on simple oversampled a d conversion in shift invariant spaces it has been found that the quantization error e for a conventional oversampled analog to digital a d conversion behaves like spl par e spl par sup 2 o spl tau sup 2 with respect to the sampling rate spl tau recently conventional a d conversion has been extended to a d conversion based on shift invariant spaces as consequences of such extension it offers rich choices to build a nonideal a d conversion system of high accuracy and low computational complexity as well as reduces the noise sensitivity and computational complexity in digital to analog d a conversion therefore it is necessary to establish the estimate of quantization error for the extended a d conversion based on shift invariant spaces in this paper we introduce a constructive method to establish an estimate of the quantization error as e sup 2 o spl tau sup 2 for oversampled a d conversion in shift invariant spaces meanwhile we demonstrate that the bit rate required to encode the converted digital signal in such a d conversion scheme only increases as the logarithm of the sampling ratio therefore the quantization error is an exponentially decaying function of the bit rate in order to establish such an estimate we need the nonuniform sampling theorem for shift invariant spaces which as the necessary preparation is studied prior to introducing the constructive method

the zak transform and sampling theorems for wavelet subspaces the zak transform is used for generalizing a sampling theorem of g waiter see ieee trans informat theory vol 38 p 881 884 1992 for wavelet subspaces cardinal series based on signal samples f a n n in z with a possibly unequal to 0 waiter s case are considered the condition number of the sampling operator and worst case aliasing errors are expressed in terms of zak transforms of scaling function and wavelet this shows that the stability of the resulting interpolation formula depends critically on a

stability and asymptotic stability in the energy space of the sum of n solitons for subcritical gkdv equations we prove in this paper the stability and asymptotic stability in h1 of a decoupled sum of n solitons for the subcritical generalized kdv equations the proof of the stability result is based on energy arguments and monotonicity of the local l2 norm note that the result is new even for p 2 the kdv equation the asymptotic stability result then follows directly from a rigidity theorem in 16

stability and instability of solitary waves of korteweg de vries type considered herein are the stability and instability properties of solitary wave solutions of a general class of equations that arise as mathematical models for the unidirectional propagation of weakly nonlinear dispersive long waves special cases for which our analysis is decisive include equations of the korteweg de vries and benjamin ono type necessary and sufficient conditions are formulated in terms of the linearized dispersion relation and the nonlinearity for the solitary waves to be stable

nonuniform sampling and reconstruction in shift invariant spaces this article discusses modern techniques for nonuniform sampling and reconstruction of functions in shift invariant spaces it is a survey as well as a research paper and provides a unified framework for uniform and nonuniform sampling and reconstruction in shift invariant subspaces by bringing together wavelet theory frame theory reproducing kernel hilbert spaces approximation theory amalgam spaces and sampling inspired by applications taken from communication astronomy and medicine the following aspects will be emphasized a the sampling problem is well defined within the setting of shift invariant spaces b the general theory works in arbitrary dimension and for a broad class of generators c the reconstruction of a function from any sufficiently dense nonuniform sampling set is obtained by efficient iterative algorithms these algorithms converge geometrically and are robust in the presence of noise d to model the natural decay conditions of real signals and images the sampling theory is developed in weighted l p spaces

shift invariant and sampling spaces associated with the fractional fourier transform domain shift invariant spaces play an important role in sampling theory multiresolution analysis and many other areas of signal and image processing a special class of the shift invariant spaces is the class of sampling spaces in which functions are determined by their values on a discrete set of points one of the vital tools used in the study of sampling spaces is the zak transform the zak transform is also related to the poisson summation formula and a common thread between all these notions is the fourier transform in this paper we extend some of these notions to the fractional fourier transform frft domain first we introduce two definitions of the discrete fractional fourier transform and two semi discrete fractional convolutions associated with them we employ these definitions to derive necessary and sufficient conditions pertaining to frft domain under which integer shifts of a function form an orthogonal basis or a riesz basis for a shift invariant space we also introduce the fractional zak transform and derive two different versions of the poisson summation formula for the frft these extensions are used to obtain new results concerning sampling spaces to derive the reproducing kernel for the spaces of fractional band limited signals and to obtain a new simple proof of the sampling theorem for signals in that space finally we present an application of our shift invariant signal model which is linked with the problem of fractional delay filtering

