on the asymptotic convergence of b spline wavelets to gabor functions a family of nonorthogonal polynomial spline wavelet transforms is considered these transforms are fully reversible and can be implemented efficiently the corresponding wavelet functions have a compact support it is proven that these b spline wavelets converge to gabor functions modulated gaussian pointwise and in all l sub p norms with 1

analysis and vlsi architecture for 1 d and 2 d discrete wavelet transform in this paper a detailed analysis of very large scale integration vlsi architectures for the one dimensional 1 d and two dimensional 2 d discrete wavelet transform dwt is presented in many aspects and three related architectures are proposed as well the 1 d dwt and inverse dwt idwt architectures are classified into three categories convolution based lifting based and b spline based they are discussed in terms of hardware complexity critical path and registers as for the 2 d dwt the large amount of the frame memory access and the die area occupied by the embedded internal buffer become the most critical issues the 2 d dwt architectures are categorized and analyzed by different external memory scan methods the implementation issues of the internal buffer are also discussed and some real life experiments are given to show that the area and power for the internal buffer are highly related to memory technology and working frequency instead of the required memory size only besides the analysis the b spline based idwt architecture and the overlapped stripe based scan method are also proposed last we propose a flexible and efficient architecture for a one level 2 d dwt that exploits many advantages of the presented analysis

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

complete parameterization of piecewise polynomial interpolation kernels every now and then a new design of an interpolation kernel appears in the literature while interesting results have emerged the traditional design methodology proves laborious and is riddled with very large systems of linear equations that must be solved analytically we propose to ease this burden by providing an explicit formula that can generate every possible piecewise polynomial kernel given its degree its support its regularity and its order of approximation this formula contains a set of coefficients that can be chosen freely and do not interfere with the four main design parameters it is thus easy to tune the design to achieve any additional constraints that the designer may care for

mimo biorthogonal partners and applications multiple input multiple output mimo biorthogonal partners arise in many different contexts one of them being multiwavelet theory they also play a central role in the theory of mimo channel equalization especially with fractionally spaced equalizers in this paper we first derive some theoretical properties of mimo biorthogonal partners we develop conditions for the existence of mimo biorthogonal partners and conditions under which fir solutions are possible in the process of constructing fir mimo biorthogonal partners we exploit the nonuniqueness of the solution this will lead to the design of flexible fractionally spaced mimo zero forcing equalizers the additional flexibility in design makes these equalizers more robust to channel noise finally other situations where mimo biorthogonal partners occur are also considered such as prefiltering in multiwavelet theory and deriving the vector version of the least squares signal projection problem

characterizing jacobians via trisecants of the kummer variety we prove welters trisecant conjecture an indecomposable principally polarized abelian variety x is the jacobian of a curve if and only if there exists a trisecant of its kummer variety k x

embedded image coding using zerotrees of wavelet coefficients the embedded zerotree wavelet algorithm ezw is a simple yet remarkably effective image compression algorithm having the property that the bits in the bit stream are generated in order of importance yielding a fully embedded code the embedded code represents a sequence of binary decisions that distinguish an image from the null image using an embedded coding algorithm an encoder can terminate the encoding at any point thereby allowing a target rate or target distortion metric to be met exactly also given a bit stream the decoder can cease decoding at any point in the bit stream and still produce exactly the same image that would have been encoded at the bit rate corresponding to the truncated bit stream in addition to producing a fully embedded bit stream the ezw consistently produces compression results that are competitive with virtually all known compression algorithms on standard test images yet this performance is achieved with a technique that requires absolutely no training no pre stored tables or codebooks and requires no prior knowledge of the image source the ezw algorithm is based on four key concepts 1 a discrete wavelet transform or hierarchical subband decomposition 2 prediction of the absence of significant information across scales by exploiting the self similarity inherent in images 3 entropy coded successive approximation quantization and 4 universal lossless data compression which is achieved via adaptive arithmetic coding

flipping structure an efficient vlsi architecture for lifting based discrete wavelet transform in this paper an efficient very large scale integration vlsi architecture called flipping structure is proposed for the lifting based discrete wavelet transform it can provide a variety of hardware implementations to improve and possibly minimize the critical path as well as the memory requirement of the lifting based discrete wavelet transform by flipping conventional lifting structures the precision issues are also analyzed by case studies of the jpeg2000 default lossy 9 7 filter an integer 9 7 filter and the 6 10 filter the efficiency of the proposed flipping structure is demonstrated

