prolongation structures and backlund transformations for the matrix korteweg de vries and the boomeron equation prolongation structures are determined for the matrix korteweg de vries and the boomeron equation by using the integrability condition for a linear system of first order equations symmetries of these prolongation structures are used to derive backlund transformations and to construct solutions for both equations

best approximation of gaussian neural networks with nodes uniformly spaced this paper is aimed at exposing the reader to certain aspects in the design of the best approximants with gaussian radial basis functions rbfs the class of functions to which this approach applies consists of those compactly supported in frequency the approximative properties of uniqueness and existence are restricted to this class functions which are smooth enough can be expanded in gaussian series converging uniformly to the objective function the uniqueness of these series is demonstrated by the context of the orthonormal basis in a hilbert space furthermore the best approximation to a given band limited function from a truncated gaussian series is analyzed by an energy based argument this analysis not only gives a theoretical proof concerned with the existence of best approximations but addresses the problems of architectural selection specifically guidance for selecting the variance and the oversampling parameters is provided for practitioners

channel capacity under sub nyquist nonuniform sampling this paper investigates the effect of sub nyquist sampling upon the capacity of an analog channel the channel is assumed to be a linear time invariant gaussian channel where perfect channel knowledge is available at both the transmitter and the receiver we consider a general class of right invertible time preserving sampling methods which includes irregular nonuniform sampling and characterize in closed form the channel capacity achievable by this class of sampling methods under a sampling rate and power constraint our results indicate that the optimal sampling structures extract out the set of frequencies that exhibits the highest signal to noise ratio among all spectral sets of measure equal to the sampling rate this can be attained through filterbank sampling with uniform sampling grid employed at each branch with possibly different rates or through a single branch of modulation and filtering followed by uniform sampling these results reveal that for a large class of channels employing irregular nonuniform sampling sets while are typically complicated to realize in practice does not provide capacity gain over uniform sampling sets with appropriate preprocessing our findings demonstrate that aliasing or scrambling of spectral components does not provide capacity gain in this scenario which is in contrast to the benefits obtained from random mixing in spectrum blind compressive sampling schemes

the euler maclaurin summation formula the sampling theorem and approximate integration over the real axis abstract the euler maclaurin summation formula is used to deduce the whittaker shannon sampling theorem for not necessarily band limited functions as well as to study numerical integration over the real axis concerning the latter error estimates are determined in case the function to be integrated is smooth but not necessarily analytic two characteristic examples are given

bounds on the aliasing error in multidimensional shannon sampling we present a pair of sharp lower and upper bounds on the 2 norm of the aliasing error in general multiband sampling representations for not necessarily bandlimited multidimensional functions these bounds improve and generalize previous bounds they also complement a uniform upper bound due to higgins 1985

a generalized sampling theorem for locally compact abelian groups we present a sampling theorem for locally compact abelian groups the sampling sets are finite unions of cosets of a closed subgroup this generalizes the well known case of non equidistant but periodic sampling on the real line for non bandlimited functions an l1 type estimate for the aliasing error is given we discuss the application of the theorem to a class of sampling sets in r s give a general algorithm for computer implementation present a detailed description of the implementation for the s dimensional torus group and point out connections to lattice rules for numerical integration

nonuniform sampling and antialiasing in image representation a unified approach to the representation and processing of a class of images which are not bandlimited but belong to the space of locally bandlimited signals is presented a nonuniform sampling theorem clark et al 1985 for functions belonging to this space is extended and a class of nonstationary stochastic processes is considered the space of locally bandlimited signals is shown to be a reproducing kernel space a generalized projection theorem can therefore be applied yielding either a continuous or a discrete projection filter the former can be used for image conditioning prior to nonuniform sampling while the latter provides a general tool for image representation by nonuniform sampling schemes the problem of finding the local bandwidth of a given signal in order to generate an optimal sampling scheme is addressed in the context of signal representation in the combined position frequency space the stochastic estimation of parameters which characterize the local bandwidth is discussed bounds on the error resulting from the utilization of nonexact position varying signal parameters are derived

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

application of basic hypergeometric series to stable analytic continuation abstract stable sampling formula using basic hypergeometric series for reconstructing analytic functions from exponentially spaced samples is considered criterion for selecting regularizing parameter and error estimates are obtained

exponential weighting algorithms for reconstruction of bandlimited signals the problem of reconstruction of bandlimited signals from discrete noisy observations is considered whittaker shannon interpolation series based estimates involving the weighting factors are proposed it is shown that they converge as the sampling rate increases to infinity the rate of convergence of the double exponential weighting algorithm is established these results are corroborated in computer simulations

the use of modulated splines for the reconstruction of band limited signals a class of kernels of short duration is developed for interpolating band limited signals the class consists of modulated splines and utilizes the rapid attenuation of their fourier transform the design is based on a minimax mean square error criterion

on lagrange interpolations and kramer s sampling theorem associated with self adjoint boundary value problems abstract it is known that kramer s sampling theorem and lagrange type interpolation generalize the celebrated whittaker shannon kotel nikov sampling theorem in two different directions however no direct connection between these two directions seems to be known in this article we show that kramer s sampling theorem gives nothing more than the lagrange type interpolations provided that the kernel function associated with kramer s theorem arises from a self adjoint boundary value problem with sample eigenvalues

