fuzzy piecewise multilinear and piecewise linear systems as universal approximators in sobolev norms a piecewise multilinear and a piecewise linear model for fuzzy systems are introduced and their approximation capabilities investigated classical results on the approximation of regular functions are improved showing that fuzzy systems may keep their semantic structure while approximating to any degree of accuracy not only sufficiently regular functions but also their derivatives

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

robust h sub spl infin and mixed h sub 2 h sub spl infin filters for equalization designs of nonlinear communication systems fuzzy interpolation approach generally it is difficult to design equalizers for signal reconstruction of nonlinear communication channels with uncertain noises in this paper we propose the h sub spl infin and mixed h sub 2 h sub spl infin filters for equalization detection of nonlinear channels using fuzzy interpolation and linear matrix inequality lmi techniques first the signal transmission system is described as a state space model and the input signal is embedded in the state vector such that the signal reconstruction estimation design becomes a nonlinear state estimation problem then the takagi sugeno fuzzy linear model is applied to interpolate the nonlinear channel at different operation points through membership functions since the statistics of noises are unknown the fuzzy h sub spl infin equalizer is proposed to treat the state estimation problem from the worst case robust point of view when the statistics of noises are uncertain but with some nominal or average information available the mixed h sub 2 h sub spl infin equalizer is employed to take the advantage of both h sub 2 optimal performance with nominal statistics of noises and the h sub spl infin robustness performance against the statistical uncertainty of noises using the lmi approach the fuzzy h sub 2 h sub spl infin equalizer detector design problem is characterized as an eigenvalue problem evp the evp can be solved efficiently with convex optimization techniques

multiresolution wavelet analysis of integer scale bessel functions we identify multiresolution subspaces giving rise via hankel transforms to bessel functions they emerge as orthogonal systems derived from geometric hilbert space considerations the same way the wavelet functions from a multiresolution scaling wavelet construction arise from a scale of hilbert spaces we study the theory of representations of the c algebra o 1 arising from this multiresolution analysis a connection with markov chains and representations of o 1 is found projection valued measures arising from the multiresolution analysis give rise to a markov trace for the quantum groups soq

orthonormal dilations of parseval wavelets we prove that any parseval wavelet frame is the projection of an orthonormal wavelet basis for a representation of the baumslag solitar group bs 1 2 langle u t utu 1 t 2 rangle we give a precise description of this representation in some special cases and show that for wavelet sets it is related to symbolic dynamics theorem 3 14 we prove that the structure of the representation depends on the analysis of certain finite orbits for the associated symbolic dynamics theorem 3 24 we give concrete examples of parseval wavelets for which we compute the orthonormal dilations in detail we construct parseval wavelet sets which have infinitely many non isomorphic orthonormal dilations

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

an interlaced extended kalman filter an estimation algorithm for a class of discrete time nonlinear systems is proposed the system structure we deal with is partitionable into in subsystems each affine w r t the corresponding part of the state vector the algorithm consists of a bank of m interlaced kalman filters and each of them estimates a part of the state considering the remaining parts as known time varying parameters whose values are evaluated by the other filters at the previous step the procedure neglects the subsystem coupling terms in the covariance matrix of the state estimation error and counteracts the errors so introduced by suitably increasing the noise covariance matrices comparisons through numerical simulations with the extended kalman filter and its modified versions proposed in the literature illustrate the good trade off provided by the algorithm between the reduction of the computational load and the estimation accuracy

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

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

classification of generalized multiresolution analyses abstract we discuss how generalized multiresolution analyses gmras both classical and those defined on abstract hilbert spaces can be classified by their multiplicity functions m and matrix valued filter functions h given a natural number valued function m and a system of functions encoded in a matrix h satisfying certain conditions a construction procedure is described that produces an abstract gmra with multiplicity function m and filter system h an equivalence relation on gmras is defined and described in terms of their associated pairs m h this classification system is applied to mras and other classical examples in l 2 r d as well as to previously studied abstract examples

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

probability measures on solenoids corresponding to fractal wavelets the measure on generalized solenoids constructed using filters by dutkay and jorgensen in 12 is analyzed further by writing the solenoid as the product of a torus and a cantor set using this decomposition key differences are revealed between solenoid measures associated with classical filters in r d and those associated with filters on inflated fractal sets in particular it is shown that the classical case produces atomic fiber measures and as a result supports both suitably defined solenoid msf wavelets and systems of imprimitivity for the corresponding wavelet representation of the generalized baumslag solitar group in contrast the fiber measures for filters on inflated fractal spaces cannot be atomic and thus can support neither msf wavelets nor systems of imprimitivity

back propagation of imprecision in a cheese ripening fuzzy model based on human sensory evaluations in the food processes taking human sensory evaluations into account is relevant for the online control of the sensory quality of food products however the operator sensory measurements are subjected to higher imperfections than the conventional sensory measurements therefore to determine the tolerated imprecision on the sensory evaluations according to the specified imprecision for the control variable is an important challenge for food processes thus from a takagi sugeno fuzzy model developed on an application dealing with the control of a cheese ripening process the proposed approach allows to determine the maximum imprecision tolerated on the cheese coat and consistency input sensory measurements in order to satisfy a specified tolerance for the ripening degree inferred with the ts fuzzy model results show that for some ripening periods the maximum imprecision tolerated on the input measurements is inferior to the human imprecision on these measurements thus the model must be used carefully or measurements have to be repeated in these periods