characterization of jacobian varieties in terms of soliton equations pour dans le demi espace superieur de siegel h g soit x c g z g z g la variete abelienne principalement polarisee correspondante et soit z z m e z gexp 2 i t mz i t m m la fonction theta de riemann pour x les deux conditions a et b pour x sont equivalentes a 1 il existe des vecteurs a 1 a 2 a 3 c g a 1 0 et une forme quadratique q t i j 1 3 q ij t i t j q ij c tel que pour c g t t exp q t t 1 a 1 t 2 a 2 t 3 a 3 donne une solution a l equation de kadomtsev petviashvili sous la forme d hirota d 1 4 3d 2 2 4d 1 d 3 0 2 le diviseur de x est irreductible b x est isomorphe a la variete jacobienne d une courbe lisse complete de genre g sur c

analysis and modification of newton s method for algebraic riccati equations when newton s method is applied to find the maximal symmetric solution of an algebraic riccati equation convergence can be guaranteed under moderate conditions in particular the initial guess need not be close to the solution the convergence is quadratic if the frechet derivative is invertible at the solution in this paper we examine the behaviour of the newton iteration when the derivative is not invertible at the solution we find that a simple modification can improve the performance of the newton iteration dramatically

singularity detection and processing with wavelets the mathematical characterization of singularities with lipschitz exponents is reviewed theorems that estimate local lipschitz exponents of functions from the evolution across scales of their wavelet transform are reviewed it is then proven that the local maxima of the wavelet transform modulus detect the locations of irregular structures and provide numerical procedures to compute their lipschitz exponents the wavelet transform of singularities with fast oscillations has a particular behavior that is studied separately the local frequency of such oscillations is measured from the wavelet transform modulus maxima it has been shown numerically that one and two dimensional signals can be reconstructed with a good approximation from the local maxima of their wavelet transform modulus as an application an algorithm is developed that removes white noises from signals by analyzing the evolution of the wavelet transform maxima across scales in two dimensions the wavelet transform maxima indicate the location of edges in images

the restricted grassmannian banach lie poisson spaces and coadjoint orbits we investigate some basic questions concerning the relationship between the restricted grassmannian and the theory of banach lie poisson spaces by using universal central extensions of lie algebras we find that the restricted grassmannian is symplectomorphic to symplectic leaves in certain banach lie poisson spaces and the underlying banach space can be chosen to be even a hilbert space smoothness of numerous adjoint and coadjoint orbits of the restricted unitary group is also established several pathological properties of the restricted algebra are pointed out

removal of the resolvent like energy dependence from interactions and invariant subspaces of a total hamiltonian the spectral problem a v z z is considered where the main hamiltonian a is a self adjoint operator of sufficiently arbitrary nature the perturbation v z b a z 1b depends on the energy z as resolvent of another self adjoint operator a the latter is usually interpreted as a hamiltonian describing an internal structure of physical system the operator b is assumed to have a finite hilbert schmidt norm the conditions are formulated when one can replace the perturbation v z with an energy independent potential w such that the hamiltonian h a w has the same spectrum more exactly a part of spectrum and the same eigenfunctions as the initial spectral problem the hamiltonian h is constructed as a solution of the non linear operator equation h a v h it is established that this equation is closely connected with the problem of searching for invariant subspaces of the hamiltonian h b aa b the orthogonality and expansion theorems are proved for eigenfunction systems of the hamiltonian h a w

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

sampling moments and reconstructing signals of finite rate of innovation shannon meets strang fix consider the problem of sampling signals which are not bandlimited but still have a finite number of degrees of freedom per unit of time such as for example nonuniform splines or piecewise polynomials and call the number of degrees of freedom per unit of time the rate of innovation classical sampling theory does not enable a perfect reconstruction of such signals since they are not bandlimited recently it was shown that by using an adequate sampling kernel and a sampling rate greater or equal to the rate of innovation it is possible to reconstruct such signals uniquely these sampling schemes however use kernels with infinite support and this leads to complex and potentially unstable reconstruction algorithms in this paper we show that many signals with a finite rate of innovation can be sampled and perfectly reconstructed using physically realizable kernels of compact support and a local reconstruction algorithm the class of kernels that we can use is very rich and includes functions satisfying strang fix conditions exponential splines and functions with rational fourier transform this last class of kernels is quite general and includes for instance any linear electric circuit we thus show with an example how to estimate a signal of finite rate of innovation at the output of an rc circuit the case of noisy measurements is also analyzed and we present a novel algorithm that reduces the effect of noise by oversampling