global well posedness in the energy space for a modified kp ii equation via the miura transform we prove global well posedness of the initial value problem for a modified kadomtsev petviashvili ii mkp ii equation in the energy space the proof proceeds in three main steps and involves several different techniques in the first step we make use of several linear estimates to solve a fourth order parabolic regularization of the mkp ii equation by a fixed point argument for regular initial data one estimate is similar to the sharp kato smoothing effect proved for the kdv equation by kenig ponce and vega 1991 then compactness arguments the energy method performed through the miura transform give the existence of a local solution of the mkp ii equation for regular data finally we approximate any data in the energy space by a sequence of smooth initial data using bourgain s result concerning the global well posedness of the kp ii equation in l 2 and the miura transformation we obtain convergence of the sequence of smooth solutions to a solution of mkp ii in the energy space

quantization noise in single loop sigma delta modulation with sinusoidal inputs an exact nonlinear difference equation is derived and solved for a simple sigma delta modulator consisting of a discrete time integrator and a binary quantizer inside a single feedback loop and an arbitrary input signal it is shown that the system can be represented as an affine operation discrete time integration of a biased input followed by a memoryless nonlinearity an extension of the transform method for the analysis of nonlinear systems is applied to obtain formulas for first and second order time average moments of the binary quantization noise including the sample mean energy and autocorrelation the results are applied to the special case of a sinusoidal input signal to evaluate these time averages and the power spectrum in the limit of large oversampling ratios the marginal moments behave as if the quantization noise had a uniform distribution the spectrum is neither white nor continuous however even in the limit of large oversampling ratios

approximating a bandlimited function using very coarsely quantized data improved error estimates in sigma delta modulation sigma delta quantization is a method of representing bandlimited signals by 0 1 sequences that are computed from regularly spaced samples of these signals as the sampling density convolving these one bit sequences with appropriately chosen kernels produces increasingly close approximations of the original signals this method is widely used for analogto digital and digital to analog conversion because it is less expensive and simpler to implement than the more familiar critical sampling followed by fine resolution quantization we present examples of how tools from number theory and harmonic analysis are employed in sharpening the error estimates in sigma delta quantization

comparison of interpolating methods for image resampling when resampling an image to a new set of coordinates for example when rotating an image there is often a noticeable loss in image quality to preserve image quality the interpolating function used for the resampling should be an ideal low pass filter to determine which limited extent convolving functions would provide the best interpolation five functions were compared a nearest neighbor b linear c cubic b spline d high resolution cubic spline with edge enhancement a 1 and e high resolution cubic spline a 0 5 the functions which extend over four picture elements c d e were shown to have a better frequency response than those which extend over one a or two b pixels the nearest neighbor function shifted the image up to one half a pixel linear and cubic b spline interpolation tended to smooth the image the best response was obtained with the high resolution cubic spline functions the location of the resampled points with respect to the initial coordinate system has a dramatic effect on the response of the sampled interpolating function the data are exactly reproduced when the points are aligned and the response has the most smoothing when the resampled points are equidistant from the original coordinate points thus at the expense of some increase in computing time image quality can be improved by resampled using the high resolution cubic spline function as compared to the nearest neighbor linear or cubic b spline functions

lp approximation by certain projection operators abstract we estimate the lp r d approximation rate 1 p provided dilates of an orthogonal projection operator of l2 r d onto a space generated by shifts of a function which has a polynomial decay rate at infinity and has stable shifts to this end we employ a quasi interpolation scheme and invoke wiener s lemma in particular this paper provides further substantiation to one of the fundamental properties of refinable functions the connection between smoothness and approximation order

stability of the line soliton of the kp ii equation under periodic transverse perturbations we prove the nonlinear stability of the kdv solitary waves considered as solutions of the kp ii equation with respect to periodic transverse perturbations our proof uses a miura transform which sends the solutions of an mkp ii equation to solutions of the kp ii equation

a sampling theorem for wavelet subspaces the classical shannon sampling theorem is extended to the subspaces used in the multiresolution analysis in wavelet theory under weak hypotheses these subspaces are first shown to have a riesz basis formed from the reproducing kernels these in turn are used to construct the sampling sequences examples are given

