wavelets and filter banks theory and design the wavelet transform is compared with the more classical short time fourier transform approach to signal analysis then the relations between wavelets filter banks and multiresolution signal processing are explored a brief review is given of perfect reconstruction filter banks which can be used both for computing the discrete wavelet transform and for deriving continuous wavelet bases provided that the filters meet a constraint known as regularity given a low pass filter necessary and sufficient conditions for the existence of a complementary high pass filter that will permit perfect reconstruction are derived the perfect reconstruction condition is posed as a bezout identity and it is shown how it is possible to find all higher degree complementary filters based on an analogy with the theory of diophantine equations an alternative approach based on the theory of continued fractions is also given these results are used to design highly regular filter banks which generate biorthogonal continuous wavelet bases with symmetries

a quasi invariance theorem for measures on banach spaces we show that for a measure on a banach space directional differentiability implies quasi translation invariance this result is shown to imply the cameron martin theorem a second application is given in which ty is the image of a gaussian measure under a suitably regular map

on the relationship between differentiability and absolute continuity of measures on ℝ n we give an elementary proof of the fact that a finite borel measure on n is absolutely continuous with a c 1 density if and only if it has directional derivatives which are continuous almost everywhere the radon nikodym derivative of a differentiable measure is given in terms of the directional derivatives

euclidean gibbs measures on loop lattices existence and a priori estimates we present a new method to prove existence and uniform a priori estimates for euclidean gibbs measures corresponding to quantum anharmonic crystals it is based first on the alternative characterization of gibbs measures in terms of their logarithmic derivatives through integration by parts formulas and second on the choice of appropriate lyapunov functionals

a generalized two dimensional analytical solution for hydrodynamic dispersion in bounded media with the first type boundary condition at the source a generalized two dimensional analytical solution is developed for hydrodynamic dispersion in a unidirectional bounded surface and subsurface media flow field from time and space dependent sources with the first type boundary condition at the source longitudinal and transverse dispersion radioactive decay and linear adsorption are considered in the mathematical analysis laplace transform and fourier analysis techniques are used simultaneously and a general equation in infinite series form for the unsteady state concentration distribution has been obtained solution for a single strip source is obtained as a special case from the general solution the equations for convective dispersive flux components are also derived the solution of bruch and street 1967 for a symmetrically located strip source is shown to be a special case of the solution expressions for concentration distribution and convective dispersive flux components for the steady state solute transport case are also presented the results are compared with two different finite element codes with a good agreement the results of this model may be used for solute transport in a unidirectional porous media flow field as well as in rivers and canals with some additional assumptions the solutions can also be used for numerical model validation

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

multivariate refinable hermite interpolant we introduce a general definition of refinable hermite interpolants and investigate their general properties we study also a notion of symmetry of these refinable interpolants results and ideas from the extensive theory of general refinement equations are applied to obtain results on refinable hermite interpolants the theory developed here is constructive and yields an easy to use construction method for multivariate refinable hermite interpolants using this method several new refinable hermite interpolants with respect to dierent dilation matrices and symmetry groups are constructed and analyzed some of the hermite interpolants constructed here are related to well known spline interpolation schemes developed in the computer aided geometric design community e g the powell sabin scheme we make

absolutely continuous flows generated by sobolev class vector fields in finite and infinite dimensions abstract we prove the existence of the global flow ut generated by a vector field a from a sobolev class w1 1 on a finite or infinite dimensional space x with a measure provided is sufficiently smooth and that a a and a where a is the divergence with respect to are exponentially integrable in addition the measure is shown to be quasi invariant under ut in the case x r n and p dx where p w1 1loc r n is a locally uniformly positive probability density a sufficient condition is exp c a exp c a p p l1 for all c in the infinite dimensional case we get analogous results for measures differentiable along sufficiently many directions examples of measures which fit our framework important for applications are symmetric invariant measures of infinite dimensional diffusions and gibbs measures typically in both cases such measures are essentially non gaussian our result in infinite dimensions significantly extends previously studied cases where was a gaussian measure finally we study flows generated by vector fields whose values are not necessarily in the cameron martin space

on interpolatory divergence free wavelets we construct interpolating divergence free multiwavelets based on cubic hermite splines we give characterizations of the relevant function spaces and indicate their use for analyzing experimental data of incompressible flow fields we also show that the standard interpolatory wavelets based on the deslauriers dubuc interpolatory scheme or on interpolatory splines cannot be used to construct compactly supported divergence free interpolatory wavelets

