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

nonlocal boundary problems for a third order one dimensional nonlinear pseudoparabolic equation abstract we consider initial boundary value problems for a third order nonlinear pseudoparabolic equation with one space dimension the boundary condition is given by an integral the function involved could exhibit singularities which distinguishes this nonlocal condition from its dirichlet or neumann counterparts by means of appropriate elliptic estimates we are able to seek solutions not only in the weighted spaces but also in the usual sobolev spaces the procedure is carried out in a unified way our results characterize a regularity of the pseudoparabolic operator that is different from that of the parabolic operator

synthesis of complete orthonormal fractional basis functions with prescribed poles in this paper fractional orthonormal basis functions that generalize the well known fixed pole rational basis functions are synthesized for a range of noninteger differentiation orders and under mild restrictions on the choice of the basis poles the synthesized basis functions are shown to be complete in the space of functions which are analytic on the open right half plane and square integrable on the imaginary axis this presents an extension of the completeness results for the fractional laguerre and kautz bases to fractional orthonormal bases with prescribed pole locations

unwarping of unidirectionally distorted epi images echo planar imaging epi is a fast nuclear magnetic resonance imaging mri method unfortunately local magnetic field inhomogeneities induced mainly by the subject s presence cause significant geometrical distortion predominantly along the phase encoding direction which must be undone to allow for meaningful further processing so far this aspect has been too often neglected in this paper the authors suggest a new approach using an algorithm specifically developed for the automatic registration of distorted epi images with corresponding anatomically correct mri images they model the deformation field with splines which gives us a great deal of flexibility while comprising the affine transform as a special case the registration criterion is least squares interestingly the complexity of its evaluation does not depend on the resolution of the control grid the spline model gives the authors good accuracy thanks to its high approximation order the short support of splines leads to a fast algorithm a multiresolution approach yields robustness and additional speedup the algorithm was tested on real as well as synthetic data and the results were compared with a manual method a wavelet based sobolev type random deformation generator was developed for testing purposes a blind test indicates that the proposed automatic method is faster more reliable and more precise than the manual one

fresnelets new multiresolution wavelet bases for digital holography we propose a construction of new wavelet like bases that are well suited for the reconstruction and processing of optically generated fresnel holograms recorded on ccd arrays the starting point is a wavelet basis of l sub 2 to which we apply a unitary fresnel transform the transformed basis functions are shift invariant on a level by level basis but their multiresolution properties are governed by the special form that the dilation operator takes in the fresnel domain we derive a heisenberg like uncertainty relation that relates the localization of fresnelets with that of their associated wavelet basis according to this criterion the optimal functions for digital hologram processing turn out to be gabor 1948 functions bringing together two separate aspects of the holography inventor s work we give the explicit expression of orthogonal and semi orthogonal fresnelet bases corresponding to polynomial spline wavelets this special choice of fresnelets is motivated by their near optimal localization properties and their approximation characteristics we then present an efficient multiresolution fresnel transform algorithm the fresnelet transform this algorithm allows for the reconstruction backpropagation of complex scalar waves at several user defined wavelength independent resolutions furthermore when reconstructing numerical holograms the subband decomposition of the fresnelet transform naturally separates the image to reconstruct from the unwanted zero order and twin image terms this greatly facilitates their suppression we show results of experiments carried out on both synthetic simulated data sets as well as on digitally acquired holograms

analytical footprints compact representation of elementary singularities in wavelet bases we introduce a family of elementary singularities that are point holder regular these singularities are self similar and are the green functions of fractional derivative operators i e by suitable fractional differentiation one retrieves a dirac function at the exact location of the singularity we propose to use fractional operator like wavelets that act as a multiscale version of the derivative in order to characterize and localize singularities in the wavelet domain we show that the characteristic signature when the wavelet interacts with an elementary singularity has an asymptotic closed form expression termed the analytical footprint practically this means that the dictionary of wavelet footprints is embodied in a single analytical form we show that the wavelet coefficients of the nonredundant decomposition can be fitted in a multiscale fashion to retrieve the parameters of the underlying singularity we propose an algorithm based on stepwise parametric fitting and the feasibility of the approach to recover singular signal representations