asymptotic stability of solitary waves we show that the family of solitary waves 1 solitons of the korteweg de vries equation partial t u u partial x u partial x 3 u 0 is asymptotically stable our methods also apply for the solitary waves of a class of generalized korteweg de vries equations partial t u partial x f u partial x 3 u 0 in particular we study the case wheref u up 1 p 1 p 1 2 3 and 3 0 withf c4 the same asymptotic stability result for kdv is also proved for the casep 2 the modified korteweg de vries equation we also prove asymptotic stability for the family of solitary waves for all but a finite number of values ofp between 3 and 4 the solitary waves are known to undergo a transition from stability to instability as the parameterp increases beyond the critical valuep 4 the solution is decomposed into a modulating solitary wave with time varying speedc t and phase t bound state part and an infinite dimensional perturbation radiating part the perturbation is shown to decay exponentially in time in a local sense relative to a frame moving with the solitary wave asp 4 the local decay or radiation rate decreases due to the presence of aresonance pole associated with the linearized evolution equation for solitary wave perturbations

transient behavior of collapsing ring solutions in the critical nonlinear schrodinger equation abstract the critical nonlinear schrodinger equation nls possesses nearly self similar ring profile solutions we address the question of whether this profile is maintained all the way until the point of singularity a perturbative analysis of the rescaled pde and the resulting self similar profile uncover slow dynamics that eventually drive the ring structure to the classical peak shaped collapse instead a numerical scheme capable of resolving self similar behavior to high resolutions confirms our analysis we also consider ring type blowup arising either from azimuthally polarized solutions of a coupled nls system or as vortex solutions of the usual nls in this case the ring profile is maintained up to the time of singularity

constructing solutions of the mkdv equation abstract using commutation methods i e n 1 supersymmetry underlying miura s transformation an explicit construction of solutions of the modified korteweg de vries equation given a solution of the ordinary korteweg de vries equation is provided

asymptotic stability of solitons of the gkdv equations with general nonlinearity we consider the generalized korteweg de vries equation gkdv partial t u partial x partial x 2 u f u 0 quad t x in 0 t times mathbb r with general c3 nonlinearity f under an explicit condition on f and c 0 there exists a solution in the energy space h1 of the type u t x qc x x0 ct called soliton in this paper under general assumptions on f and qc we prove that the family of solitons around qc is asymptotically stable in some local sense in h1 i e if u t is close to qc for all t 0 then u t locally converges in the energy space to some qc as t note in particular that we do not assume the stability of qc this result is based on a rigidity property of the gkdv equation around qc in the energy space whose proof relies on the introduction of a dual problem these results extend the main results in martel siam j math anal 38 759 781 2006 martel and merle j math pures appl 79 339 425 2000 arch ration mech anal 157 219 254 2001 nonlinearity 1 55 80 devoted to the pure power case

irregular sampling for spline wavelet subspaces spline wavelets spl psi sub m t are important in time frequency localization due to i spl psi sub m can be arbitrarily close to the optimal case as m is sufficiently large ii spl psi sub m has compact support and simple analytic expression which lead to effective computation although the spline wavelet subspaces are so simple walter s well known sampling theorem does not hold if the order of spline m is even moreover when irregular sampling is considered in these spaces it is hard to determine the sampling density which is a serious problem in applications in this correspondence a general sampling theorem is obtained for m spl ges 3 in the sense of iterative construction and the sampling density spl delta sub m is estimated

on simple oversampled a d conversion in l sup 2 r the accuracy of oversampled analog to digital a d conversion the dependence of accuracy on the sampling interval spl tau and on the bit rate r are characteristics fundamental to a d conversion but not completely understood these characteristics are studied for oversampled a d conversion of band limited signals in l sup 2 r we show that the digital sequence obtained in the process of oversampled a d conversion describes the corresponding analog signal with an error which tends to zero as spl tau sup 2 in energy provided that the quantization threshold crossings of the signal constitute a sequence of stable sampling in the respective space of band limited functions further we show that the sequence of quantized samples can be represented in a manner which requires only a logarithmic increase in the bit rate with the sampling frequency r o log spl tau and hence that the error of oversampled a d conversion actually exhibits an exponential decay in the bit rate as the sampling interval tends to zero

single bit oversampled a d conversion with exponential accuracy in the bit rate a scheme for simple oversampled analog to digital a d conversion using single bit quantization is presented the scheme is based on recording positions of zero crossings of the input signal added to a deterministic dither function this information can be represented in a manner such that the bit rate increases only logarithmically with the oversampling factor r the input band limited signal can be reconstructed from this information locally with o 1 r pointwise error resulting in an exponentially decaying distortion rate characteristic in the course of studying the accuracy of the proposed a d conversion scheme some new results are established about reconstruction of band limited signals from irregular samples using linear combination of functions with fast decay schemes for local interpolation of band limited signals from quantized irregular samples are also proposed

