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

on stabilized finite element methods for the reissner mindlin plate model stabilized finite element formulation for the reissner mindlin plate model is considered physical interpretation for the stabilization procedure for low order elements is established explicit interpolation functions for linear and bilinear stabilized mitc elements are derived some numerical examples including buckling and frequency analyses are presented copyright 2007 john wiley sons ltd

reconstruction of uniformly sampled sequence from nonuniformly sampled transient sequence using symmetric extension in this correspondence reconstruction of a uniformly sampled sequence from a nonuniformly sampled transient sequence using symmetric extension is described first a relationship between the discrete fourier transform dft of a uniformly sampled sequence and the dft of a nonuniformly sampled sequence is obtained from the relationship the formula to reconstruct the dft of a uniformly sampled sequence from the dft of a nonuniformly sampled sequence is derived when the nonuniform sampling ratios are known second a symmetric extension of the nonuniformly sampled sequence is described to avoid discontinuity that adds high frequency content in the dft finally experimental results are presented

performance of linear field reconstruction techniques with noise and uncertain sensor locations we consider a wireless sensor network sampling a bandlimited field described by a limited number of harmonics sensor nodes are irregularly deployed over the area of interest or subject to random displacement in addition sensors measurements are affected by noise our goal is to obtain a high quality reconstruction of the field with the mean square error mse of the estimate as performance metric in particular we analytically derive the performance of several reconstruction estimation techniques based on linear filtering for each technique we obtain the mse as well as its asymptotic expression in the case where the number of field harmonics and the number of sensors grow to infinity while their ratio is kept constant through numerical simulations we show the validity of the asymptotic analysis even for a small number of sensors we provide some novel guidelines for the design of sensor networks when many parameters such as field bandwidth number of sensors reconstruction quality and sensor displacement characteristics to be traded off

banach spaces related to integrable group representations and their atomic decompositions i abstract we present a general theory of banach spaces which are invariant under the action of an integrable group representation and give their atomic decompositions with respect to coherent states i e the atoms arise from a single element under the group action several well known decomposition theories are contained as special examples and are unified under the aspect of group theory

on the structure and interpolation properties of quasi shift invariant spaces abstract the structure of certain types of quasi shift invariant spaces which take the form v x span l 2 x j j z for an infinite discrete set x x j r is investigated additionally the relation is explored between pairs x and y such that interpolation of functions in v x via interpolants in v y solely from the samples of the original function is possible and stable some conditions are given for which the sampling problem is stable and for which recovery of functions from their interpolants from a family of spaces v y is possible

a new finite element formulation for computational fluid dynamics viii the galerkin least squares method for advective diffusive equations abstract galerkin least squares finite element methods are presented for advective diffusive equations galerkin least squares represents a conceptual simplification of supg and is in fact applicable to a wide variety of other problem types a convergence analysis and error estimates are presented

fir filter design by convex optimization using directed iterative rank refinement algorithm the advances in convex optimization techniques have offered new formulations of design with improved control over the performance of fir filters by using lifting techniques the design of a length l fir filter can be formulated as a convex semidefinite program sdp in terms of an l times l matrix that must be rank 1 although this formulation provides means for introducing highly flexible design constraints on the magnitude and phase responses of the filter convex solvers implementing interior point methods almost never provide a rank 1 solution matrix to obtain a rank 1 solution we propose a novel directed iterative rank refinement dirr algorithm where at each iteration a matrix is obtained by solving a convex optimization problem the semidefinite cost function of that convex optimization problem favors a solution matrix whose dominant singular vector is on a direction determined in the previous iterations analytically it is shown that the dirr iterations provide monotonic improvement and the global optimum is a fixed point of the iterations over a set of design examples it is illustrated that the dirr requires only a few iterations to converge to an approximately rank 1 solution matrix the effectiveness of the proposed method and its flexibility are also demonstrated for the cases where in addition to the magnitude constraints the constraints on the phase and group delay of filter are placed on the designed filter