direct limits multiresolution analyses and wavelets a multiresolution analysis for a hilbert space realizes the hilbert space as the direct limit of an increasing sequence of closed subspaces in a previous paper we showed how conversely direct limits could be used to construct hilbert spaces which have multiresolution analyses with desired properties in this paper we use direct limits and in particular the universal property which characterizes them to construct wavelet bases in a variety of concrete hilbert spaces of functions our results apply to the classical situation involving dilation matrices on l2 rn the wavelets on fractals studied by dutkay and jorgensen and hilbert spaces of functions on solenoids

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

construction and reconstruction of tight framelets and wavelets via matrix mask functions abstract the paper develops construction procedures for tight framelets and wavelets using matrix mask functions in the setting of a generalized multiresolution analysis gmra we show the existence of a scaling vector of a gmra such that its first component exhausts the spectrum of the core space near the origin the corresponding low pass matrix mask has an especially advantageous form enabling an effective reconstruction procedure of the original scaling vector we also prove a generalization of the unitary extension principle for an infinite number of generators this results in the construction scheme for tight framelets using low pass and high pass matrix masks generalizing the classical mra constructions we prove that our scheme is flexible enough to reconstruct all possible orthonormal wavelets as an illustration we exhibit a pathwise connected class of non msf non mra wavelets sharing the same wavelet dimension function

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

generalized filters the low pass condition and connections to multiresolution analyses abstract we study generalized filters that are associated to multiplicity functions and homomorphisms of the dual of an abelian group these notions are based on the structure of generalized multiresolution analyses we investigate when the ruelle operator corresponding to such a filter is a pure isometry and then use that characterization to study the problem of when a collection of closed subspaces which satisfies all the conditions of a gmra except the trivial intersection condition must in fact have a trivial intersection in this context we obtain a generalization of a theorem of bownik and rzeszotnik

modified extended kalman filtering in this paper we compare the performance of the extended kalman filter ekf with that of a proposed filter called modified extended kalman filter mekf in terms of the actual state and the estimated state as a function of time and the integral squared error as a function of noise intensity both for dynamics and measurement noise all the simulation results indicate that the performance of the mekf is better than ekf for illustration numerical examples are presented using both ekf and mekf

nonlinear filtering of systems governed by ito differential equations with jump parameters abstract in this paper we present a system of coupled stochastic infinite dimensional differential equations of zakai type which describes the evolution of the unnormalized conditional density of a conditional markov process this generalizes the nonlinear filtering equation of zakai

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

harmonic analysis of iterated function systems with overlap an iterated function system ifs is a system of contractive mappings i y y i 1 n finite where y is a complete metric space every such ifs has a unique up to scale equilibrium measure also called the hutchinson measure and we study the hilbert space l2 in this paper we extend previous work on ifss without overlap our method involves systems of operators generalizing the more familiar cuntz relations from operator algebra theory and from subband filter operators in signal processing these cuntz like operator systems were used in recent papers on wavelet analysis by baggett jorgensen merrill and packer contemp math 345 11 25 2004 where they serve as a first step to generating wavelet bases of parseval type alias normalized tight frames i e wavelet bases with redundancy similarly it was shown in work by dutkay and jorgensen rev mat iberoam 22 131 180 2006 that the iterative operator approach works well for generating wavelets on fractals from ifss without overla

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

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

hilbert spaces built on a similarity and on dynamical renormalization we develop a hilbert space framework for a number of general multiscale problems from dynamics the aim is to identify a spectral theory for a class of systems based on iterations of a non invertible endomorphism we are motivated by the more familiar approach to wavelet theory which starts with the two to one endomorphism r z z2 in the one torus t a wavelet filter and an associated transfer operator this leads to a scaling function and a corresponding closed subspace v0 in the hilbert space l2 r using the dyadic scaling on the line r one has a nested family of closed subspaces vn n z with trivial intersection and with dense union in l2 r more generally we achieve the same outcome but in different hilbert spaces for a class of non linear problems in fact we see that the geometry of scales of subspaces in hilbert space is ubiquitous in the analysis of multiscale problems e g martingales complex iteration dynamical systems graph iterated function systems of affine type and subshifts in

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

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

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

fuzzy control with fuzzy inputs this paper is concerned with the use of fuzzy inputs in fuzzy logic controllers a precise representation of fuzzy logic controllers by means of mappings is used to introduce different ways for dealing with fuzzy inputs two types of fuzzy inputs are presented and their potential use in fuzzy control is discussed the proposed concepts are applied to control a first order process with a pi controller this simple process is chosen to clearly illustrate the behavior of the closed loop system using fuzzy inputs for fuzzy reference and fuzzy measurement finally a nonlinear process is used to illustrate the effects of fuzzy inputs on a more complex system although it is sometimes speculated that fuzzy inputs may improve the behavior of fuzzy controllers experiments developed in this paper show this point is not straightforward and that the relevance of fuzzy inputs should be questioned in closed loop fuzzy control