wavelets fractals and radial basis functions wavelets and radial basis functions rbfs lead to two distinct ways of representing signals in terms of shifted basis functions rbfs unlike wavelets are nonlocal and do not involve any scaling which makes them applicable to nonuniform grids despite these fundamental differences we show that the two types of representation are closely linked together through fractals first we identify and characterize the whole class of self similar radial basis functions that can be localized to yield conventional multiresolution wavelet bases conversely we prove that for any compactly supported scaling function spl phi x there exists a one sided central basis function spl rho sub x that spans the same multiresolution subspaces the central property is that the multiresolution bases are generated by simple translation of spl rho sub without any dilation we also present an explicit time domain representation of a scaling function as a sum of harmonic splines the leading term in the decomposition corresponds to the fractional splines a recent continuous order generalization of the polynomial splines

asymptotic error expansion of wavelet approximations of smooth functions ii we generalize earlier results concerning an asymptotic error expansion of wavelet approximations the properties of the monowavelets which are the building blocks for the error expansion are studied in more detail and connections between spline wavelets and euler and bernoulli polynomials are pointed out the expansion is used to compare the error for different wavelet families we prove that the leading terms of the expansion only depend on the multiresolution subspaces v j and not on how the complementary subspaces w j are chosen consequently for a fixed set of subspaces v j the leading terms do not depend on the fact whether the wavelets are orthogonal or not we also show that daubechies orthogonal wavelets need in general one level more than spline wavelets to obtain an approximation with a prescribed accuracy these results are illustrated with numerical examples

stochastic models for sparse and piecewise smooth signals we introduce an extended family of continuous domain stochastic models for sparse piecewise smooth signals these are specified as solutions of stochastic differential equations or equivalently in terms of a suitable innovation model the latter is analogous conceptually to the classical interpretation of a gaussian stationary process as filtered white noise the two specific features of our approach are 1 signal generation is driven by a random stream of dirac impulses poisson noise instead of gaussian white noise and 2 the class of admissible whitening operators is considerably larger than what is allowed in the conventional theory of stationary processes we provide a complete characterization of these finite rate of innovation signals within gelfand s framework of generalized stochastic processes we then focus on the class of scale invariant whitening operators which correspond to unstable systems we show that these can be solved by introducing proper boundary conditions which leads to the specification of random spline type signals that are piecewise smooth these processes are the poisson counterpart of fractional brownian motion they are nonstationary and have the same 1 type spectral signature we prove that the generalized poisson processes have a sparse representation in a wavelet like basis subject to some mild matching condition we also present a limit example of sparse process that yields a map signal estimator that is equivalent to the popular tv denoising algorithm

on the shiftability of dual tree complex wavelet transforms the dual tree complex wavelet transform dt bbcwt is known to exhibit better shift invariance than the conventional discrete wavelet transform we propose an amplitude phase representation of the dt ibbc wt which among other things offers a direct explanation for the improvement in the shift invariance the representation is based on the shifting action of the group of fractional hilbert transform fht operators which extends the notion of arbitrary phase shifts from sinusoids to finite energy signals wavelets in particular in particular we characterize the shift ability of the dt bbcwt in terms of the shifting property of the fhts at the heart of the representation are certain fundamental invariances of the fht group namely that of translation dilation and norm which play a decisive role in establishing the key properties of the transform it turns out that these fundamental invariances are exclusive to this group next by introducing a generalization of the bedrosian theorem for the fht operator we derive an explicitly understanding of the shifting action of the fht for the particular family of wavelets obtained through the modulation of lowpass functions e g the shannon and gabor wavelet this in effect links the corresponding dual tree transform with the framework of windowed fourier analysis finally we extend these ideas to the multidimensional setting by introducing a directional extension of the fht the fractional directional hilbert transform in particular we derive a signal representation involving the superposition of direction selective wavelets with appropriate phase shifts which helps explain the improved shift invariance of the transform along certain preferential directions

uncertainty principles in finitely generated shift invariant spaces with additional invariance author s tessera romain wang haichao abstract we consider finitely generated shift invariant spaces sis with additional invariance in l 2 r d we prove that if the generators and their translates form a frame then they must satisfy some stringent restrictions on their behavior at infinity part of this work non trivially generalizes recent results obtained in the special case of a principal shift invariant spaces in l 2 r whose generator and its translates form a riesz basis