on kramer s sampling theorem associated with general strum liouville problems and lagrange interpolation kramer s sampling theorem which is a generalization of shannon s sampling theorem and its relationship with lagrange interpolation is studied kramer s sampling theorem is extended to the case where the interval a b is infinite and it is shown that any function that has a sampling expansion in the scope of kramer s theorem also has a lagrange type interpolation expansion provided that the kernel associated with kramer s theorem arises from a second order sturm liouville boundary value problem whether it is regular or singular this new technique not only reproduces many of the known sampling expansions such as those for the cosine the finite hankel and the continuous legendre transforms but also generates new ones such as those for the continuous jacobi laguerre and hermite transforms

spatio temporal bifurcations in plasma drift waves abstract experimental data from an experiment on drift waves in plasma is presented the experiment provides a space time diagnostic and has a control parameter that permits the study of the transition from a stable plasma to a turbulent plasma the biorthogonal decomposition is used to analyse the data we introduce the notion of complex modulation for two dimensional systems we decompose the real physical system into complex modulated monochromatic travelling waves and give a simple model describing the speed doubling observed in the data as the control parameter increases

symmetry classes of variable coefficient nonlinear schrodinger equations a variable coefficient nonlinear schrodinger vcnls equation involving three arbitrary complex functions of space time in 1 1 dimensions is analysed from the point of view of its symmetries all equations of the type studied having non trivial lie point symmetry groups g are identified the symmetry group is shown to be at most five dimensional and only when the equation is equivalent to the nls equation itself or to a rather special complex ginzburg landau equation lie point transformations are used to obtain solutions of specific vcnls equations that should be of interest in nonlinear optics or other branches of physics

spatiotemporal chaos in nonintegrable three wave interactions abstract the nonliear three wave interaction is ubiquitous having many physical manifestations the conservative equations are integrable by inverse scattering transforms and can possess soliton solutions two nonintegrable three wave models were numerically simulated spatiotemporal chaos where coherent structures evolve and interact chaotically was observed in the long time large system limit correlation functions were measured the behaviour was analysed by using perturbation theory around the inverse scattering transform solutions of the nonperturbed integrable equations saturation amplitudes correlation times and lengths were estimated

commutation methods applied to the mkdv equation an explicit construction of solutions of the modified korteweg de vries equation given a solution of the ordinary korteweg de vries equation is provided our theory is based on commutation methods i e n 1 supersymmetry underlying miura s transformation that links solutions of the two evolution equations

the distribution of sampling rates for signals with equally wide equally spaced spectral bands the distribution of available sampling rates is analysed for a particular type of multiband signal which has bands of equal width centered at harmonics of a fundamental frequency

approximation of wide sense stationary stochastic processes by shannon sampling series in this paper the convergence behavior of the symmetric and the nonsymmetric shannon sampling series is analyzed for bandlimited continuous time wide sense stationary stochastic processes that have absolutely continuous spectral measure it is shown that the nonsymmetric sampling series converges in the mean square sense uniformly on compact subsets of the real axis if and only if the power spectral density of the process fulfills a certain integrability condition moreover if this condition is not fulfilled then the pointwise mean square approximation error of the nonsymmetric sampling series and the supremum of the mean square approximation error over the real axis of the symmetric sampling series both diverge this shows that there is a significant difference between the convergence behavior of the symmetric and the nonsymmetric sampling series

nonuniform interpolation of noisy signals using support vector machines the problem of signal interpolation has been intensively studied in the information theory literature in conditions such as unlimited band nonuniform sampling and presence of noise during the last decade support vector machines svm have been widely used for approximation problems including function and signal interpolation however the signal structure has not always been taken into account in svm interpolation we propose the statement of two novel svm algorithms for signal interpolation specifically the primal and the dual signal model based algorithms shift invariant mercer s kernels are used as building blocks according to the requirement of bandlimited signal the sine kernel which has received little attention in the svm literature is used for bandlimited reconstruction well known properties of general svm algorithms sparseness of the solution robustness and regularization are explored with simulation examples yielding improved results with respect to standard algorithms and revealing good characteristics in nonuniform interpolation of noisy signals

sampling theory for not necessarily band limited functions a historical overview shannon s sampling theorem is one of the most powerful results in signal analysis the aim of this overview is to show that one of its roots is a basic paper of de la vallee poussin of 1908 the historical development of sampling theory from 1908 to the present especially the matter dealing with not necessarily band limited functions which includes the duration limited case actually studied in 1908 is sketched emphasis is put on the study of error estimates as well as on the delicate point wise behavior of sampling sums at discontinuity points of the signal to be reconstructed

an improved γ riccati backlund transformation and its applications for the inhomogeneous nonlinear schrodinger model from plasma physics and nonlinear optics abstract the inhomogeneous nonlinear schrodinger type nls model from certain plasmas and optical fibers is investigated with symbolic computation an improved riccati backlund transformation r bt is presented which can generate successively a hierarchy of solutions through algebraic manipulations based on the improved r bt the darboux transformation is obtained the analytic one two soliton like solutions are presented and the physical characteristics of the influences of the coefficient parameters on the propagation of the soliton pulses are discussed graphically

crystal precipitation with discrete initial data abstract we consider a model of crystal precipitation described mathematically as a conservation law with nonlinear nonlocal flux function the initial data consist of a finite number of dirac measures it is proved that as t the radii of the crystals tend to a critical radius