integrated harnack inequalities on lie groups we show that the logarithmic derivatives of the convolution heat kernels on a uni modular lie group are exponentially integrable this result is then used to prove an integrated harnack inequality for these heat kernels it is shown that this integrated harnack inequality is equivalent to a version of wang s harnack inequality a key feature of all of these inequalities is that they are dimension independent finally we show these inequalities imply quasi invariance properties of heat kernel measures for two classes of infinite dimensional lie groups

image coding using wavelet transform a scheme for image compression that takes into account psychovisual features both in the space and frequency domains is proposed this method involves two steps first a wavelet transform used in order to obtain a set of biorthogonal subclasses of images the original image is decomposed at different scales using a pyramidal algorithm architecture the decomposition is along the vertical and horizontal directions and maintains constant the number of pixels required to describe the image second according to shannon s rate distortion theory the wavelet coefficients are vector quantized using a multiresolution codebook to encode the wavelet coefficients a noise shaping bit allocation procedure which assumes that details at high resolution are less visible to the human eye is proposed in order to allow the receiver to recognize a picture as quickly as possible at minimum cost a progressive transmission scheme is presented it is shown that the wavelet transform is particularly well adapted to progressive transmission

a new class of biorthogonal wavelet systems for image transform coding we construct general biorthogonal coifman wavelet systems a new class of compactly supported biorthogonal wavelet systems with vanishing moments equally distributed for a scaling function and wavelet pair a time domain design method is employed and closed form expressions for the impulse responses and the frequency responses of the corresponding dual filters are derived the resulting filter coefficients are all dyadic fractions which is an attractive feature in the realization of multiplication free discrete wavelet transform even ordered systems in this family are symmetric which correspond to linear phase dual filters in particular three filterbanks fbs in this family are systematically verified to have competitive compression potential to the 9 7 tap biorthogonal wavelet fb by cohen et al 1992 which is currently the most widely used one in the field of wavelet transform coding in addition the proposed fb s have much smaller computational complexity in terms of floating point operations required in transformation and therefore indicate a better tradeoff between compression performance and computational complexity

wavelet stabilization of the lagrange multiplier method we propose here a stabilization strategy for the lagrange multiplier formulation of dirichlet problems the stabilization is based on the use of equivalent scalar products for the spaces h 1 2 partial omega and h 1 2 partial omega which are realized by means of wavelet functions the resulting stabilized bilinear form is coercive with respect to the natural norm associated to the problem a uniformly coercive approximation of the stabilized bilinear form is constructed for a wide class of approximation spaces for which an optimal error estimate is provided finally a formulation is presented which is obtained by eliminating the multiplier by static condensation this formulation is closely related to the nitsche s method for solving dirichlet boundary value problems

a priori estimates for symmetrizing measures and their applications to gibbs states abstract we prove existence and uniform a priori estimates for tempered gibbs states of certain classical lattice systems with unbounded spins nonharmonic pair potentials and infinite radius of interaction we use an alternative characterization of gibbs measures in terms of their radon nikodym derivatives w r t local shifts of the configuration space and corresponding integration by parts formulas

composite wavelet bases for operator equations this paper is concerned with the construction of biorthogonal wavelet bases defined on a union of parametric images of the unit n cube these bases are to satisfy certain requirements imposed by applications to a class of operator equations acting on such domains this covers also elliptic boundary value problems although this study is primarily motivated by our previous analysis of wavelet methods for pseudo differential equations with special emphasis on boundary integral equations in this case it is natural to model the boundary surface as a union of parametric images of the unit cube it will be shown how to construct wavelet bases on the surface which are composed of wavelet bases defined on each surface patch here the relevant properties are the validity of norm equivalences in certain ranges of sobolev scales as well as appropriate moment conditions

necessary and sufficient conditions for constructing orthonormal wavelet bases this paper proves a previous conjecture of the author characterizing sequences h l2 z that yield orthonormal wavelet bases of l2 r in terms of the multiplicity of the eigenvalue 1 of an operator associated to h the proof utilizes a result of cohen characterizing these sequences in terms of the real zeros of their fourier transforms the mapping from sequences to wavelets is shown to define a continuous mapping from a subset of l2 z into l2 r related conjectures are discussed