self similarity part ii optimal estimation of fractal processes in a companion paper see self similarity part i splines and operators we characterized the class of scale invariant convolution operators the generalized fractional derivatives of order gamma we used these operators to specify regularization functionals for a series of tikhonov like least squares data fitting problems and proved that the general solution is a fractional spline of twice the order we investigated the deterministic properties of these smoothing splines and proposed a fast fourier transform fft based implementation here we present an alternative stochastic formulation to further justify these fractional spline estimators as suggested by the title the relevant processes are those that are statistically self similar that is fractional brownian motion fbm and its higher order extensions to overcome the technical difficulties due to the nonstationary character of fbm we adopt a distributional formulation due to gel fand this allows us to rigorously specify an innovation model for these fractal processes which rests on the property that they can be whitened by suitable fractional differentiation using the characteristic form of the fbm we then derive the conditional probability density function pdf p bh t y where y bh k n k kisinz are the noisy samples of the fbm bh t with hurst exponent h we find that the conditional mean is a fractional spline of degree 2h which proves that this class of functions is indeed optimal for the estimation of fractal like processes the result also yields the optimal minimum mean square error mmse parameters for the smoothing spline estimator as well as the connection with kriging and wiener filtering

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

approximation from shift invariant subspaces of l 2 ℝ d a complete characterization is given of closed shift invariant subspaces of l2 rd which provide a specified approximation order when such a space is principal i e generated by a single function then this characterization is in terms of the fourier transform of the generator as a special case we obtain the classical strang fix conditions but without requiring the generating function to decay at infinity the approximation order of a general closed shift invariant space is shown to be already realized by a specifiable principal subspace

b spline signal processing ii efficiency design and applications for pt i see ibid vol 41 no 2 p 821 33 1993 a class of recursive filtering algorithms for the efficient implementation of b spline interpolation and approximation techniques is described in terms of simplicity of realization and reduction of computational complexity these algorithms compare favorably with conventional matrix approaches a filtering interpretation low pass filter followed by an exact polynomial spline interpolator of smoothing spline and least squares approximation methods is proposed these techniques are applied to the design of digital filters for cubic spline signal processing an efficient implementation of a smoothing spline edge detector is proposed it is also shown how to construct a cubic spline image pyramid that minimizes the loss of information in passage from one resolution level to the next in terms of common measures of fidelity this data structure appears to be superior to the gaussian laplacian pyramid

subdivisions with infinitely supported mask in this paper we investigate the convergence of subdivision schemes associated with masks being polynomially decay sequences two scale vector refinement equations are the email protected x email protected email protected za a f email protected email protected r where the vector of functions f f 1 f r t is in l 2 r r and email protected a a a z is polynomially decay sequence of rxr matrices called refinement mask associated with the mask a is a linear operator on l 2 r r given byq af x email protected email protected za a f email protected email protected r f f 1 f r email protected l 2 r r by using same methods in b han r q jia characterization of riesz bases of wavelets generated from multiresolution analysis manuscript b han refinable functions and cascade algorithms in weighted spaces with infinitely supported masks manuscript r q jia q t jiang z w shen convergence of cascade algorithms associated with nonhomogeneous refinement equations proc amer math soc 129 2001 415 427 r q jia convergence of vector subdivision schemes and construction of biorthogonal multiple wavelets in advances in wavelet hong kong 1997 springer singapore 1998 pp 199 227 a characterization of convergence of the sequences q a nf n 1 2 in the l 2 norm is given which extends the main results in r q jia s d riemenschneider d x zhou vector subdivision schemes and multiple wavelets math comp 67 1998 1533 1563 on convergence of the subdivision schemes associated with a finitely supported mask to the case in which mask a is polynomially decay sequence as an application we also obtain a characterization of smoothness of solutions of the refinement equation mentioned above for the case r 1

