include refinement of hessian of scalar valued translation invariant kernels and transformation kernels existence and properties of operator valued reproducing kernels preserved during the refinement process are also investigated

oversampling and reconstruction functions with compact support assume that a sequence of samples of a filtered version of a function in a shift invariant space is given this paper deals with the existence of a sampling formula involving these samples and having reconstruction functions with compact support this is done in the light of the generalized sampling theory by using the oversampling technique a necessary and sufficient condition is given in terms of the smith canonical form of a polynomial matrix finally we prove that the aforesaid oversampled formulas provide nice approximation schemes with respect to the uniform norm

non linear propagation of broadband noise signals measurements of aircraft flyover noise have strongly suggested that non linear propagation effects can significantly modify the noise spectrum the data show that over ranges of 500 to 1000 m the attenuation in the range 5 to 10 khz may fall considerably short of that predicted on the basis of linear propagation if the noise is sufficiently intense such distances are comparable to those used in aircraft noise certification measurements and the frequencies involved have an important influence on the perceived noise level therefore a scheme is required for predicting the finite amplitude propagation of random signals over a wide range of source and atmospheric conditions taylor series solutions to a generalized burgers equation are generated for stationary noise signals the expansions are in powers of the range variable and any frequency dependence of attenuation and dispersion is allowed the limitations of such an approach are discussed and it is found that the number of terms needed increases rapidly with range for propagation over a long range when non linear effects are important it is recommended that one of the differential equations derived for the power spectrum be used together with a closure hypothesis individual terms in the series may be treated separately as local spectrum evolution equations valid at all points along the propagation path with appropriate modelling of the non linear terms such an equation can form a basis for aircraft noise prediction schemes

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

wavelets of multiplicity r a multiresolution approximation km m z of l2 r is of multi plicity r 0 if there are r functions x r whose translates form a riesz basis for v in the general theory we derive necessary and sufficient conditions for the translates of x r y x y r to form a riesz basis for v the resulting reconstruction and decomposition sequences lead to the construction of dual bases for v0 and its orthogonal complement w0 in vx the general theory is applied in the construction of spline wavelets with mul tiple knots algorithms for the construction of these wavelets for some special cases are given

algorithms for designing wavelets to match a specified signal algorithms for designing a mother wavelet spl psi x such that it matches a signal of interest and such that the family of wavelets 2 sup j 2 spl psi 2 sup j x k forms an orthonormal riesz basis of l sup 2 spl rscr are developed the algorithms are based on a closed form solution for finding the scaling function spectrum from the wavelet spectrum many applications require wavelets that are matched to a signal of interest most current design techniques however do not design the wavelet directly they either build a composite wavelet from a library of previously designed wavelets modify the bases in an existing multiresolution analysis or design a scaling function that generates a multiresolution analysis with some desired properties in this paper two sets of equations are developed that allow us to design the wavelet directly from the signal of interest both sets impose bandlimitedness resulting in closed form solutions the first set derives expressions for continuous matched wavelet spectrum amplitudes the second set of equations provides a direct discrete algorithm for calculating close approximations to the optimal complex wavelet spectrum the discrete solution for the matched wavelet spectrum amplitude is identical to that of the continuous solution at the sampled frequencies an interesting byproduct of this work is the result that meyer s spectrum amplitude construction for an orthonormal bandlimited wavelet is not only sufficient but necessary specific examples are given which demonstrate the performance of the wavelet matching algorithms for both known orthonormal wavelets and arbitrary signals

relations between multiresolution analysis and quantum mechanics we discuss a procedure to construct multiresolution analyses mra of l2 r starting from a given seed function h s which should satisfy some conditions our method originally related to the quantum mechanical hamiltonian of the fractional quantum hall effect is shown to be model independent the role of a canonical map between certain canonically conjugate operators is discussed this clarifies our previous procedure and makes much easier most of the original formulas producing a convenient framework to produce examples of mra