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

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

multiresolution properties of the wavelet galerkin operator this paper extends recent results by the author j math phys 31 1898 1990 j math phys 32 57 1991 that show a scaling parameter sequence h yields an orthonormal wavelet basis for l2 r if and only if an associated operator sh has eigenvalue 1 with multiplicity 1 the operator transforms a sequence a by sh a k 2 m n h m h n a 2k m n a correspondence is derived between sh and galerkin projection operators related to the multiresolution analysis defined by the orthonormal wavelet basis the spectrum of sh is characterized in terms of the fourier modulus of the unique scaling function that satisfies x 2 nh n 2x n this characterization yields several results including a direct alternate proof that the eigenvalue 1 of sh has multiplicity 1

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

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

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

tight frames of compactly supported affine wavelets this paper extends the class of orthonormal bases of compactly supported wavelets recently constructed by daubechies commun pure appl math 41 909 1988 for each integer n 1 a family of wavelet functions having support 0 2n 1 is constructed such that jk x 2j 2 2jx k kj k z is a tight frame of l2 r i e for every f l2 r f c jk jk f jk for some c 0 this family is parametrized by an algebraic subset vn of r4n furthermore for n 2 a proper algebraic subset wn of vn is specified such that all points in vn outside of wn yield orthonormal bases the relationship between these tight frames and the theory of group representations and coherent states is discussed

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

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

affine systems in l2 rd the analysis of the analysis operator abstract discrete affine systems are obtained by applying dilations to a given shift invariant system the complicated structure of the affine system is due first and foremost to the fact that it is not invariant under shifts affine frames carry the additional difficulty that they are global in nature it is the entire interaction between the various dilation levels that determines whether the system is a frame and not the behaviour of the system within one dilation level we completely unravel the structure of the affine system with the aid of two new notions the affine product and a quasi affine system this leads to a characterization of affine frames the induced characterization of tight affine frames is in terms of exact orthogonality relations that the wavelets should satisfy on the fourier domain several results such as a general oversampling theorem follow from these characterizations most importantly the affine product can be factored during a multiresolution analysis construction and this leads to a complete characterization of all tight frames that can be constructed by such methods moreover this characterization suggests very simple sufficient conditions for constructing tight frames from multiresolution of particular importance are the facts that the underlying scaling function does not need to satisfy any a priori conditions and that the freedom offered by redundancy can be fully exploited in these constructions an

explicit analytical formulation and exact inversion of decomposable fuzzy systems with singleton consequents abstract this paper is devoted to the inversion of fuzzy systems expressed by fuzzy rules with singleton consequents if input variables are described using strong triangular partitions as pointed out in recent works such fuzzy systems can be decomposed into collections of multi linear subsystems in this paper an analytical formulation of the system output is explicitly developed and directly used in order to determine solutions to the inversion problem based on this analytical methodology an algorithm is proposed for computing inverse solutions as the inversion is handled analytically the exactness of the obtained solutions is guaranteed furthermore according to the decomposability of the studied fuzzy systems all inverse solutions are found finally whatever the fuzzy system under consideration there is no need to study its invertibility beforehand since the algorithm is able to handle all possible situations no solution one unique solution multiple solutions an infinity of solutions the proposed approach can be easily extended to other types of fuzzy systems provided that decomposability is preserved in other words with regard to exact inversion which often plays a key role in engineering applications such as control or diagnosis decomposability is probably the first criterion that should be considered when choosing a specific fuzzy system structure

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

h sub spl infin fuzzy estimation for a class of nonlinear discrete time dynamic systems this paper studies h sub spl infin fuzzy filtering design for state estimation of nonlinear discrete time systems with bounded but unknown disturbances first the takagi and sugeno 1985 fuzzy model is proposed to represent a nonlinear system next using a linear matrix inequality lmi approach the h sub spl infin fuzzy filtering problems are characterized in terms of a linear matrix inequality problem lmip the lmip can be efficiently solved using convex optimization techniques simulation examples are given to illustrate the design procedure and the applicability of the proposed method the results indicate that the proposed method is suitable for practical applications

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

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

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

decomposition property of fuzzy systems and its applications this paper presents the decomposition property of fuzzy systems using a simple constructive decomposition procedure that is by properly dividing the input space into sub input spaces a general fuzzy system is decomposed into several sub fuzzy systems which are the simplest fuzzy systems in the sub input spaces based on the decomposition property of fuzzy systems the analysis of fuzzy systems can be divided into two steps first analyze the properties of the simplest fuzzy systems and then use the decomposition property to extend the results to general fuzzy systems using this idea two applications of the decomposition property are given the first is the application to the representation capability analysis of fuzzy systems the second is the application to the analysis of a class of nonlinear control systems then based on the piecewise affine fuzzy system model the existence condition and the design of a stable control for a class of single input single output siso nonlinear systems are presented

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