an orthogonal family of quincunx wavelets with continuously adjustable order we present a new family of two dimensional and three dimensional orthogonal wavelets which uses quincunx sampling the orthogonal refinement filters have a simple analytical expression in the fourier domain as a function of the order spl lambda which may be noninteger we can also prove that they yield wavelet bases of l sub 2 r sup 2 for any spl lambda 0 the wavelets are fractional in the sense that the approximation error at a given scale a decays like o a sup spl lambda they also essentially behave like fractional derivative operators to make our construction practical we propose a fast fourier transform based implementation that turns out to be surprisingly fast in fact our method is almost as efficient as the standard mallat algorithm for separable wavelets

fractional differential mask a fractional differential based approach for multiscale texture enhancement in this paper we intend to implement a class of fractional differential masks with high precision thanks to two commonly used definitions of fractional differential for what are known as grumwald letnikov and riemann liouville we propose six fractional differential masks and present the structures and parameters of each mask respectively on the direction of negative x coordinate positive x coordinate negative y coordinate positive y coordinate left downward diagonal left upward diagonal right downward diagonal and right upward diagonal moreover by theoretical and experimental analyzing we demonstrate the second is the best performance fractional differential mask of the proposed six ones finally we discuss further the capability of multiscale fractional differential masks for texture enhancement experiments show that for rich grained digital image the capability of nonlinearly enhancing complex texture details in smooth area by fractional differential based approach appears obvious better than by traditional integral based algorithms

polyharmonic smoothing splines and the multidimensional wiener filtering of fractal like signals motivated by the fractal like behavior of natural images we develop a smoothing technique that uses a regularization functional which is a fractional iterate of the laplacian this type of functional was initially introduced by duchon for the approximation of nonuniformily sampled multidimensional data he proved that the general solution is a smoothing spline that is represented by a linear combination of radial basis functions rbfs unfortunately this is tedious to implement for images because of the poor conditioning of rbfs and their lack of decay here we present a much more efficient method for the special case of a uniform grid the key idea is to express duchon s solution in a fractional polyharmonic b spline basis that spans the same space as the rbfs this allows us to derive an algorithm where the smoothing is performed by filtering in the fourier domain next we prove that the above smoothing spline can be optimally tuned to provide the mmse estimation of a fractional brownian field corrupted by white noise this is a strong result that not only yields the best linear filter wiener solution but also the optimal interpolation space which is not bandlimited it also suggests a way of using the noisy data to identify the optimal parameters order of the spline and smoothing strength which yields a fully automatic smoothing procedure we evaluate the performance of our algorithm by comparing it against an oracle wiener filter which requires the knowledge of the true noiseless power spectrum of the signal we find that our approach performs almost as well as the oracle solution over a wide range of conditions

fractional wavelet transform for the quantitative spectral resolution of the composite signals of the active compounds in a two component mixture fractional calculus is a powerful tool that has been applied successfully for the analysis of the complex systems one interesting example of a complex mixture is given by the multicomponent pharmaceutical samples having constant matrix content the main aim of this study is to develop a new approach based on the combined use of the fractional wavelet transform fwt and the continuous wavelet transform cwt in order to quantify atorvastatin ato and amlodipine aml in their mixtures without requiring a chemical pretreatment in the first step the absorption spectra of the compounds and their samples were processed by the fwt method in the next step the cwt approach was applied to the fractional wavelet spectra obtained in the above step the aim of the application of fwt is data reduction corresponding to the spectra of compounds and their commercial samples in the following step the cwt was used for the quantitative resolution of the composite signals of the analyzed compounds after method validation the proposed signal processing methods based on the combined use of the fwt and the cwt were successfully applied to the resolution of the composite spectra for the quantitation of atorvastatin ato and amlodipine aml in tablets

denoising functional mr images a comparison of wavelet denoising and gaussian smoothing we present a general wavelet based denoising scheme for functional magnetic resonance imaging fmri data and compare it to gaussian smoothing the traditional denoising method used in fmri analysis one dimensional wavelab thresholding routines were adapted to two dimensional 2 d images and applied to 2 d wavelet coefficients to test the effect of these methods on the signal to noise ratio snr we compared the snr of 2 d fmri images before and after denoising using both gaussian smoothing and wavelet based methods we simulated a fmri series with a time signal in an active spot and tested the methods on noisy copies of it the denoising methods were evaluated in two ways by the average temporal snr inside the original activated spot and by the shape of the spot detected by thresholding the temporal snr maps denoising methods that introduce much smoothness are better suited for low snrs but for images of reasonable quality they are not preferable because they introduce heavy deformations wavelet based denoising methods that introduce less smoothing preserve the sharpness of the images and retain the original shapes of active regions we also performed statistical parametric mapping on the denoised simulated time series as well as on a real fmri data set false discovery rate control was used to correct for multiple comparisons the results show that the methods that produce smooth images introduce more false positives the less smoothing wavelet based methods although generating more false negatives produce a smaller total number of errors than gaussian smoothing or wavelet based methods with a large smoothing effect