the weyl correspondence and time frequency analysis describes the weyl correspondence and its properties showing how it gives a window independent definition of time frequency concentration for use in models in signal detection the definition of concentration is justified by showing that it gives reasonable answers in certain intuitive cases the weyl correspondence expresses a linear transformation as a weighted superposition of time frequency shifts of the signal and then authors explain why this is not the same as transforming a signal into the time frequency domain multiplying by a weight in the transform domain and taking the inverse the investigation into time frequency concentration and the weyl correspondence is justified by a new result the authors show that convolving the wigner distribution with a general smoothing function is equivalent to evaluating a weighted sum of spectrograms this is a new interpretation of the process of smoothing the wigner distribution to reduce cross terms it relates smoothing of the wigner distribution to the multiple window technique pioneered by thomson 1982

gauge and backlund transformations for the variable coefficient higher order modified korteweg de vries equation a family of higher order modified korteweg de vries equations with variable coefficients t ho mkdv is introduced a one to one correspondence between a real solution of these equations and a complex solution of the variable coefficient higher order korteweg de vries t ho kdv equations is established through a complex miura transformation an auto backlund transformation for these t ho mkdv equations is derived from that of the t ho kdv equations the associated gauge transformations of the corresponding akns systems are presented they enable one to construct a hierarchy of solutions of the t ho mkdv equations from a known hierarchy without solving the differential equations for the wave functions except the first one a new family of higher order evolution equations with an auto backlund transformation is also derived in connection with the gauge transformation of the t ho mkdv equations

downsampling of bounded bandlimited signals and the bandlimited interpolation analytic properties and computability downsampling and the computation of the bandlimited interpolation of discrete time signals are two important concepts in signal processing in this paper we analyze the downsampling operation regarding its impact on the existence and computability of the bounded bandlimited interpolation we assume that the discrete time signal is obtained by downsampling the samples of a bounded bandlimited signal that vanishes at infinity and we study two problems first we investigate the existence of the bounded bandlimited interpolation for such discrete time signals from a signal theoretic perspective and show that there exist signals for which the bounded bandlimited interpolation does not exist second we analyze the algorithmic generation of the bounded bandlimited interpolation using the concept of turing computability turing computability models what is theoretically implementable on a digital computer interestingly it turns out that even if the bounded bandlimited interpolation exists analytically it is not always computable which implies that there exists no algorithm on a digital computer that can always compute it computability is important in order that the approximation error be controlled if a signal is not computable we cannot ascertain whether the computed signal is sufficiently close to the true signal i e we cannot verify every approximation accuracy

sampling of deterministic signals and systems in this paper we analyze sampling approximations of stable linear time invariant systems for input signals in the paley wiener space pw 1 the goal is to approximate a transform tf by using only the samples of the signal f which do not necessarily have to be equidistant we completely characterize the stable linear time invariant lti systems t and sampling patterns for which the approximation process converges to tf for all signals in pw 1 furthermore we prove that for every complete interpolating sequence there exist a stable lti system and a signal in pw 1 for which the approximation error increases unboundedly as the number of samples that are used for the approximation is increased this shows that a more flexible choice of the sampling points instead of equidistant sampling where such a divergence behavior is known to exist is not sufficient to overcome the divergence thus sampling based signal processing has fundamental limits the calculations further indicate that oversampling cannot resolve the convergence problems

a two channel system approximation for bandlimited functions the approximation of stable linear time invariant lti systems is studied for the paley wiener space mathcal pw pi 1 of bandlimited functions with absolutely integrable fourier transform for pointwise sampling it is known that there exist stable lti systems and functions such that the approximation process diverges regardless of the oversampling factor recently it was shown that the divergence can be overcome by using more general measurement functionals that are based on a complete orthonormal system however this approach requires the approximation process to have an increased bandwidth in this paper a two channel approximation process is presented that is uniformly convergent for all stable lti systems and all functions in mathcal pw pi 1 an advantage of the two channel approach compared with the one channel approach is the reduction of the approximation bandwidth which can be exactly the same as the input function bandwidth

comonotone parametric c1 interpolation of nongridded data we present a local scheme for constructing a c 1 interpolating function comonotone with a set of data having a tensor product topology the constructed interpolant is a parametric surface with piecewise bicubic components which locally maintain the monotonicity of the data along curves parallel to the related coordinate lines error estimates and graphical examples are provided

an impossibility result for linear signal processing under thresholding in this paper we analyze the approximation of the outputs of linear time invariant systems by sampling series that use only the samples of the input signal the samples are disturbed by the threshold operator which sets all samples with an absolute value smaller than some threshold to zero we do the analysis for the space of paley wiener signals with absolutely integrable fourier transform and show for the hilbert transform that the peak approximation error can grow arbitrarily large for some signals in this space when the threshold approaches zero this behavior is counterintuitive because one would expect a better behavior if the threshold was decreased since we consider oversampling and all kernels from a certain meaningful set the results are valid not only for one specific approximation process but for a whole class of approximation processes furthermore we give a game theoretic interpretation of the problem in the setting of a game against nature and show that nature has a universal strategy to win this game

