distinct variants of the kdv equation with compact and noncompact structures in this paper we examine three distinct variants of the generalized kdv equation both the compact and the noncompact structures are treated for these variants we also emphasize the different physical structures of the two classes the focusing branch and the defocusing branch

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

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

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

substitutions and interval exchange transformations of rotation class we propose a method for obtaining the symbolic orbits of interval exchange transformations of rotation class over n intervals by composing a finite set of basic substitutions i e by doing simple parallel rewriting based on surface theory this method is shown to be closely related to rauzy induction sturmian objects are known to correspond to interval exchange transformations over 2 intervals in this respect our n intervals case is shown to be also related to continued fractions and the obtained words have also linear complexity

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

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 computational approach to soliton solutions of the kadomtsev petviashvili equation in this paper we present a computational approach to develop soliton solutions of the nonlinear kadomtsev petviashvili equation our approach rests mainly on the adomian decomposition method to include few components of the decomposition series the proposed framework is presented in a general way so that it can be used in nonlinear evolution equations of the same type numerical examples are tested to illustrate the proposed scheme

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

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

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

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

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

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

real numbers having ultimately periodic representations in abstract numeration systems using a genealogically ordered infinite regular language we know how to represent an interval of r numbers having an ultimately periodic representation play a special role in classical numeration systems the aim of this paper is to characterize the numbers having an ultimately periodic representation in generalized systems built on a regular language the syntactical properties of these words are also investigated finally we show the equivalence of the classical expansions with our generalized representations in some special case related to a pisot number

general compactons solutions for the focusing branch of the nonlinear dispersive k n n equations in higher dimensional spaces in this paper we will study the focusing branch of the genuinely nonlinear dispersive k n n equation that exhibits compactons solitons with finite wavelengths the equation will be studied in one two and three dimensional spaces general formulas for compactons solutions are developed for all positive integers n n 1 two sets of entirely new formulas one set produces compactons for even integers n whereas the second set produces compactons and anticompactons for odd integers n n 1 are established

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

itineraries of rigid rotations and diffeomorphisms of the circle we examine the itinerary of email protected s 1 r z under the rotation by email protected email protected q the motivating question is if we are given only the itinerary of 0 relative to email protected s 1 a finite union of closed intervals can we recover a and i we prove that the itineraries do determine a and i up to certain equivalences then we present elementary methods for finding a and i moreover if g s 1 s 1 is a c 2 orientation preserving diffeomorphism with an irrational rotation number then we can use the orbit itinerary to recover the rotation number up to certain equivalences

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

generalized boussinesq type of equations with compactons solitons and periodic solutions generalized boussinesq type of equations with positive and negative exponents are examined the analysis depends mainly on the sine cosine ansatz it is formally shown that these nonlinear models give rise to compactons solitary patterns solitons and periodic solutions depending on the exponents and the coefficients of the derivatives of u x t the presented scheme reveals quite a number of remarkable features that will be helpful for identical problems

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

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

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

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

languages d0l systems sets of curves and surface automorphisms we introduce extend and apply some relationships between formal language theory and surface theory first we show how boundaries of languages topologized with the cantor metric can be mapped to sets of curves on surfaces namely laminations second we present how and when endomorphisms of free monoids i e substitutions can be mapped to automorphisms of surfaces so that d0l systems correspond to iterations of these automorphisms third we apply these ideas to construct sets of non periodic irreducible automorphisms of surfaces following 14 35 43 so that the involved proofs do not use any differential or algebraic tools but accordingly substitution and d0l system theory of 11 12 31 32 37 mainly the decidability for a d0l language 12 to be strongly repetitive or not this construction also yields effective symbolic descriptions of the stable sets of curves associated with these automorphisms

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