on a nonlocal problem for nonlinear pseudoparabolic equations abstract for a nonlinear pseudoparabolic equation with one space dimension we consider its initial boundary value problem on an interval the boundary condition on the left end is of dirichlet type the right end condition is replaced by a nonlocal one because it is given by an integral the function involved could exhibit singularities which distinguishes this nonlocal condition from its dirichlet counterpart based on an elliptic estimate and an iteration method we established the well posedness of solutions in a weighted sobolev space

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

fast parametric elastic image registration we present an algorithm for fast elastic multidimensional intensity based image registration with a parametric model of the deformation it is fully automatic in its default mode of operation in the case of hard real world problems it is capable of accepting expert hints in the form of soft landmark constraints much fewer landmarks are needed and the results are far superior compared to pure landmark registration particular attention has been paid to the factors influencing the speed of this algorithm the b spline deformation model is shown to be computationally more efficient than other alternatives the algorithm has been successfully used for several two dimensional 2 d and three dimensional 3 d registration tasks in the medical domain involving mri spect ct and ultrasound image modalities we also present experiments in a controlled environment permitting an exact evaluation of the registration accuracy test deformations are generated automatically using a random hierarchical fractional wavelet based generator

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

generalized smoothing splines and the optimal discretization of the wiener filter we introduce an extended class of cardinal l sup l splines where l is a pseudo differential operator satisfying some admissibility conditions we show that the l sup l spline signal interpolation problem is well posed and that its solution is the unique minimizer of the spline energy functional spl par ls spl par sub l2 sup 2 subject to the interpolation constraint next we consider the corresponding regularized least squares estimation problem which is more appropriate for dealing with noisy data the criterion to be minimized is the sum of a quadratic data term which forces the solution to be close to the input samples and a smoothness term that privileges solutions with small spline energies here too we find that the optimal solution among all possible functions is a cardinal l sup l spline we show that this smoothing spline estimator has a stable representation in a b spline like basis and that its coefficients can be computed by digital filtering of the input signal we describe an efficient recursive filtering algorithm that is applicable whenever the transfer function of l is rational which corresponds to the case of exponential splines we justify these algorithms statistically by establishing an equivalence between l sup l smoothing splines and the minimum mean square error mmse estimation of a stationary signal corrupted by white gaussian noise in this model based formulation the optimum operator l is the whitening filter of the process and the regularization parameter is proportional to the noise variance thus the proposed formalism yields the optimal discretization of the classical wiener filter together with a fast recursive algorithm it extends the standard wiener solution by providing the optimal interpolation space we also present a bayesian interpretation of the algorithm

diophantine approximation and badly approximable sets let x d be a metric space and d a compact subspace of x which supports a non atomic finite measure m we consider natural classes of badly approximable subsets of loosely speaking these consist of points in which stay clear of some given set of points in x the classical set bad of badly approximable numbers in the theory of diophantine approximation falls within our framework as do the sets bad i j of simultaneously badly approximable numbers under various natural conditions we prove that the badly approximable subsets of have full hausdorff dimension applications of our general framework include those from number theory classical complex p adic and formal power series and dynamical systems iterated function schemes rational maps and kleinian groups

the near shift invariance of the dual tree complex wavelet transform revisited abstract the dual tree complex wavelet transform dt c wt is an enhancement of the conventional discrete wavelet transform dwt due to a higher degree of shift invariance and a greater directional selectivity finding its applications in signal and image processing this paper presents a quantitative proof of the superiority of the dt c wt over the dwt in case of modulated wavelets