second generation wavelet collocation method for the solution of partial differential equations an adaptive numerical method for solving partial differential equations is developed the method is based on the whole new class of second generation wavelets wavelet decomposition is used for grid adaptation and interpolation while a new o n hierarchical finite difference scheme which takes advantage of wavelet multilevel decomposition is used for derivative calculations the treatment of nonlinear terms and general boundary conditions is a straightforward task due to the collocation nature of the algorithm in this paper we demonstrate the algorithm for one particular choice of second generation wavelets namely lifted interpolating wavelets on an interval with uniform regular sampling the main advantage of using second generation wavelets is that wavelets can be custom designed for complex domains and irregular sampling thus the strength of the new method is that it can be easily extended to the whole class of second generation wavelets leaving the freedom and flexibility to choose the wavelet basis depending on the application

on the least asymmetric wavelets the asymmetry of daubechies 1988 1992 scaling functions and wavelets can be diminished by minimizing a special second moment in time for the wavelet generating discrete time filter the moment is involved in an uncertainty relation for discrete time signals other measures of asymmetry are addressed as well and corresponding results are compared

element by element construction of wavelets satisfying stability and moment conditions in this paper we construct a class of locally supported wavelet bases for c0 lagrange finite element spaces on possibly nonuniform meshes on n dimensional domains or manifolds the wavelet bases are stable in the sobolev spaces mathcal h s for s frac 3 2 s leq 1 on lipschitz manifolds and the wavelets can in principle be arranged to have any desired order of vanishing moments as a consequence these bases can be used e g for constructing an optimal solver of discretized mathcal h s elliptic problems for s in above ranges the construction of the wavelets consists of two parts an implicit part involves some computations on a reference element which for each type of finite element space have to be performed only once in addition there is an explicit part which takes care of the necessary adaptations of the wavelets to the actual mesh the only condition we need for this construction to work is that the refinements of initial elements are uniform we will show that the wavelet bases can be implemented efficiently

interpolatory hermite splines on rectangular domains abstract based on bittner and urban s construction of interpolatory multiwavelets kai bittner karsten urban on interpolatory divergence free wavelets math comput 76 258 2007 903 929 we use the truncation method to obtain interpolatory multiwavelets on rectangular domains in this paper in addition the characterization for the corresponding functional spaces is given

approximation by translates of refinable functions the functions f 1 x dots f r x are refinable if they are combinations of the rescaled and translated functions f i 2x k this is very common in scientific computing on a regular mesh the space v 0 of approximating functions with meshwidth h 1 is a subspace of v 1 with meshwidth h 1 2 these refinable spaces have refinable basis functions the accuracy of the computations depends on p the order of approximation which is determined by the degree of polynomials 1 x dots x p 1 that lie in v 0 most refinable functions such as scaling functions in the theory of wavelets have no simple formulas the functions f i x are known only through the coefficients c k in the refinement equation scalars in the traditional case r times r matrices for multiwavelets the scalar sum rules that determine p are well known we find the conditions on the matrices c k that yield approximation of order p from v 0 these are equivalent to the strang fix conditions on the fourier transforms hat f i omega but for refinable functions they can be explicitly verified from the c k

a discrete transform and decompositions of distribution spaces abstract we study a representation formula of the form q q q for a distribution on r n this formula is obtained by discretizing and localizing a standard littlewood paley decomposition the map taking to the sequence q q with q running over the dyadic cubes in r n is called the transform the functions q and q have a particularly simple form moreover most of the familiar distribution spaces lp spaces 1 r n we also consider pointwise multipliers for the characteristic function of a domain we obtain boundedness results for a general class of domains which properly includes lipschitz domains several interpolation methods are easily analyzed via the sequence spaces for real interpolation we obtain among other things an extension to the case p 0 this in turn gives a new approach to the traditional atomic decomposition of hardy spaces

uniform refinement of curves abstract we propose and analyze a class of algorithms for the generation of curves and surfaces these algorithms encompass some well known methods of subdivision for bernstein bezier curves de casteljau s algorithm and b spline curves lane and riesenfeld s algorithm several results concerning properties of the limiting curves as well as related questions are discussed