towards a multiresolution approach to linear control we develop a multiresolution approximation framework for linear control we construct a multiresolution analysis of the set of input functions of a linear system the approximation of an input u at a scale j is defined as the input uj of minimal energy such that the trajectories of the system associated with u and uj coincide on a grid of step length 2 j we propose a set of wavelet functions which generate this multiresolution analysis these functions called control theoretic wavelets satisfy useful properties for the representation of control inputs of a linear system as an example of application of our multiresolution approximation framework we propose a method for efficient encoding of control inputs with regard to several criteria

multilevel preconditioning this paper is concerned with multilevel techniques for preconditioning linear systems arising from galerkin methods for elliptic boundary value problems a general estimate is derived which is based on the characterization of besov spaces in terms of weighted sequence norms related to corresponding multilevel expansions the result brings out clearly how the various ingredients of a typical multilevel setting affect the growth rate of the condition numbers in particular our analysis indicates how to realize even uniformly bounded condition numbers for example the general results are used to show that the bramble pasciak xu preconditioner for piecewise linear finite elements gives rise to uniformly bounded condition numbers even when the refinements of the underlying triangulations are highly nonuniform furthermore they are applied to a general multivariate setting of refinable shift invariant spaces in particular covering those induced by various types of wavelets

local reproducible smoothing without shrinkage a simple local smoothing filter is defined for curves or surfaces combining the advantages of gaussian smoothing and fourier curve description unlike gaussian filters the filter described has no shrinkage problem repeated application of the filter does not yield a curve or surface smaller than the original but simply reproduces the approximate result that would have been obtained by a single application at the largest scale unlike fourier description the filter is local in space the wavelet transform of y meyer 1989 is also shown to have these properties

wavelet transform based adaptive filters analysis and new results in this paper the wavelet transform is used in an adaptive filtering structure the coefficients of the adaptive filter are updated by the help of the least mean square lms algorithm first the wavelet transform based adaptive filter wtaf is described and it is analyzed for its wiener optimal solution then the performance of the wtaf is studied by the help of learning curves for three different convergence factors 1 constant convergence factor 2 time varying convergence factor and 3 exponentially weighted convergence factor the exponentially weighted convergence factor is proposed to introduce scale based variation to the weight update equation it is shown for two different sets of data that the rate of convergence increases significantly for all three wtaf structures as compared to that of time domain lms the high convergence rates of the wtaf give us reason to expect that it will perform well in tracking rapid changes in a signal

extending the scope of wavelet regression methods by coefficient dependent thresholding abstract various aspects of the wavelet approach to nonparametric regression are considered with the overall aim of extending the scope of wavelet techniques to irregularly spaced data to regularly spaced datasets of arbitrary size to heteroscedastic and correlated data and to data that contain outliers the core of the methodology is an algorithm for finding ail of the variances and within level covariances in the wavelet table of a sequence with given covariance structure if the original covariance matrix is band limited then the algorithm is linear in the length of the sequence the variance calculation algorithm allows data on any set of independent variable values to be treated by first interpolating to a fine regular grid of suitable length and then constructing a wavelet expansion of the gridded data various thresholding methods are discussed and investigated exact risk formulas for the mean square error of the methodology for given design are derived good performance is obtained by nois

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

posterior probability intervals for wavelet thresholding summary we use cumulants to derive bayesian credible intervals for wavelet regression estimates the first four cumulants of the posterior distribution of the estimates are expressed in terms of the observed data and integer powers of the mother wavelet functions these powers are closely approximated by linear combinations of wavelet scaling functions at an appropriate finer scale hence a suitable modification of the discrete wavelet transform allows the posterior cumulants to be found efficiently for any given data set johnson transformations then yield the credible intervals themselves simulations show that these intervals have good coverage rates even when the underlying function is inhomogeneous where standard methods fail in the case where the curve is smooth the performance of our intervals remains competitive with established nonparametric regression methods

perfect reconstruction filtering with unitary operators and projections abstract perfect reconstruction filter banks wavelets and multiresolution analysis have generated many new results in harmonic analysis and function theory traditionally these techniques are based on families of convolutions and decimation interpolation operations when viewed as linear algebraic operations those building blocks can be generalized to unitary operators and related projections we have developed a general approach to perfect reconstruction and multiresolution analysis that uses polynomial functions of unitary operators and closely related projection operations this general framework is powerful in that it easily captures much of the known theory as special cases and gives a simple mechanism for constructing arbitrary perfect reconstruction families of operators using the general framework we derive necessary and sufficient conditions for the perfect reconstruction property even for higher dimensional non separable sampling lattices and operators the necessary conditions can be interpreted in terms of the spectral mapping theorem and although explicity stated are not directly computable