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

a criterion for jacobi varieties is a 2 1 map onto its image the kummer wirtinger variety kw x associated with x the following is well known cf 6 cf also 10 assume that x is the polarized jacobian of a smooth curve c and consider c as embedded in x by an arbitrary element of picl c let d be an effective divisor of degree 3 on c and call y the subscheme of x defined by d put s s y the abelian sum of the components of the corresponding 0 cycle of degree 3 on x then for any d ex with 2d e c s there exists a line 1 in pn such that the translated subscheme d y satisfies d y c 4 1 l in other words 1 is a trisecant or a flex of the kummer wirtinger variety and the image of d y lies in the intersection of 1 with kw x in his paper 3 r c gunning has shown how to use the above property in order to characterize jacobians gunning s result can be stated as follows

perturbation of spectra and spectral subspaces we consider the problem of variation of spectral subspaces for linear self adjoint operators with emphasis on the case of off diagonal perturbations we prove a number of new optimal results on the shift of the spectrum and obtain sharp estimates on the norm of the difference of two spectral projections associated with isolated parts of the spectrum of the perturbed and unperturbed operators respectively

linear relations among holomorphic quadratic differentials and induced siegel s metric on mg we find the explicit form of the volume form on the moduli space of non hyperelliptic riemann surfaces induced by the siegel metric a long standing question in string theory this question is related to the explicit form of the g 2 g 3 2 linearly independent relations among the 2 fold products of holomorphic abelian differentials that are provided in the case of canonical curves of genus g 4 such relations can be completely expressed in terms of determinants of the standard normalized holomorphic abelian differentials remarkably it turns out that the induced volume form is the kodaira spencer map of the square of the bergman reproducing kernel

self similarity part i splines and operators the central theme of this pair of papers parts i and ii in this issue is self similarity which is used as a bridge for connecting splines and fractals the first part of the investigation is deterministic and the context is that of l splines these are defined in the following terms s t is a cardinal l spline iff l s t sigma kisinza k delta t k where l is a suitable pseudodifferential operator our starting point for the construction of self similar splines is the identification of the class of differential operators l that are both translation and scale invariant this results into a two parameter family of generalized fractional derivatives parttau gamma where gamma is the order of the derivative and tau is an additional phase factor we specify the corresponding l splines which yield an extended class of fractional splines the operator parttau gamma is used to define a scale invariant energy measure the squared l2 norm of the gammath derivative of the signal which provides a regularization functional for interpolating or fitting the noisy samples of a signal we prove that the corresponding variational or smoothing spline estimator is a cardinal fractional spline of order 2gamma which admits a stable representation in a b spline basis we characterize the equivalent frequency response of the estimator and show that it closely matches that of a classical butterworth filter of order 2gamma we also establish a formal link between the regularization parameter lambda and the cutoff frequency of the smoothing spline filter omega0aplambda 2gamma finally we present an efficient computational solution to the fractional smoothing spline problem it uses the fast fourier transform and takes advantage of the multiresolution properties of the underlying basis functions

on applications of schauderʼs fixed point theorem for the solution of the non symmetric algebraic riccati equation abstract in this article we give sufficient conditions for the existence of contractive solutions for the non symmetric algebraic riccati equation the conditions are given in terms of the matrix of the coefficients only we also provide sufficient conditions for unique contractive solution which is a unique minimal norm solution the results are generalized to contractive solutions with respect to generalized unit matrix balls