moving average restoration of bandlimited signals from noisy observations the purpose of this paper is to describe the extension of the whittaker shannon sampling theorem to reconstruction of bandlimited functions in the presence of zero mean uncorrelated noise it is shown that the classical whittaker shannon sampling scheme is not consistent in the case of noisy measurements and new reconstruction algorithms based on the moving average smoothing are proposed the weak and strong consistency of the algorithms is established and the rate of convergence is investigated the theory is verified in the computer simulations

numerical solution of certain classes of transport equations in any dimension by shannon sampling a method is developed for computing solutions to some class of linear and nonlinear transport equations hyperbolic partial differential equations with smooth solutions in any dimension which exploits shannon sampling widely used in information theory and signal processing the method can be considered a spectral or a wavelet method strictly related to the existence of characteristics but allows in addition for some precise error estimates in the reconstruction of continuous profiles from discrete data non dissipativity and in some case parallelizability are other features of this approach monotonicity preserving cubic splines are used to handle nonuniform sampling several numerical examples in dimension one or two pertaining to single linear and nonlinear integro differential equations as well as to certain systems are given

gauge transformation and the higher order korteweg de vries equation to the family of higher order korteweg de vries kdv equations a new family of higher order mkdv equations depending on a parameter is constructed they are connected by an 2 dependent miura transformation a backlund transformation is also established furthermore a new gauge transformation is associated to this backlund transformation it enables one to derive auto backlund transformations for the families of higher order 2 mkdv and kdv equations respectively for illustration four generations of explicit solutions of the second order kdv equation are presented

spatiotemporal analysis of complex signals theory and applications we present a space time description of regular and complex phenomena which consists of a decomposition of a spatiotemporal signal into orthogonal temporal modes that we call chronos and orthogonal spatial modes that we call topos this permits the introduction of several characteristics of the signal three characteristic energies and entropies one temporal one spatial and one global and a characteristic dimension although the technique is general we concentrate on its applications to hydrodynamic problems specifically the transition to turbulence we consider two cases of application a coupled map lattice as a dynamical system model for spatiotemporal complexity and the open flow instability on a rotating disk in the latter we show a direct relation between the global entropy and the different instabilities that the flow undergoes as reynolds number increases

an improved method for darboux transformation for the coupled inhomogeneous nonlinear schrodinger system from plasma physics and nonlinear optics abstract the coupled inhomogeneous nonlinear schrodinger type system which can be used to control soliton propagation and interaction in certain plasmas and optical fibers is investigated an improved method for darboux transformation dt is presented in more general forms by constructing an improved riccati type backlund transformation r bt with the n th iterated r bt or the n th iterated dt which is a compact representation for the n soliton like solutions and can generate a series of analytic solutions from a pair of the seed solutions through algebraic manipulations the analytic one two soliton like solutions are provided with the choice of parameters for the soliton solutions the dynamical characteristics of the influences of the inhomogeneous parameters on the propagation of the soliton pulses are discussed graphically

on the zeros of the generalized euler frobenius laurent polynomial and reconstruction of cardinal splines of polynomial growth from local average samples abstract let h t be a nonnegative measurable function supported in 1 2 1 2 and m d t 1 2 1 2 1 2 1 2 1 2 1 2 t d 1 times be the central b spline of degree d we show that the roots of the generalized euler frobenius laurent polynomial defined by e h d z n z h m d n z n are simple negative and all are different from 1 as a consequence of this result we show that for every sequence y n n z r z of samples having polynomial growth and nonnegative measurable function h supported in 1 2 1 2 there is a unique spline f of degree d with polynomial growth satisfying f h n y n n z the presented work answers affirmatively the open problem posed in perez villalon and portal 2012 9

dyadic calculus and sampling theorems for functions with multidimensional domain i general theory the theory of dyadic walsh analysis for functions of several variables is presented to give a rather complete presentation of dyadic analysis a multidimensional version of the dyadic derivative introduced by gibbs and butzer wagner is developed moreover using walsh fourier transform techniques connections between dyadic differentiability and smoothness conditions are established

on the minimax capacity loss under sub nyquist universal sampling this paper investigates the information rate loss in analog channels when the sampler is designed to operate independent of the instantaneous channel occupancy specifically a multiband linear time invariant gaussian channel under universal sub nyquist sampling is considered the entire channel bandwidth is divided into n subbands of equal bandwidth at each time only k constant gain subbands are active where the instantaneous subband occupancy is not known at the receiver and the sampler we study the information loss through an information rate lossmetric that is the gap of achievable rates caused by the lack of instantaneous subband occupancy information we characterize the minimax information rate loss for the sub nyquist regime provided that the number n of subbands and the snr are both large the minimax limits depend almost solely on the band sparsity factor and the undersampling factor modulo some residual terms that vanish as n and snr grow our results highlight the power of randomized sampling methods i e the samplers that consist of random periodic modulation and low pass filters which are able to approach the minimax information rate loss with exponentially high probability

orthogonal sampling formulas a unified approach this paper intends to serve as an educational introduction to sampling theory basically sampling theory deals with the reconstruction of functions signals through their values samples on an appropriate sequence of points by means of sampling expansions involving these values in order to obtain such sampling expansions in a unified way we propose an inductive procedure leading to various orthogonal formulas this procedure which we illustrate with a number of examples closely parallels the theory of orthonormal bases in a hilbert space all intermediate steps will be described in detail so that the presentation is self contained the required mathematical background is a basic knowledge of hilbert space theory finally despite the introductory level some hints are given on more advanced problems in sampling theory which we motivate through the examples