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

multiresolution analysis on local fields abstract in this paper we first establish the fundamental framework of multiresolution on local field in wavelet analysis accurately orthonormal systems consisting of integer translations u k k p of a single function l2 k are characterized theorems 1 and 2 the concept of multiresolution analysis on local field definition 3 and construction of corresponding wavelet vectors theorem 3 are given the definition of integral periodicity of a function on local field definition 4 is also given an example is finally presented

multi resolution multi scale topology optimization a new paradigm abstract the purpose of this work is to present a new concept multi resolution multi scale topology optimization the key idea of the present strategy is that design optimization should be performed progressively from low to high resolution not at a single resolution level to achieve the multi resolution strategy design optimization is formulated in a wavelet based variable space not in a direct density variable space the major advantages of the multi resolution design optimization include 1 topologically simple and close to the global optimum structures may be obtained without any explicit constraint and 2 the convergence is not sensitive to mathematical programming methods for the efficient numerical implementation of the multi resolution approach the side constraints imposed on the direct density variables are removed by mapping the density variables into intermediate variables these intermediate variables are then wavelet transformed to new design variables it is addressed that the present multi resolution topology optimization can resolve major numerical instability problems such as mesh dependencies and local minima the usefulness of the multi scale nature of the wavelets in the present multi resolution multi scale optimization formulation is also discussed

stochastic transforms for jump diffusion processes combined with related backward stochastic differential equations abstract this study considers three stochastic transforms for jump diffusion processes the stochastic laplace transform stochastic fourier transform and stochastic wavelet transform first we introduce the stochastic laplace transform for processes adapted by brownian filtration as a solution for complex number valued backward stochastic differential equations bsdes this transform can be explained well by the girsanov theorem which also allows us to define the stochastic laplace transform for jump diffusion processes directly based on this perspective we give natural definitions of the stochastic fourier transform and stochastic wavelet transform the advantages of these stochastic transforms are all related to the uniqueness of the processes compared with the classical transforms the newly introduced parameters guarantee the uniqueness of the stochastic transforms for the adapted processes while they also agree with the corresponding parameters in the classical transforms which can represent the frequency property of the processes in addition these three stochastic transforms can also be regarded as the solutions of related bsdes with jumps

adaptive learning for event modeling and characterization adaptive learning of specific patterns or events of interest has been an area of significant research for various applications in the last two decades in developing diagnostic evaluation and safety monitoring applications of a propulsion system it is critical to detect characterize and model events of interest it is a challenging task since the detection system should allow adaptive characterization of potential events of interest and correlate them to learn new models for future detection for online health monitoring and diagnostic evaluation in this paper a novel framework is established using a hierarchical adaptive clustering approach with fuzzy membership functions to characterize specific events of interest from the measured and processed features raw engine measurement data is first analyzed using the wavelet transform to provide features for localization of frequency information for use in the classification system a method combining hierarchical and fuzzy k means clustering is then applied to a set of selected measurements and computed features to determine the events of interest during engine operations experimental results have shown that the proposed approach is effective and computationally efficient to detect characterize and model new events of interest from data collected through continuous operations

multiscale autoregressive processes i schur levinson parametrizations in many applications it is of interest to analyze and recognize phenomena occurring at different scales the recently introduced wavelet transforms provide a time and scale decomposition of signals that offers the possibility of such analysis a corresponding statistical framework to support the development of optimal multiscale statistical signal processing algorithms is described the theory of multiscale signal representation leads naturally to models of signals on trees and this provides the framework for investigation in particular the class of isotropic processes on homogeneous trees is described and a theory of autoregressive models is developed in this context this leads to generalizations of schur and levinson recursions associated properties of the resulting reflection coefficients and the initial pieces in a system theory for multiscale modeling