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

riccati equation and the problem of decoherence the block operator matrix theory is used to investigate the problem of a single qubit we establish a connection between the riccati operator equation and the possibility of obtaining an exact reduced dynamics for the qubit in question the model of the half spin particle in the rotating magnetic field coupling with the external environment is discussed we show that the model defined in such a way can be reduced to a time independent problem

a new concept for block operator matrices the quadratic numerical range abstract in this paper a new concept for 2 2 block operator matrices the quadratic numerical range is studied the main results are a spectral inclusion theorem an estimate of the resolvent in terms of the quadratic numerical range factorization theorems for the schur complements and a theorem about angular operator representations of spectral invariant subspaces which implies e g the existence of solutions of the corresponding riccati equations and a block diagonalization all results are new in the operator as well as in the matrix case

existence and uniqueness of contractive solutions of some riccati equations abstract in this paper we establish criteria for the existence and uniqueness of contractive solutions k of the riccati equation kbk ka dk c 0 under the assumption that the spectra of a and d are disjoint

a new flexible bi orthogonal filter design for multiresolution filterbanks with application to image compression we propose a new fast method with great potential for multiresolution pyramid decomposition of signals and images the method allows unusual flexibility in choosing a filter for any task involving the multiresolution analysis and synthesis using our method one can choose any low pass filter for the multiresolution filtering this method enabled us to choose the best filters for set partitioning in hierarchical trees spiht image compression for the corresponding support sizes the compression results for our seven tap filters are better than those of the 9 7 wavelet filters and approximately the same as those of the 10 18 filters while at the same time our seven tap filters are faster than 10 18 filters

a characterization of non hyperelliptic jacobi varieties this improves the main result of z ran s paper 6 and theorem 7 p 476 of his earlier paper 7 we remark that since any reducible principally polarized abelian variety is a trivial solution to the hypotheses in mumford s problem the main statement in 6 is not entirely correct both works are the main source of inspiration for the present one in contrast with z ran s techniques we use general facts about line bundles on abelian varieties secondly an eventual use of r c gunning s recent result 2 brings the proof to a quick end

a fast wavelet based reconstruction method for magnetic resonance imaging in this work we exploit the fact that wavelets can represent magnetic resonance images well with relatively few coefficients we use this property to improve magnetic resonance imaging mri reconstructions from undersampled data with arbitrary k space trajectories reconstruction is posed as an optimization problem that could be solved with the iterative shrinkage thresholding algorithm ista which unfortunately converges slowly to make the approach more practical we propose a variant that combines recent improvements in convex optimization and that can be tuned to a given specific k space trajectory we present a mathematical analysis that explains the performance of the algorithms using simulated and in vivo data we show that our nonlinear method is fast as it accelerates ista by almost two orders of magnitude we also show that it remains competitive with tv regularization in terms of image quality

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

eigenvalues of spl darr 2 h and convergence of the cascade algorithm this paper is about the eigenvalues and eigenvectors of spl darr 2 h the ordinary fir filter h is a convolution with a vector h h o h n which is the impulse response the operator spl darr 2 downsamples the output y h x keeping the even numbered components y 2n where h is represented by a constant diagonal matrix this is a toeplitz matrix with h k on its kth diagonal the odd numbered rows are removed in spl darr 2 h the result is a double shift between rows yielding a block toeplitz matrix with 1 spl times 2 blocks iteration of the filter is governed by the eigenvalues if the transfer function h z spl sigma h k z sup k has a zero of order p at z 1 corresponding to spl omega spl pi then spl darr 2 h has p special eigenvalues 1 2 1 4 1 2 sup p we show how each additional zero at spl pi divides all eigenvalues by 2 and creates a new eigenvector for spl lambda 1 2 this eigenvector solves the dilation equation spl phi t 2 spl sigma h k spl phi 2t k at the integers t n the left eigenvectors show how 1 t t sup p 1 can be produced as combinations of spl phi t k the dilation equation is solved by the cascade algorithm which is an infinite iteration of m spl darr 2 2h convergence in l sup 2 is governed by the eigenvalues of t spl darr 2 2hh sup t corresponding to the response 2h z h z sup 1 we find a simple proof of the necessary and sufficient condition for convergence