efficient reconstruction algorithms using shifted lattices in this paper we consider sampling theory for irregular sampling sets of minimal density which are structured an efficient method for the reconstruction of band limited discrete signals from sampling sets which are unions of shifted lattices is developed these sets are not necessarily periodic a signal can be reconstructed from its samples provided the sampling set and the spectrum of the signal satisfy certain compatibility conditions while explicit reconstruction formulas for unions of sampling lattices are possible it is more convenient to use a recursive algorithm a numerical example utilizing nonperiodic sampling sets implemented in matlab is given the theory and the related algorithm has been applied to speech and image processing examples

dyadic calculus and sampling theorems for functions with multidimensional domain ii applications to dyadic sampling representations part ii is concerned with applications of the general theory of multidimensional dyadic analysis to dyadic sampling theorems for various types of signals together with rates of approximation and orders of magnitude for the aliasing error respectively the theory developed in part i the contents of which are assumed to be known here is now applied in order to study various types of sampling theorems it may be mentioned that there does not exist any paper at all dealing with multidimensional dyadic sampling theorems this can for example be seen in the survey paper by a j jerri 1977 proc inst electr engrs 65 1565 1596 nevertheless in the classical multidimensional fourier case there are some papers dealing with sampling representations for functions of several variables see e g prosser 1966 petersen and middleton 1962 and miyakawa 1959

asymptotic maximum likelihood estimator performance for chaotic signals in noise the performance of the maximum likelihood estimator for a 1 d chaotic signal in white gaussian noise is derived it is found that the estimator is inconsistent and therefore the usual asymptotic distribution large data record length is invalid however for high signal to noise ratios snrs the maximum likelihood estimator is asymptotically unbiased and attains the cramer rao lower bound

sampling expansions associated with kamke problems the present paper is devoted to the derivation of sampling expansions for entire functions which are represented as integral transforms where a differential operator is acting on the kernels the situation generalizes the results obtained in sampling theory associated with boundary value problems to the case when the differential equation has the form n y lambda p y where n and p are two differential expressions of orders n and p respectively n p and lambda is the eigenvalue parameter both self adjoint and non self adjoint cases will be considered with examples in which the boundary conditions are strongly regular

backlund transformation for the non isospectral and variable coefficient non linear schrodinger equation a non linear schrodinger equation nlse with a non isospectral lax pair and variable coefficients is studied this equation is shown to be an integrability condition for an akns system a gamma riccati system and a lax pair these three systems are proved to be equivalent they are all invariant under some generalisation of the konno wadati transformation related to this nlse there is also a variable coefficients evolution equation for the function gamma derived from the gamma riccati system a backlund transformation for this gamma equation has been constructed with this a new backlund transformation for the nlse is presented so that successive generation of new solutions is reduced to integration only

sampling approximation of continuous functions with multidimensional domain corresp after evaluating shannon s sampling theorem for functions of several variables the aliasing error is estimated in dependence upon smoothness conditions as a main tool the concept of multidimensional convolution integrals is introduced in particular those of de la vallee poussin which are also helpful in constructing generalized sampling series having better aliasing bounds

the application of random reference sequences to the reconstruction of clipped differentiable signals the estimation of analog waveforms s t from their clipped contaminated samples is considered for differentiable signals s t appropriate estimates are established and their performance is investigated error bounds and convergence rates are derived which improve with the degree of differentiability of the signal s t the analytical results are supplemented by a simulation study

quasi random sampling for signal recovery abstract the problem of reconstruction of band limited signals from nonuniformly sampled and and noisy observations is studied it is proposed to sample a signal at quasi random points that form a deterministic sequence with properties resembling a random variable being uniformly distributed such quasi random points can be easily and efficiently generated the reconstruction method based on the modified whittaker shannon cardinal expansion is proposed and its asymptotical properties are examined in particular the sufficient conditions for the convergence of the mean integrated squared error are found the key difference between the proposed reconstruction algorithm and the classical whittaker shannon scheme is in treating the sampling rate and the reconstruction rate differently this distinction is necessary to ensure consistency of the reconstruction algorithm in the presence of noise

recovering band limited signals under noise we consider the problem of recovering a band limited signal f t from noisy data yk f k spl tau spl gt epsilon sub k where spl tau is the sampling rate starting from the truncated whittaker shannon cardinal expansion with or without sampling windows both cases yield inconsistent estimates of f t we propose estimators that are convergent to f t in the pointwise and uniform sense the basic idea is to cut down high frequencies in the data and to use suitable oversampling spl tau spl les spl pi spl omega spl omega being the bandwidth maximum frequency of f t the simplest estimator we propose is given by f spl circ sub n t spl tau spl sigma t k spl tau spl les n spl tau yksin spl omega t k spl tau spl pi t k spl tau t spl les n spl tau generalizations of f spl circ sub n including sampling windows are also examined the main aim is to examine the mean squared error mse properties of such estimators in order to determine the optimal choice of the sampling rate spl tau yielding the fastest possible rate of convergence the best rate for the mse we obtain is o in n n