the structure of finitely generated shift invariant spaces in l2 rd a simple characterization is given of finitely generated subspaces of l2 rd which are invariant under translation by any multi integer and is used to give conditions under which such a space has a particularly nice generating set namely a basis and more than that a basis with desirable properties such as stability orthogonality or linear independence the last property makes sense only for local spaces i e shift invariant spaces generated by finitely many compactly supported functions and special attention is paid to such spaces as an application we prove that the approximation order provided by a given local space is already provided by the shift invariant space generated by just one function with this function constructible as a finite linear combination of the finite generating set for the whole space hence compactly supported this settles a question of some 20 years standing

deterministic analysis of oversampled a d conversion and decoding improvement based on consistent estimates discusses the deterministic analysis of oversampled a d conversion adc the properties derivable from such an analysis and the consequences on reconstruction using nonlinear decoding given a band limited input x producing a quantized version c the authors consider the set of all input signals that are band limited and produce c they call any element of this set a consistent estimate of x regardless of the type of encoder simple predictive or noise shaping they show that this set is convex and as a consequence any nonconsistent estimate can be improved they also show that the classical linear decoding estimates are not necessarily consistent numerical tests performed on simple adc single loop and multiloop spl sigma spl delta modulation show that consistent estimates yield a mean square error mse that decreases asymptotically with the oversampling ratio faster than the linear decoding mse by approximately 3 db octave this implies an asymptotic mse of the order of spl oscr r sup spl minus 2n 2 instead of spl oscr r sup spl minus 2n 1 in linear decoding where r is the oversampling ratio and n the order of the modulator methods of improvement of nonconsistent estimates based on the deterministic knowledge of the quantized signal are proposed for simple adc predictive adc single loop and multiloop spl sigma spl delta modulation

mean squared error sampling and reconstruction in the presence of noise one of the main goals of sampling theory is to represent a continuous time function by a discrete set of samples here we treat the class of sampling problems in which the underlying function can be specified by a finite set of samples our problem is to reconstruct the signal from nonideal noisy samples which are modeled as the inner products of the signal with a set of sampling vectors contaminated by noise to mitigate the effect of the noise and the mismatch between the sampling and reconstruction vectors the samples are linearly processed prior to reconstruction considering a statistical reconstruction framework we characterize the strategies that are mean squared error mse admissible meaning that they are not dominated in terms of mse by any other linear reconstruction we also present explicit designs of admissible reconstructions that dominate a given inadmissible method adapting several classical estimation approaches to our particular sampling problem we suggest concrete admissible reconstruction methods and compare their performance the results are then specialized to the case in which the samples are processed by a digital correction filter

a sampling theorem for shift invariant subspace a sampling theorem for regular sampling in shift invariant subspaces is established the sufficient necessary condition for which it holds is found then the theorem is modified to the shift sampling in shift invariant subspaces by using the zak transform finally some examples are presented to show the generality of the theorem

spectral analysis of quantization noise in a single loop sigma delta modulator with dc input an exact discrete time analysis of the moments and spectra of the quantization noise of a discrete time single loop sigma delta modulator with a dc input is presented an exact difference equation for the discrete time nonlinear system is used to evaluate the first and second order moments and power spectrum of the binary quantizer noise and the binary quantizer output for a single loop sigma delta encoder with a dc input it is shown that the sample mean and power of the binary quantization noise are consistent with the common uniform distribution assumption but that the autocorrelation and power spectrum are not consistent with the white noise assumption the results are used to evaluate the overall sample average mean squared quantization error as a function of the decimation filter used

estimate of aliasing error for non smooth signals prefiltered by quasi projections into shift invariant spaces an ideal analog to digital a d converter prefilters a signal by an ideal lowpass filter recent research on a d conversion based on shift invariant spaces reveals that prefiltering signals by quasiprojections into shift invariant spaces provides more flexible choices in designing an a d conversion system of high accuracy this paper focuses on the accuracy of such prefiltering in which the aliasing error e sub f sup spl lambda is found to behave like e sub f sup spl lambda sub 2 o spl lambda sup spl alpha with respect to the dilation spl lambda of the underlying shift invariant space provided that the input signal f is lipschitz spl alpha continuous a formula to calculate the coefficient of the decay rate is also figured out in this paper

