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

new objective functions for waveform inversion inversion is an organized search for parameter values that maximize or minimize an objective function referred to here as a processor this note derives three new seismic processors that require neither prior deconvolution nor knowledge of the source receiver wavelet the most powerful of these is the fourwise processor as it is applicable to data sets from multiple shots and receivers even when each shot has a different unknown signature and each receiver has a different unknown impulse response somewhat less powerful than the fourwise processor is the pairwise processor which is applicable to a data set consisting of two or more traces with the same unknown wavelet but possibly different gains when only one seismogram exists the partition processor can be used the partition processor is also applicable when there is only one shot receiver and each receiver shot has a different signature in fourwise and pairwise inversions the unknown wavelets may be arbitrarily long in time and need not be mi

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

fast scattered data approximation with neumann and other boundary conditions abstract an important problem in applications such as signal and image procesing is the approximation of a function f from a finite set of randomly scattered data f x j a common and powerful approach is to construct a trigonometric least squares approximation based on the set of exponentials e 2 i kx this leads to fast numerical algorithms but suffers from disturbing boundary effects due to the underlying periodicity assumption on the data which is rarely satisfied in practice to overcome this drawback we impose neumann boundary conditions on the data this implies the use of cosine polynomials cos kx as basis functions we show that using cosine polynomials leads to a least squares problem involving certain toeplitz plus hankel matrices and derive estimates on the condition number of these matrices unlike other toeplitz plus hankel matrices these matrices cannot be diagonalized by the discrete cosine transform dct but they still allow a fast matrix vector multiplication via dct which gives rise to fast conjugate gradient type algorithms we show how the results can be generalized to higher dimensions we also consider anti symmetric boundary conditions leading to sine polynomials as proper trigonometric basis finally we demonstrate the performance of the proposed methods by an application to a two dimensional geophysical scattered data problem

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

computationally attractive reconstruction of bandlimited images from irregular samples an efficient method for the reconstruction of bandlimited images and the approximation of arbitrary images from nonuniform sampling values is developed the novel method is based on the observation that the reconstruction problem can be formulated as linear system of equations using two dimensional 2 d trigonometric polynomials where the matrix is of block toeplitz type with toeplitz blocks this system is solved iteratively by the conjugate gradient cg method we show that the use of so called adaptive weights in the establishment of the block toeplitz matrix can be seen as efficient preconditioning the superiority of the new method over conventional approaches is demonstrated by numerical experiments

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

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

a new algorithm for solving toeplitz systems of equations abstract we present some recurrences that are the basis for an algorithm to invert an n n toeplitz system of linear equations with computational complexity o n log 2 n the recurrences used are closely related to the zohar trench and bareiss algorithms but do not have any obvious connection with other asymptotically fast algorithms for the inversion of toeplitz systems

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

efficient sampling set selection for bandlimited graph signals using graph spectral proxies we study the problem of selecting the best sampling set for bandlimited reconstruction of signals on graphs a frequency domain representation for graph signals can be defined using the eigenvectors and eigenvalues of variation operators that take into account the underlying graph connectivity smoothly varying signals defined on the nodes are of particular interest in various applications and tend to be approximately bandlimited in the frequency basis sampling theory for graph signals deals with the problem of choosing the best subset of nodes for reconstructing a bandlimited signal from its samples most approaches to this problem require a computation of the frequency basis i e the eigenvectors of the variation operator followed by a search procedure using the basis elements this can be impractical in terms of storage and time complexity for real datasets involving very large graphs we circumvent this issue in our formulation by introducing quantities called graph spectral proxies defined using the powers of the variation operator in order to approximate the spectral content of graph signals this allows us to formulate a direct sampling set selection approach that does not require the computation and storage of the basis elements we show that our approach also provides stable reconstruction when the samples are noisy or when the original signal is only approximately bandlimited furthermore the proposed approach is valid for any choice of the variation operator thereby covering a wide range of graphs and applications we demonstrate its effectiveness through various numerical experiments

discrete irregular sampling with larger gaps abstract discrete irregular sampling is the problem of recovering a band limited discrete signal from its nonuniformly sampled values we introduce a new method for discrete irregular sampling it features a new sampling geometry where the sampling points are chosen to come in pairs a frequency domain formulation reduces the problem to a system of linear equations for which the matrix is of toeplitz type and is positive definite as soon as there are enough samples an explicit estimate of the condition number of that matrix is given under some condition on the maximum sampling gap this condition allows gaps larger than the nyquist gap we recombine the new method and the conjugate method to deduce an efficient iterative reconstruction scheme for band limited discrete signals numerical results are provided to illustrate the efficiency of the method