multidimensional manhattan sampling and reconstruction this paper introduces manhattan sampling in two and higher dimensions and proves sampling theorems for them in 2 d manhattan sampling which takes samples densely along a manhattan grid of lines can be viewed as sampling on the union of two rectangular lattices one dense horizontally and the other vertically with the coarse spacing of each being a multiple of the fine spacing of the other the sampling theorem shows that the images bandlimited to the union of the nyquist regions of the two rectangular lattices can be recovered from their manhattan samples and an efficient procedure for doing so is given such recovery is possible even though there is an overlap among the spectral replicas induced by manhattan sampling in three and higher dimensions there are many possible configurations for manhattan sampling each consisting of the union of special rectangular lattices called bi step lattices this paper identifies them proves a sampling theorem showing that the images bandlimited to the union of the nyquist regions of the bi step rectangular lattices are recoverable from manhattan samples presents an efficient onion peeling procedure for doing so and shows that the union of nyquist regions is as large as any bandlimited region such that all images supported by such can be stably reconstructed from samples taken at the rate of the manhattan sampling it also develops a special representation for the bi step lattices with a number of useful properties while most of this paper deals with continuous space images manhattan sampling of discrete space images is also considered for infinite as well as finite support images

initialization and inner product computations of wavelet transform by interpolatory subdivision scheme the initialization of wavelet transforms and the inner product computations of wavelets with their derivatives are very important in many applications in this correspondence the interpolatory subdivision scheme iss is proposed to solve these problems efficiently we introduce a general procedure to compute the exact values of derivatives of the interpolatory fundamental function and then derive a fast recursive algorithm for the realization of the initialization and inner product evaluations error analysis of the algorithm and its comparison with other approaches are discussed numerical experiments demonstrate the high performance of the algorithm

an improved nyquist shannon irregular sampling theorem from local averages the nyquist shannon sampling theorem is on the reconstruction of a band limited signal from its uniformly sampled samples the higher the signal bandwidth gets the more challenging the uniform sampling may become to deal with this problem signal reconstruction from local averages has been studied in the literature in this paper we obtain an improved nyquist shannon sampling theorem from general local averages in practice the measurement apparatus gives a weighted average over an asymmetrical interval as a special case for local averages from symmetrical interval we show that the sampling rate is much lower than that of a result by grochenig moreover we obtain two exact dual frames from local averages one of which improves a result by sun and zhou at the end of this paper as an example application of local average sampling we consider a reconstruction algorithm the piecewise linear approximations

irregular sampling theorems and series expansions of band limited functions abstract we present a new approach to the problem of irregular sampling of band limited functions that is based on the approximation and factorization of convolution operators a special case of our main result is the following theorem if r n is compact g l 1 rn a band limited function with g t 0 on and xi i i and yj j j are two discrete sets in rn which are dense enough then every band limited function f lwp 1 p our methods work a for a very general class of function spaces not only weighted lwp spaces b in all dimensions even in general locally compact abelian groups c to provide the correct behavior of the coefficients in contrast to the traditional estimates currently used by engineers furthermore the method is constructive and stable with respect to input errors such as round off errors truncation effors jitter errors or aliasing errors

approximation error of the whittaker cardinal series in terms of an averaged modulus of smoothness covering discontinuous signals the whittaker shannon kotel nikov sampling theorem enables one to reconstruct signals f bandlimited to w w from its sampled values f k w k z in terms of swf t k f kw sinc wt k f t t r if f is continuous but not bandlimited one normally considers limw swf t in the supremum norm together with aliasing error estimates expressed in terms of the modulus of continuity of f or its derivatives since in practice signals are however often discontinuous this paper is concerned with the convergence of swf to f in the lp r norm for 1 p the classical modulus of continuity being replaced by the averaged modulus of smoothness r f w 1 m r p the major theorem enables one to sample any bounded signal f belonging to a certain subspace p of lp r the jump discontinuities of which may even form a set of measure zero on r a corollary gives the counterpart of the approximate sampling theorem now in the lp norm the averaged modulus so far only studied for functions defined on a compact interval a b had first to be extended to functions defined on the whole real axis r basic tools are the de la vallee poussin means and a semi discrete hilbert transform

sampling and interpolation for a lacunary spectrum we reduce the problem of constructing a sampling and interpolating set for the space of functions with limited multi band spectra to a problem of invertibility of certain convolution operators on a system of intervals and obtain an example of such a set located in a horizontal strip along the real axis we also study the question of sampling of a signal with two banded spectra via its values at the union of two arithmetic progressions

efficient numerical methods in non uniform sampling theory summary we present a new second generation reconstruction algorithm for irregular sampling i e for the problem of recovering a band limited function from its non uniformly sampled values the efficient new method is a combination of the adaptive weights method which was developed by the two first named authors and the method of conjugate gradients for the solution of positive definite linear systems the choice of adaptive weights can be seen as a simple but very efficient method of preconditioning further substantial acceleration is achieved by utilizing the toeplitztype structure of the system matrix this new algorithm can handle problems of much larger dimension and condition number than have been accessible so far furthermore if some gaps between samples are large then the algorithm can still be used as a very efficient extrapolation method across the gaps

equivalence transformations and differential invariants of a generalized nonlinear schrodinger equation by using lie s invariance infinitesimal criterion we obtain the continuous equivalence transformations of a class of nonlinear schrodinger equations with variable coefficients we construct the differential invariants of order 1 starting from a special equivalence subalgebra we apply these latter ones to find the most general subclass of variable coefficient nonlinear schrodinger equations which can be mapped by means of an equivalence transformation of to the well known cubic schrodinger equation we also provide the explicit form of the transformation