on the distribution of points in projective space of bounded height in this paper we consider the uniform distribution of points in compact metric spaces we assume that there exists a probability measure on the borel subsets of the space which is invariant under a suitable group of isometries in this setting we prove the analogue of weyl s criterion and the erdos tur an inequality by using orthogonal polynomials associated with the space and the measure in particular we discuss the special case of pro jective space over completions of number elds in some detail an invariant measure in these projective spaces is introduced and the explicit formulas for the orthogonal polynomials in this case are given finally using the analo gous erdos tur an inequality we prove that the set of all projective points over the number eld with bounded arakelov height is uniformly distributed with respect to the invariant measure as the bound increases

a moment problem for one dimensional nonlinear pseudoparabolic equation we are concerned with a moment problem for a nonlinear pseudoparabolic equation with one space dimension on an interval the boundary conditions are imposed in terms of the zero order moment and the first order moment based on an elliptic estimate and an iteration method we established the well posedness of solutions in the usual sobolev space we are able to get regularity of the solution so that both solution and its derivative with respect to the time variable belong to the same sobolev space with respect to the space variable this feature is different from problems with parabolic equations where the regularity order of solution is higher than that of the time derivative with respect to the space variable previous results reflected only this parabolic nature for the pseudoparabolic equation

wspm wavelet based statistical parametric mapping recently we have introduced an integrated framework that combines wavelet based processing with statistical testing in the spatial domain in this paper we propose two important enhancements of the framework first we revisit the underlying paradigm i e that the effect of the wavelet processing can be considered as an adaptive denoising step to improve the parameter map followed by a statistical detection procedure that takes into account the non linear processing of the data with an appropriate modification of the framework we show that it is possible to reduce the bias of the method with respect to the best linear estimate providing conservative results that are closer to the original data second we propose an extension of our earlier technique that compensates for the lack of shift invariance of the wavelet transform we demonstrate experimentally that both enhancements have a positive effect on performance in particular we present a reproducibility study for multi session data that compares wspm against spm with different amounts of smoothing the full approach is available as a toolbox named wspm for the spm2 software it takes advantage of multiple options and features of spm such as the general linear model

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

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

quantitative fourier analysis of approximation techniques ii wavelets for pt i see ibid vol 47 no 10 p 2783 95 1999 in a previous paper we proposed a general fourier method that provides an accurate prediction of the approximation error irrespective of the scaling properties of the approximating functions here we apply our results when these functions satisfy the usual two scale relation encountered in dyadic multiresolution analysis as a consequence of this additional constraint the quantities introduced in our previous paper can be computed explicitly as a function of the refinement filter this is in particular true for the asymptotic expansion of the approximation error for biorthonormal wavelets as the scale tends to zero one of the contributions of this paper is the computation of sharp asymptotically optimal upper bounds for the least squares approximation error another contribution is the application of these results to b splines and daubechies 1988 1992 scaling functions which yields explicit asymptotic developments and upper bounds thanks to these explicit expressions we can quantify the improvement that can be obtained by using b splines instead of daubechies wavelets in other words we can use a coarser spline sampling and achieve the same reconstruction accuracy as daubechies specifically we show that this sampling gain converges to spl pi as the order tends to infinity

on continuous solutions of a generalized cauchy riemann system with more than one singularity abstract we study the solvability of the riemann hilbert problem for a generalized cauchy riemann system with several singularities and reveal several new phenomenon for the number of continuous solutions we shall show that it depends not only on the index but also on the location and type of the singularities moreover it does not depend continuously on the coefficients of the equation

almost no points on a cantor set are very well approximable we prove that almost no numbers in cantor9s middle thirds set are very well approximable by rationals more generally we discuss diophantine properties of almost every point where almost every is understood relative to any measure on the real line satisfying a decay condition introduced in the work of veech

a note on badly approximabe sets in projective space recently ghosh and haynes j reine angew math 712 39 50 2016 proved a khintchine type result for the problem of diophantine approximation in certain projective spaces in this note we complement their result by observing that a jarnik type result also holds for badly approximable points in real projective space in particular we prove that the natural analogue in projective space of the classical set of badly approximable numbers has full hausdorff dimension when intersected with certain compact subsets of real projective space furthermore we also establish an analogue of khintchine s theorem for convergence relating to intrinsic approximation of points in these compact sets

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