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

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

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

source independent full waveform inversion of seismic data a set of seismic trace data is collected in an input data set that is first fourier transformed in its entirety into the frequency domain a normalized wavefield is obtained for each trace of the input data set in the frequency domain normalization is done with respect to the frequency response of a reference trace selected from the set of seismic trace data the normalized wavefield is source independent complex and dimensionless the normalized wavefield is shown to be uniquely defined as the normalized impulse response provided that a certain condition is met for the source this property allows construction of the inversion algorithm disclosed herein without any source or source coupling information the algorithm minimizes the error between data normalized wavefield and the model normalized wavefield the methodology is applicable to any 3 d seismic problem and damping may be easily included in the process

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

the eigenvalues of matrices that occur in certain interpolation problems the eigenvalues of the matrices that occur in certain finite dimensional interpolation problems are directly related to their well posedness and strongly depend on the distribution of the interpolation knots that is on the sampling set we study this dependency as a function of the sampling set itself and give accurate bounds for the eigenvalues of the interpolation matrices the bounds can be evaluated in as few as four arithmetic operations and therefore they greatly simplify the assessment of sampling sets regarding numerical stability the accuracy and usefulness of the bounds are illustrated with examples

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

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

topological pressure via saddle points let a be a compact locally maximal invariant set of a c 2 diffeomorphism f m m on a smooth riemannian manifold m in this paper we study the topological pressure p top with respect to the dynamical system f for a wide class of holder continuous potentials and analyze its relation to dynamical as well as geometrical properties of the system we show that under a mild non uniform hyperbolicity assumption the topological pressure of is entirely determined by the values of on the saddle points of f in a moreover it is enough to consider saddle points with large lyapunov exponents we also introduce a version of the pressure for certain non continuous potentials and establish several variational inequalities for it finally we deduce relations between expansion and escape rates and the dimension of a our results generalize several well known results to certain non uniformly hyperbolic systems

a levinson galerkin algorithm for regularized trigonometric approximation trigonometric polynomials are widely used for the approximation of a smooth function from a set of nonuniformly spaced samples if the samples are perturbed by noise a good choice for the polynomial degree of the trigonometric approximation becomes an essential issue to avoid overfitting and underfitting of the data standard methods for trigonometric least squares approximation assume that the degree for the approximating polynomial is known a priori which is usually not the case in practice we derive a multilevel algorithm that recursively adapts to the least squares solution of suitable degree we analyze under which conditions this multilevel approach yields the optimal solution the proposed algorithm computes the solution in at most mathcal o rm m 2 operations m being the polynomial degree of the approximation and r being the number of samples by solving a family of nested toeplitz systems it is shown how the presented method can be extended to multivariate trigonometric approximation we demonstrate the performance of the algorithm by applying it in echocardiography to the recovery of the boundary of the left ventricle of the heart

equality of pressures for diffeomorphisms preserving hyperbolic measures for a diffeomorphism which preserves a hyperbolic measure the potential varphi u log rm jac df e u is studied various types of pressure of varphi u are introduced it is shown that these pressures satisfy a corresponding variational principle

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

recovery of signals from nonuniform samples using iterative methods an iterative method to recover a bandlimited signal from its ideal nonuniform samples is proposed the convergence of iterations is proved and general regions for convergence are found it is shown that the iterative method is also applicable to other forms of nonuniform sampling i e natural sampling and interpolated sampling such as sample and hold signal simulation results show that this method works effectively and fairly fast and the errors after a few iterations are negligible if a particular sufficient condition is satisfied or the sampling rate is higher than the nyquist rate

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

an iterative method to compensate for the interpolation distortion the author shows that iterative techniques can be used to compensate for sample and hold or in general interpolation distortion it is proven that this iterative method converges to the true signal in the limit the simulation results of the iterative technique are compared to results of the modular method simulation results show that the iterative method is a powerful tool for reducing distortion compared to other well known techniques

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

numerical analysis of the non uniform sampling problem we give an overview of recent developments in the problem of reconstructing a band limited signal from nonuniform sampling from a numerical analysis view point it is shown that the appropriate design of the finite dimensional model plays a key role in the numerical solution of the nonuniform sampling problem in the one approach often proposed in the literature the finite dimensional model leads to an ill posed problem even in very simple situations the other approach that we consider leads to a well posed problem that preserves important structural properties of the original infinite dimensional problem and gives rise to efficient numerical algorithms furthermore a fast multilevel algorithm is presented that can reconstruct signals of unknown bandwidth from noisy nonuniformly spaced samples we also discuss the design of efficient regularization methods for ill conditioned reconstruction problems numerical examples from spectroscopy and exploration geophysics demonstrate the performance of the proposed methods