approximation of discontinuous signals by sampling kantorovich series abstract in this paper the behavior of the sampling kantorovich operators has been studied when discontinuous functions signals are considered in the above sampling series moreover the rate of approximation for the family of the above operators is estimated when uniformly continuous and bounded signals are considered finally several examples of duration limited kernels which satisfy the assumptions of the present theory have been provided and also the problem of the linear prediction by sampling values from the past is analyzed

the sampling theorem dirichlet series and hankel transforms abstract some very surprising relations between fundamental theorems and formulas of signal analysis of analytic number theory and of applied analysis are presented it is shown that generalized forms of the classical whittaker kotelnikov shannon sampling theorem as well as of the brown butzer splettstoser approximate sampling expansion for non band limited signal functions can be deduced via the theory of dirichlet series with functional equations from a new summation formula for hankel transforms this counterpart to poisson s summation formula is shown to be essentially equivalent to the famous functional equation of riemann s zeta function to the modular relation of the theta function to the nielsen doetsch summation formula for bessel functions and to the partial fraction expansion of the periodic hilbert kernel

testing parametric assumptions on band or time limited signals under noise this paper considers the problem of testing parametric assumptions on signals f from which only noisy observations y sub k f spl tau k spl epsi sub k are available and where the signal is assumed to be either band limited or time limited to this end the signal is reconstructed by an estimator based on the whittaker shannon ws sampling theorem with oversampling as test statistic the minimal l sub 2 distance between the estimated signal and the parametric model is used to construct appropriate tests the asymptotic distribution of the test statistic is derived both under the hypothesis of the validity of the parametric model and under fixed local alternatives as a byproduct the asymptotic distribution of the integrated square error of the estimator is computed which is of interest by itself e g for the analysis of a cross validated bandwidth selector

pointwise error bounds for orthogonal cardinal spline approximation for orthogonal cardinal spline approximation closed form expressions of the reproducing kernel and the peano kernels in terms of exponential splines are proved concrete and sharp pointwise error bounds are deduced for low degree splines

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

there exists no globally uniformly convergent reconstruction for the paley wiener space cal pw _ pi 1 of bandlimited functions sampled at nyquist rate the bandlimited interpolation of signals and the convergence behavior of the shannon sampling series are discussed in order to show that it is desirable to have a uniformly convergent reconstruction for as large a space of signals as possible in this paper general sampling series are analyzed for the frequently utilized paley wiener space pw pi 1 which is the largest space in the scale of paley wiener spaces the analysis is done not only for the shannon sampling series but for a whole class of axiomatically defined reconstruction processes it is shown that for this very general class which contains all common sampling series including the shannon sampling series a uniformly convergent reconstruction is not possible for the space pw pi 1 moreover a universal signal is identified that causes the divergence behavior for all sampling series finally a lower and an upper bound are derived and used to describe the asymptotic behavior of the peak value of the finite sampling series

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

sub nyquist sampling of short pulses we develop sub nyquist sampling systems for analog signals comprised of several possibly overlapping finite duration pulses with unknown shapes and time positions efficient sampling schemes when either the pulse shape or the locations of the pulses are known have been previously developed to the best of our knowledge stable and low rate sampling strategies for continuous signals that are superpositions of unknown pulses without knowledge of the pulse locations have not been derived the goal in this paper is to fill this gap we propose a multichannel scheme based on gabor frames that exploits the sparsity of signals in time and enables sampling multipulse signals at sub nyquist rates moreover if the signal is additionally essentially multiband then the sampling scheme can be adapted to lower the sampling rate without knowing the band locations we show that with proper preprocessing the necessary gabor coefficients can be recovered from the samples using standard methods of compressed sensing in addition we provide error estimates on the reconstruction and analyze the proposed architecture in the presence of noise

postfiltering versus prefiltering for signal recovery from noisy samples we consider the extension of the whittaker shannon ws reconstruction formula to the case of signals sampled in the presence of noise and which are not necessarily band limited observing that in this situation the classical sampling expansion yields inconsistent reconstruction we introduce a class of signal recovery methods with a smooth correction of the interpolation series two alternative data smoothing methods are examined based either on a global postfiltering or a local data presmoothing we assess the accuracy of the methods by the global l sub 2 error both band limited and non band limited signals are considered a general class of correlated noise processes is taken into account the weak and strong rates of convergence of the algorithms are established and their relative efficiency is discussed the influence of noise memory and its moment structure on the accuracy is thoroughly examined

complete characterization of stable bandlimited systems under quantization and thresholding in this paper we analyze the approximation behavior of sampling series where the sample values taken equidistantly at nyquist rate are disturbed either by the nonlinear threshold operator or the nonlinear quantization operator we perform the analysis for several spaces of bandlimited signals and completely characterize the spaces for which an approximation is possible additionally we study the approximation of outputs of stable linear time invariant systems by sampling series with disturbed samples for signals in pw pi 1 we show that there exist stable systems that become unstable under thresholding and quantization and that the approximation error is unbounded irrespective of how small the quantization step size is chosen further we give a necessary and sufficient condition for the pointwise and the uniform convergence of the series surprisingly this condition is the well known condition for bounded input bounded output bibo stability finally we discuss the special case of finite impulse response fir filters and give an upper bound for the approximation error