reconstruction algorithms in irregular sampling a constructive solution of the irregular sampling problem for band limited functions is given we show how a band limited function can be com pletely reconstructed from any random sampling set whose density is higher than the nyquist rate and give precise estimates for the speed of convergence of this iteration method variations of this algorithm allow for irregular sampling with derivatives reconstruction of band limited functions from local averages and irregular sampling of multivariate band limited functions in the irregular sampling problem one is asked whether and how a band limited function can be completely reconstructed from its irregularly sam pled values f xi this has many applications in signal and image processing seismology meteorology medical imaging etc finding constructive solutions of this problem has received considerable attention among mathematicians and engineers the mathematical literature provides several uniqueness results 1 2 17 18 19 it is now part of the folklore that for stable sampling the sampling rate must be at least the nyquist rate 18 these results as deep as they are have had little impact for the applied sciences because they were not constructive if the sampling set is just a perturbation of the regular oversampling then a reconstruction method has been obtained in a seminal paper by duffin and schaeffer 6 see also 29 if for some l 0 a 0 and o 0 the sampling points xk k e z satisfy a xk ok a k i then the norm equivalence a ir f x 2dx with w n o this norm equivalence implies that it is possible to reconstruct through an iterative procedure the so called frame method most of the later work on constructive methods consists of variations of this method 3 21 22 26 the above conditions on the sampling set exclude random irregular sampling sets e g sets with regions of higher sampling density a partial but undesirable remedy to handle highly irregular sampling sets would be to force the above conditions by throwing away information on part of the points and accept a very slow convergence of the iteration

lyapunov stability of ground states of nonlinear dispersive evolution equations on presente une nouvelle demonstration de la stabilite orbitale des solitons d etat fondamental de l equation de schrodinger non lineaire pour une grande classe de non linearites on demontre la stabilite de l onde solitaire pour l equation de korteweg de vries generalisee

approximating a bandlimited function using very coarsely quantized data a family of stable sigma delta modulators of arbitrary order digital signal processing has revolutionized the storage and transmission of audio and video signals as well as still images in consumer electronics and in more scientific settings such as medical imaging the main advantage of digital signal processing is its robustness although all the operations have to be implemented with of necessity not quite ideal hardware the a priori knowledge that all correct outcomes must lie in a very restricted set of well separated numbers makes it possible to recover them by rounding off appropriately bursty errors can compromise this scenario as is the case in many communication channels as well as in memory storage devices making the perfect data unrecoverable by rounding off in this case knowledge of the type of expected contamination can be used to protect the data prior to transmission or storage by encoding them with error correcting codes this is done entirely in the digital domain these advantages have contributed to the present widespread use of digital signal processing many signals however are not digital but analog in nature audio signals for instance correspond to functions f t modeling rapid pressure oscillations which depend on the continuous time t i e t ranges over or an interval in and not over a discrete set and the range of f typically also fills an interval in f or this reason the first step in any digital processing of such signals must consist in a conversion of the analog signal to the digital world usually abbreviated as a d conversion for different types of signals different a d schemes are used in this paper we restrict our attention to a particular class of a d conversion schemes adapted to audio signals note that at the end of the chain after the signal has been processed stored retrieved transmitted all in digital form it needs to be reconverted to an analog signal that can be understood by a human hearing system we thus need a d a conversion there

minimum rate sampling and reconstruction of signals with arbitrary frequency support we examine the question of reconstruction of signals from periodic nonuniform samples this involves discarding samples from a uniformly sampled signal in some periodic fashion we give a characterization of the signals that can be reconstructed at exactly the minimum rate once a nonuniform sampling pattern has been fixed we give an implicit characterization of the reconstruction system and a design method by which the ideal reconstruction filters may be approximated we demonstrate that for certain spectral supports the minimum rate can be approached or achieved using reconstruction schemes of much lower complexity than those arrived at by using spectral slicing as in earlier work previous work on multiband signals have typically been those for which restrictive assumptions on the sizes and positions of the bands have been made or where the minimum rate was approached asymptotically we show that the class of multiband signals which can be reconstructed exactly is shown to be far larger than previously considered when approaching the minimum rate this freedom allows us in certain cases to have a far less complex reconstruction system