approximation power of biorthogonal wavelet expansions this paper looks at the effect of the number of vanishing moments on the approximation power of wavelet expansions the strang fix conditions imply that the error for an orthogonal wavelet approximation at scale a 2 sup i globally decays as a sup n where n is the order of the transform this is why for a given number of scales higher order wavelet transforms usually result in better signal approximations we prove that this result carries over for the general biorthogonal case and that the rate of decay of the error is determined by the order properties of the synthesis scaling function alone we also derive asymptotic error formulas and show that biorthogonal wavelet transforms are equivalent to their corresponding orthogonal projector as the scale goes to zero these results strengthen sweldens earlier analysis and confirm that the approximation power of biorthogonal and semi orthogonal wavelet expansions is essentially the same finally we compare the asymptotic performance of various wavelet transforms and briefly discuss the advantages of splines we also indicate how the smoothness of the basis functions is beneficial in reducing the approximation error

wavelets and dilation equations a brief introduction wavelets are new families of basis functions that yield the representation f x sum b jk w 2 j x k their construction begins with the solution phi x to a dilation equation with coefficients c k then w comes from phi and the basis comes by translation and dilation of w it is shown in part 1 how conditions on the c k lead to approximation properties and orthogonality properties of the wavelets part 2 describes the recursive algorithms also based on the c k that decompose and reconstruct f the object of wavelets is to localize as far as possible in both time and frequency with efficient algorithms

an inflectionary tangent to the kummer variety and the jacobian condition in this paper we deal with a degenerate version of the trisecant conjecture let x theta be an indecomposable principally polarized abelian variety and let theta be a symmetric representative of the polarization we shall denote by theta a non zero section of the corresponding sheaf the linear system vert 2 theta vert is base point free and it is independent of the choice of theta the image of the morphism k x rightarrow vert 2 theta vert associated with the base point free linear system vert 2 theta vert is a projective variety which is called the kummer variety of x theta welters conjectured that the existence of one trisecant line to the kummer variety characterizes the jacobians it is well known that the kummer variety of a jacobian has a rich geometry in terms of trisecants and flexes we prove that if there exists an inflectionary tangent l to the kummer variety associated with x theta then x theta is a jacobian provided that there are no set theoretical d invariant components of the scheme d theta theta cap d theta 0 where d is an invariant vector field on x associated to l

biorthogonal partners and applications two digital filters h z and f z are said to be biorthogonal partners of each other if their cascade h z f z satisfies the nyquist or zero crossing property biorthogonal partners arise in many different contexts such as filterbank theory exact and least squares digital interpolation and multiresolution theory they also play a central role in the theory of equalization especially fractionally spaced equalizers in digital communications we first develop several theoretical properties of biorthogonal partners we also develop conditions for the existence of biorthogonal partners and fir biorthogonal pairs and establish the connections to the riesz basis property we then explain how these results play a role in many of the above mentioned applications

interpolation revisited medical images application based on the theory of approximation this paper presents a unified analysis of interpolation and resampling techniques an important issue is the choice of adequate basis functions the authors show that contrary to the common belief those that perform best are not interpolating by opposition to traditional interpolation the authors call their use generalized interpolation they involve a prefiltering step when correctly applied the authors explain why the approximation order inherent in any basis function is important to limit interpolation artifacts the decomposition theorem states that any basis function endowed with approximation order ran be expressed as the convolution of a b spline of the same order with another function that has none this motivates the use of splines and spline based functions as a tunable way to keep artifacts in check without any significant cost penalty the authors discuss implementation and performance issues and they provide experimental evidence to support their claims

a representation of all solutions of the control algebraic riccati equation for infinite dimensional systems we obtain a representation of all self adjoint solutions of the control algebraic riccati equation associated to the infinite dimensional state linear system a b c under the following assumptions a generates a c0 group the system is output stabilizable strongly detectable and the dual riccati equation has an invertible self adjoint non negative solution

determinantal characterization of canonical curves and combinatorial theta identities we characterize genus g canonical curves by the vanishing of combinatorial products of g 1 determinants of brill noether matrices this also implies the characterization of canonical curves in terms of g 2 g 3 2 theta identities a remarkable mechanism based on a basis of h 0 k c expressed in terms of szego kernels reduces such identities to a simple rank condition for matrices whose entries are logarithmic derivatives of theta functions such a basis together with the fay trisecant identity also leads to the solution of the question of expressing the determinant of brill noether matrices in terms of theta functions without using the problematic klein fay section

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