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

prediction by samples from the past with error estimates covering discontinuous signals there are several reasons why the classical sampling theorem is rather impractical for real life signal processing first the sinc kernel is not very suitable for fast and efficient computation it decays much too slowly second in practice only a finite number n of sampled values are available so that the representation of a signal f by the finite sum would entail a truncation error which decreases rather slowly for n due to the first drawback third band limitation is a definite restriction due to the nonconformity of band and time limited signals further the samples needed extend from the entire past to the full future relative to some time t t0 this paper presents an approach to overcome these difficulties the sinc function is replaced by certain simple linear combinations of shifted b splines only a finite number of samples from the past need be available this deterministic approach can be used to process arbitrary not necessarily bandlimited nor differentiable signals and even not necessarily continuous signals best possible error estimates in terms of an lp average modulus of smoothness are presented several typical examples exhibiting the various problems involved are worked out in detail

the shannon sampling series and the reconstruction of signals in terms of linear quadratic and cubic splines the sine kernel function of the sampling series is replaced by spline functions having compact support all built up from the b splines m n the resulting generalized sampling series reduces to finite sums so that no truncation error occurs moreover the approximation error generally decreases more rapidly than for the classical series when w tends to infinity 1 w being the distance between the sampling points for the kernel function varphi t 5m 4 t 4m 5 t with support tfrac 5 2 tfrac 5 2 e g it decreases with order o w 4 provided the signal has a fourth order derivative for the classical series the order is o w 4 log w seven characteristic examples are treated in detail

on restoring band limited signals the problem of reconstruction of band limited signals from discrete and noisy data is studied the reconstruction schemes employing cardinal expansions are proposed and their asymptotical properties are examined in particular the conditions for the convergence of the mean integrated squared error are found and the rate of convergence is evaluated the main difference between the proposed reconstruction scheme and the classical one is in treating the sampling rate and the reconstruction rate differently this distinction is necessary to ensure consistency of the reconstruction scheme in the presence of noise

a sampling theorem and wintner s results on fourier coefficients etablissement d un theoreme d echantillonnage pour les coefficients de fourier d une fonction completant les resultats de wintner

sampling theorems associated with fourth and higher order self adjoint eigenvalue problems abstract consider a regular self adjoint eigenvalue problem of n th order and the associated whittaker shannon kramer sampling series the samples being taken at the eigenvalues and the sinc function being replaced by the generalized fourier coefficients of the kernel arising from the eigenvalue problem with respect to the complete set of orthogonal eigenfunctions the major aim of this paper is to show that each such sampling expansion can be written as a lagrange interpolation series provided the kernel satisfies suitable conditions two concrete fourth order eigenvalue problems are examined a regular one as an application of the general result and a singular one the problem whether the two resulting sampling series are really new or can be reduced to already known results is in part open

lattice theoretic analysis of time sequential sampling of spatiotemporal signals i we consider the sampling of bandlimited spatiotemporal signals subject to the time sequential ts constraint that only one spatial position can be sampled at any given time using the powerful techniques of lattice theory we develop a new unifying theory linking ts sampling with generalized multidimensional sampling the results have a geometric nature involving simultaneous packing of the spectral and spatial supports in their respective domains we provide a complete characterization of ts lattice patterns and extending the study to temporally nonuniform patterns analyze their minimum and average temporal sampling rates unlike previous studies of ts sampling our results apply to very general multidimensional spatial and spectral supports we present tight bounds on the temporal parameters of those ts sampling patterns that produce zero aliasing error the use of a ts sampler for bufferless source coding of spatiotemporal signals is considered as an attractive possibility

accurate function sinc interpolation and derivative estimations over finite intervals function expansion in terms of finite sinc bases is considered for finite intervals we considered proper transformation and weighted barycentric methods we adjusted the transformation function to comply with the truncation of the expansion this removes the unbounded behavior of the terminal derivatives and yields accurate results efficient weighted barycentric expansions are presented to improve the accuracies of function and derivative estimations a new expression for the estimation of the second derivative is derived the derived expressions are successfully applied to accurately locate the minimum of the solution and solve a singularly perturbed problem

sampling and ergodic theorems for weakly almost periodic signals the theory of abstract harmonic analysis on commutative groups is used to prove sampling and ergodic theorems concerning a particular class of finite power signals which are known as weakly almost periodic the analysis brings to light some noteworthy differences between finite energy and finite power signal sampling it is shown that the bandwidth of the fourier transform of a weakly almost periodic signal is generally larger than the bandwidth of the power spectrum of the signal consequently the signal power spectrum by itself does not generally provide enough information to determine the value of the time domain nyquist rate that is the minimum sampling rate necessary for exact signal reconstruction in the time domain on the other hand it is also shown that the minimum sampling rate needed to obtain alias free spectral estimates is determined by the bandwidth of the power spectrum and consequently may be lower than the time domain nyquist rate finally the sampling and ergodic theorems established in this paper are used in an analysis of averaged periodogram estimates of the power spectrum of a weakly almost periodic signal it is shown that the value of the time shift between consecutive windows may contribute to the asymptotic bias of the estimates