the gardner equation and the l 2 stability of the n soliton solution of the korteweg de vries equation multi soliton solutions of the korteweg de vries equation kdv are shown to be globally l2 stable and asymptotically stable in the sense of martel and merle the proof is surprisingly simple and combines the gardner transform which links the gardner and kdv equations together with the martel merle tsai and martel merle recent results on stability and asymptotic stability in the energy space applied this time to the gardner equation as a by product the results of maddocks sachs and merle vega are improved in several directions

on sampling theorem wavelets and wavelet transforms the classical shannon sampling theorem has resulted in many applications and generalizations from a multiresolution point of view it provides the sine scaling function in this case for a band limited signal its wavelet series transform wst coefficients below a certain resolution level can be exactly obtained from the samples with a sampling rate higher than the nyquist rate the authors study the properties of cardinal orthogonal scaling functions cosf which provide the standard sampling theorem in multiresolution spaces with scaling functions as interpolants they show that cosf with compact support have and only have one possibility which is the haar pulse they present a family of cosf with exponential decay which are generalizations of the haar function with these cosf an application is the computation of wst coefficients of a signal by the mallat 1989 algorithm they present some numerical comparisons for different scaling functions to illustrate the advantage of cosf for signals which are not in multiresolution spaces they estimate the aliasing error in the sampling theorem by using uniform samples

on sampling in shift invariant spaces a necessary and sufficient condition for sampling in the general framework of shift invariant spaces is derived then this result is applied respectively to the regular sampling and the perturbation of regular sampling in shift invariant spaces a simple necessary and sufficient condition for regular sampling in shift invariant spaces is attained furthermore an improved estimate for the perturbation is derived for the perturbation of regular sampling in shift invariant spaces the derived estimate is easy to calculate and shown to be optimal in some shift invariant spaces the algorithm to calculate the reconstruction frame is also presented

behavior of the quantization operator for bandlimited nonoversampled signals the process of quantization generates a loss of information and thus the original signal cannot be reconstructed exactly from the quantized samples in general however it is desirable to keep the error as small as possible in this paper the quantization error is quantified in terms of several distortion measures all these measures employ the difference between the original signal and the reconstructed signal which is obtained by bandlimited interpolation of the quantized samples we assume that the signals are bandlimited and that the samples are taken at nyquist rate it is shown that for signals in the paley wiener space pw 1 the supremum of the reconstructed signal and hence the quantization error cannot be bounded in the sense that there exists a bounded subset of pw 1 on which both quantities can increase unboundedly this unexpected behavior is due to the nonlinearity of the quantization operator and the slow decay of the sinc function the nonlinearity is essential for this behavior because every linear operator that fulfills a certain property of the quantization operator would otherwise have to be bounded furthermore it is proven that for a fixed signal the possible quantization error increases as the quantization step size tends to zero the treatment of the quantization error in this paper is completely deterministic

a general sampling theory for nonideal acquisition devices the authors first describe the general class of approximation spaces generated by translation of a function spl psi x and provide a full characterization of their basis functions they then present a general sampling theorem for computing the approximation of signals in these subspaces based on a simple consistency principle the theory puts no restrictions on the system input which can be an arbitrary finite energy signal bandlimitedness is not required in contrast to previous approaches this formulation allows for an independent specification of the sampling analysis and approximation synthesis spaces in particular when both spaces are identical the theorem provides a simple procedure for obtaining the least squares approximation of a signal they discuss the properties of this new sampling procedure and present some examples of applications involving bandlimited and polynomial spline signal representations they also define a spectral coherence function that measures the similarity between the sampling and approximation spaces and derive a relative performance bound for the comparison with the least squares solution

reduction of the mse in r times oversampled a d conversion o 1 r to o 1 r sup 2 in oversampled analog to digital conversion the usual reconstruction method using lowpass filtering leads to a mean squared error mse inversely proportional to the oversampling ratio r the authors prove under certain assumptions and with periodic analog input signals that optimal reconstruction achieves an mse with an oversampling ratio dependence order of at least o 1 r sup 2 that is an mse slope of spl minus 6 db per octave of oversampling is obtained rather than the conventional spl minus 3 db octave slope of classical schemes

weighted sampling and reconstruction in weighted reproducing kernel spaces in this paper we discuss sampling and reconstruction of signals in the weighted reproducing kernel space associated with an idempotent integral operator we show that any signal in that space can be stably reconstructed from its weighted samples taken on a relatively separated set with sufficiently small gap we also develop an iterative reconstruction algorithm for the reconstruction of a signal from its weighted samples taken on a relatively separated set with sufficiently small gap