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

on the eigenvalues of matrices for the reconstruction of missing uniform samples in this correspondence we derive the relationship between the eigenvalues associated with the matrices of the minimum dimension time domain and frequency domain approaches used for reconstructing missing uniform samples the dependency of the eigenvalues of the weighted toeplitz matrix on positive weights are explored simple bounds for the maximum and minimum eigenvalues of the weighted toeplitz matrix are also presented alternative matrices possessing the same nonzero eigenvalues as that of the weighted toeplitz matrix are provided we verify the theory by the examples presented

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

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

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

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

a discrete theory of irregular sampling abstract we derive algorithms for the iterative reconstruction of discrete band limited signals from their irregular samples they converge at a geometric rate and all constants are computable explicitly we also treat the related problem of interpolation of trigonometric polynomials and give estimates for the condition number of certain positive definite toeplitz matrices

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

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

acceleration of the frame algorithm shows how polynomial acceleration techniques which have been developed for the solution of large linear systems can be employed to improve and accelerate the frame algorithm these methods permit a reduction in the number of necessary iterations by an order of magnitude when the frame algorithm is slow the author gives several examples from the theory of irregular sampling from wavelet theory and from gabor theory where these methods are probably mandatory for efficient reconstruction

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

crosshole seismic inversion with normalized full waveform amplitude data we investigate a simple scheme for full waveform amplitude spectrum inversion of crosshole seismic data with an unknown source wavelet the method is based on our 2d 2 5d finite element method of acoustic wave modeling the normalized amplitude data defined as the spectral ratio of the original trace amplitude to the average amplitude for the entire common shot gather are used for full waveform inversion in the frequency domain in essence the normalization of amplitudes removes the source wavelet and is easily carried out in the time domain or frequency domain from crosshole seismic surveying two synthetic models simulating respectively a dipping fracture model and a complicated sedimentary structure are examined with the inversion scheme the numerical results show that clear images of the targets can be obtained by using the single and multiple frequency data for comparison two other amplitude inversion schemes in which the source signatures are known and estimated with an iterative procedure were carried out for the same models the results show that the iteratively source estimated procedure also produces satisfactory images of the velocity structure and yields an approximate amplitude of the source wavelet with the multifrequency data three such schemes yield very competitive results

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

image restoration by the method of convex projections part 1ߞtheory a projection operator onto a closed convex set in hilbert space is one of the few examples of a nonlinear map that can be defined in simple abstract terms moreover it minimizes distance and is nonexpansive and therefore shares two of the more important properties of ordinary linear orthogonal projections onto closed linear manifolds in this paper we exploit the properties of these operators to develop several iterative algorithms for image restoration from partial data which permit any number of nonlinear constraints of a certain type to be subsumed automatically their common conceptual basis is as follows every known property of an original image f is envisaged as restricting it to lie in a well defined closed convex set thus m such properties place f in the intersection e0 ei of the corresponding closed convex sets e1 e2 em given only the projection operators pi onto the individual ei s i 1 m we restore f by recursive means clearly in this approach the realization of the pi s in a hilbert space setting is one of the major synthesis problems section i describes the geometrical significance of the three main theorems in considerable detail and most of the underlying ideas are illustrated with the aid of simple diagrams section ii presents rules for the numerical implementation of 11 specific projection operators which are found to occur frequently in many signal processing applications and the appendix contains proofs of all the major results

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

reconstruction of irregularly sampled discrete time bandlimited signals with unknown sampling locations the purpose of this paper is to develop methods that can reconstruct a bandlimited discrete time signal from an irregular set of samples at unknown locations we define a solution to the problem using first a geometric and then an algebraic point of view we find the locations of the irregular set of samples by treating the problem as a combinatorial optimization problem we employ an exhaustive method and two descent methods the random search and cyclic coordinate methods the numerical simulations were made on three types of irregular sets of locations random sets sets with jitter around a uniform set and periodic nonuniform sets furthermore for the periodic nonuniform set of locations we develop a fast scheme that reduces the computational complexity of the problem by exploiting the periodic nonuniform structure of the sample locations in the dft