an alternating direction galerkin technique for simulation of contaminant transport in complex groundwater systems a general contaminant transport simulation technique based on a galerkin finite element representation in space and an alternating direction time stepping scheme is developed the formulation which is in terms of natural coordinates following the principal directions of hydraulic conductivity of the medium decouples the spatial components of the equations leading to tridiagonal matrices the numerical accuracy can be easily controlled in each principal direction through the well established peclet and courant numbers results are found to be comparable to those from the classic finite element scheme but the cost of computation is generally much less because the computational effort varies only linearly with the grid size the technique is thus well suited for transport simulations with complex layered anisotropic groundwater systems requiring large and detailed grids

une approche gibbsienne des diffusions browniennes infini dimensionnelles nous montrons qu il y a correspondance biunivoque entre les lois de diffusions browniennes infini dimensionnelles et les etats de gibbs sur l espace des trajectoires omega c 0 1 mathbb z d ce resultat est ensuite applique aux problemes de transition de phases du retournement du temps et a l etude des mesures reversibles le principal outil est une caracterisation des etats de gibbs par une version infini dimensionnelle de la formule d integration par parties du calcul de malliavin

lifting scheme for biorthogonal multiwavelets originated from hermite splines we present new multiwavelet transforms of multiplicity 2 for manipulation of discrete time signals the transforms are implemented in two phases 1 pre post processing which transforms the scalar signal into a vector signal and back and 2 wavelet transforms of the vector signal both phases are performed in a lifting manner we use the cubic interpolatory hermite splines as a predicting aggregate in the vector wavelet transform we present new pre post processing algorithms that do not degrade the approximation accuracy of the vector wavelet transforms we describe two types of vector wavelet transforms that are dual to each other but have similar properties and three pre post processing algorithms as a result we get fast biorthogonal algorithms to transform discrete time signals that are exact on sampled cubic polynomials the bases for the transform are symmetric and have short support

an adaptive multilevel wavelet collocation method for elliptic problems an adaptive multilevel wavelet collocation method for solving multi dimensional elliptic problems with localized structures is described the method is based on multi dimensional second generation wavelets and is an extension of the dynamically adaptive second generation wavelet collocation method for evolution problems int j comp fluid dyn 17 2003 151 wavelet decomposition is used for grid adaptation and interpolation while a hierarchical finite difference scheme which takes advantage of wavelet multilevel decomposition is used for derivative calculations the multilevel structure of the wavelet approximation provides a natural way to obtain the solution on a near optimal grid in order to accelerate the convergence of the solver an iterative procedure analogous to the multigrid algorithm is developed the overall computational complexity of the solver is o n where n is the number of adapted grid points the accuracy and computational efficiency of the method are demonstrated for the solution of two and three dimensional elliptic test problems

wavelet bases in h div and h curl some years ago compactly supported divergence free wavelets have been constructed which also give rise to a stable biorthogonal wavelet splitting of h div these bases have successfully been used both in the analysis and numerical treatment of the stokes and navier stokes equations in this paper we construct stable wavelet bases for the stream function spaces h curl moreover curl free vector wavelets are constructed and analysed the relationship between h div and h curl are expressed in terms of these wavelets we obtain discrete orthogonal hodge decompositions our construction works independently of the space dimension but in terms of general assumptions on the underlying wavelet systems in l 2 omega that are used as building blocks we give concrete examples of such bases for tensor product and certain more general domains omega subset er n as an application we obtain wavelet multilevel preconditioners in h div and h curl email urban dragon ian pv cnr it keywords h div h curl stream function spaces wavelets

the convergence of wavelet expansion with divergence free properties in vector valued besov spaces using the biorthogonal b spline wavelets we investigate the convergence property of wavelets expansion in the related vector valued besov spaces especially divergence free and non divergence free wavelets are added and discussed as a by product characterization of relevant besov spaces is given as well it is noted that the key point for characterization is to prove the convergence of projection operators in relevant besov spaces besides we can get convergence and characterization at the same time

wavelet adaptive method for second order elliptic problems boundary conditions and domain decomposition summary wavelet methods allow to combine high order accuracy multilevel preconditioning techniques and adaptive approximation in order to solve efficiently elliptic operator equations one of the main difficulty in this context is the efficient treatment of non homogeneous boundary conditions in this paper we propose a strategy that allows to append such conditions in the setting of space refinement i e adaptive discretizations of second order problems our method is based on the use of compatible multiscale decompositions for both the domain and its boundary and on the possibility of characterizing various function spaces from the numerical properties of these decompositions in particular this allows the construction of a lifting operator which is stable for a certain range of smoothness classes and preserves the compression of the solution in the wavelet basis an explicit construction of the wavelet bases and the lifting is proposed on fairly general domains based on c 0 conforming domain decomposition techniques

phase space localization theorem for ondelettes it is proven that an orthonormal basis of ondelettes can never have exponential localization in both position space and momentum space