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

a stable nonuniform sampling expansion involving derivatives the work of d a linden and n m abramson inf control vol 3 p 26 31 1960 and vol 4 p 95 6 1961 is completed in the sense that a full investigation of the convergence properties of the expansion is made by adapting the work of n levinson am math soc colloq pub vol 26 1940 on nonharmonic fourier series and general results from the theory of b splines the author extends the result to the case of sampling with uniformly spaced samples a stability property is proved

generalized linear inversion of reflection seismic data generalized linear inversion sometimes known as model perturbation nonlinear regression or inverse modeling is applied to synthetic and real seismic data sets with the objective of obtaining an impedance profile as a function of time the impedances solved for are parameterized in a manner that describes the unknown earth using fewer variables than previous seismic generalized linear inversion techniques in this application only single traces of common midpoint cmp processed data will be inverted the method of generalized linear inversion gli presented here is designed to improve on the shortcomings of recursive inversion with respect to relative and absolute scale of the impedance results resolution of impedance boundaries and distortion from residual wavelet effects in obtaining these goals other advantageous aspects of gli were discovered for example it is insensitive to noise in many cases and it will allow an interpreter to fix the impedance of any number of known lithologies in an in

local set based graph signal reconstruction signal processing on graph is attracting more and more attentions for a graph signal in the low frequency subspace the missing data associated with unsampled vertices can be reconstructed through the sampled data by exploiting the smoothness of the graph signal in this paper the concept of local set is introduced and two local set based iterative methods are proposed to reconstruct bandlimited graph signal from sampled data in each iteration one of the proposed methods reweights the sampled residuals for different vertices while the other propagates the sampled residuals in their respective local sets these algorithms are built on frame theory and the concept of local sets based on which several frames and contraction operators are proposed we then prove that the reconstruction methods converge to the original signal under certain conditions and demonstrate the new methods lead to a significantly faster convergence compared with the baseline method furthermore the correspondence between graph signal sampling and time domain irregular sampling is analyzed comprehensively which may be helpful to future works on graph signals computer simulations are conducted the experimental results demonstrate the effectiveness of the reconstruction methods in various sampling geometries imprecise priori knowledge of cutoff frequency and noisy scenarios

irregular sampling toeplitz matrices and the approximation of entire functions of exponential type in many applications one seeks to recover an entire function of exponential type from its non uniformly spaced samples whereas the mathematical theory usually addresses the question of when such a function in l 2 r can be recovered numerical methods operate with a finite dimensional model the numerical reconstruction or approximation of the original function amounts to the solution of a large linear system we show that the solutions of a particularly efficient discrete model in which the data are fit by trigonometric polynomials converge to the solution of the original infinite dimensional reconstruction problem this legitimatizes the numerical computations and explains why the algorithms employed produce reasonable results the main mathematical result is a new type of approximation theorem for entire functions of exponential type from a finite number of values from another point of view our approach provides a new method for proving sampling theorems

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

parabolic radon transform sampling and efficiency a good choice of the sampling in the transform domain is essential for a successful application of the parabolic radon transform the parabolic radon transform is computed for each temporal frequency and is essentially equivalent to the nonuniform fourier transform this leads to new and useful insights in the parabolic radon transform using nonuniform fourier theory we derive a minimum sampling interval for the curvature parameter and a maximum curvature range for which stability is guaranteed for general irregular sampling a significantly smaller sampling interval requires stabilization if diagonal stabilization is used no gain in resolution is obtained in contrast to conventional implementations the curvature sampling interval is proposed to be inversely proportional to the temporal frequency this results in improved quality of the transform and yields significant savings in computation time

a subadditive thermodynamic formalism for mixing repellers the thermodynamical description of fractals that has recently attracted much interest both experimentally and theoretically in the study of dynamical systems is in some ways limited being essentially an additive theory the author presents a subadditive thermodynamic formalism for which he derives a variational principle and shows how it may be used to study the dynamics of non conformal transformations in particular the author discusses an analogue of bowen s formula for the dimension of a mixing repeller

sampling in paley wiener spaces on combinatorial graphs a notion of paley wiener spaces on combinatorial graphs is introduced it is shown that functions from some of these spaces are uniquely determined by their values on some sets of vertices which are called the uniqueness sets such uniqueness sets are described in terms of poincare wirtinger type inequalities a reconstruction algorithm of paley wiener functions from uniqueness sets which uses the idea of frames in hilbert spaces is developed special consideration is given to the n dimensional lattice homogeneous trees and eigenvalue and eigenfunction problems on finite graphs

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

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

some large deviation results for dynamical systems we prove some large deviation estimates for continuous maps of compact metric spaces and apply them to attractors in differentiable dynamics rate of escape problems and to shift spaces

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

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