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

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

integration by parts and quasi invariance for heat kernel measures on loop groups abstract integration by parts formulas are established both for wiener measure on the path space of a loop group and for the heat kernel measures on the loop group the wiener measure is defined to be the law of a certain loop group valued brownian motion and the heat kernel measures are time t t 0 distributions of this brownian motion a corollary of either of these integrations by parts formulas is the closability of the pre dirichlet form considered by b k driver and t lohrenz 1996 j functional anal 140 381 448 we also show that the heat kernel measures are quasi invariant under right under right and left translations by finite energy loops

an adaptive wavelet collocation method for fluid structure interaction at high reynolds numbers two mathematical approaches are combined to calculate high rey nolds number incompressible fluid structure interaction a wavelet method to dynamically adapt the computational grid to flow intermittency and obstacle motion and brinkman penalization to enforce solid boundaries of arbitrary complexity we also implement a wavelet based multilevel solver for the poisson problem for the pressure at each time step the method is applied to two dimensional flow around fixed and moving cylinders for reynolds numbers in the range 3 times 10 1 le re le 10 5 the compression ratios of up to 1000 are achieved for the first time it is demonstrated in actual dynamic simulations that the compression scales like re 1 2 over five orders of magnitude while computational complexity scales like re this represents a significant improvement over the classical complexity estimate of re 9 4 for two dimensional turbulence

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

the wavelet transform time frequency localization and signal analysis two different procedures for effecting a frequency analysis of a time dependent signal locally in time are studied the first procedure is the short time or windowed fourier transform the second is the wavelet transform in which high frequency components are studied with sharper time resolution than low frequency components the similarities and the differences between these two methods are discussed for both schemes a detailed study is made of the reconstruction method and its stability as a function of the chosen time frequency density finally the notion of time frequency localization is made precise within this framework by two localization theorems

lattice structures for optimal design and robust implementation of two channel perfect reconstruction qmf banks a lattice structure and an algorithm are presented for the design of two channel qmf quadrature mirror filter banks satisfying a sufficient condition for perfect reconstruction the structure inherently has the perfect reconstruction property while the algorithm ensures a good stopband attenuation for each of the analysis filters implementations of such lattice structures are robust in the sense that the perfect reconstruction property is preserved in spite of coefficient quantization the lattice structure has the hierarchical property that a higher order perfect reconstruction qmf bank can be obtained from a lower order perfect reconstruction qmf bank simply by adding more lattice sections several numerical examples are provided in the form of design tables

a generalized two dimensional analytical solute transport model in bounded media for flux type finite multiple sources this paper presents a generalized two dimensional analytical solute transport model with unidirectional flow field from time and space dependent sources in a bounded homogeneous medium using the flux type or third type or cauchy boundary condition at the inlet location of the medium the solute transport equation incorporates terms accounting for advection dispersion linear equilibrium sorption and first order decay general solutions were determined for arbitrary input boundary conditions with the help of fourier analysis and laplace transform techniques after presenting a generally applicable solution special solutions are given for a single and two finite sources having different inlet solute flux distributions the expressions for convective dispersive flux components are also derived expressions for concentration distribution and convective dispersive flux components for steady state transport case are also presented comparison of the results of this analytical model and a finite difference numerical model showed very good agreement the solutions may be used for predicting solute concentrations in a unidirectional porous media flow field as well as in rivers and canals the solutions may also be used for verification of more comprehensive numerical models and laboratory or field determination of solute transport parameters

painless nonorthogonal expansions in a hilbert space h discrete families of vectors hj with the property that f j hj f hj for every f in h are considered this expansion formula is obviously true if the family is an orthonormal basis of h but also can hold in situations where the hj are not mutually orthogonal and are overcomplete the two classes of examples studied here are i appropriate sets of weyl heisenberg coherent states based on certain non gaussian fiducial vectors and ii analogous families of affine coherent states it is believed that such quasiorthogonal expansions will be a useful tool in many areas of theoretical physics and applied mathematics