self similarity part i splines and operators the central theme of this pair of papers parts i and ii in this issue is self similarity which is used as a bridge for connecting splines and fractals the first part of the investigation is deterministic and the context is that of l splines these are defined in the following terms s t is a cardinal l spline iff l s t sigma kisinza k delta t k where l is a suitable pseudodifferential operator our starting point for the construction of self similar splines is the identification of the class of differential operators l that are both translation and scale invariant this results into a two parameter family of generalized fractional derivatives parttau gamma where gamma is the order of the derivative and tau is an additional phase factor we specify the corresponding l splines which yield an extended class of fractional splines the operator parttau gamma is used to define a scale invariant energy measure the squared l2 norm of the gammath derivative of the signal which provides a regularization functional for interpolating or fitting the noisy samples of a signal we prove that the corresponding variational or smoothing spline estimator is a cardinal fractional spline of order 2gamma which admits a stable representation in a b spline basis we characterize the equivalent frequency response of the estimator and show that it closely matches that of a classical butterworth filter of order 2gamma we also establish a formal link between the regularization parameter lambda and the cutoff frequency of the smoothing spline filter omega0aplambda 2gamma finally we present an efficient computational solution to the fractional smoothing spline problem it uses the fast fourier transform and takes advantage of the multiresolution properties of the underlying basis functions

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

a practical guide to the recovery of wavelet coefficients from fourier measurements in a series of recent papers b adcock a c hansen and c poon appl comput harmon anal 36 2014 pp 387 415 b adcock m gataric and a c hansen siam j imaging sci 7 2014 pp 1690 1723 adcock et al siam j math anal 47 2015 pp 1196 1233 it was shown that one can optimally recover the wavelet coefficients of an unknown compactly supported function from pointwise evaluations of its fourier transform via the method of generalized sampling while these papers focused on the optimality of generalized sampling in terms of its stability and error bounds the current paper explains how this optimal method can be implemented to yield a computationally efficient algorithm in particular we show that generalized sampling has a computational complexity of mathcal o left m n log n right when recovering the first n boundary corrected wavelet coefficients of an unknown compactly supported function from m n fourier samples therefore due to the linear correspondences bet

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

filterbank reconstruction of bandlimited signals from nonuniform and generalized samples this paper introduces a filterbank interpretation of various sampling strategies which leads to efficient interpolation and reconstruction methods an identity which is referred to as the interpolation identity is developed and is used to obtain particularly efficient discrete time systems for interpolation of generalized samples as well as a class of nonuniform samples to uniform nyquist samples either for further processing in that form or for conversion to continuous time the interpolation identity also leads to new sampling strategies including an extension of papoulis 1977 generalized sampling expansion

sampling curves with finite rate of innovation in this paper we extend the theory of sampling signals with finite rate of innovation fri to a specific class of two dimensional curves which are defined implicitly as the zeros of a mask function here the mask function has a parametric representation as a weighted summation of a finite number of complex exponentials and therefore has finite rate of innovation an associated edge image which is discontinuous on the predefined parametric curve is proved to satisfy a set of linear annihilation equations we show that it is possible to reconstruct the parameters of the curve i e to detect the exact edge positions in the continuous domain based on the annihilation equations robust reconstruction algorithms are also developed to cope with scenarios with model mismatch moreover the annihilation equations that characterize the curve are linear constraints that can be easily exploited in optimization problems for further image processing e g image up sampling we demonstrate one potential application of the annihilation algorithm with examples in edge preserving interpolation experimental results with both synthetic curves as well as edges of natural images clearly show the effectiveness of the annihilation constraint in preserving sharp edges and improving snrs

on the relationship between uniform and recurrent nonuniform discrete time sampling schemes recurrent nonuniform discrete time signal samples can be regarded as a combination of k mutual delayed sequences of uniform discrete time signal samples taken at one kth of the nyquist sampling rate this paper introduces a new alternative discrete time analysis model of the recurrent nonuniform sampling scenario this model can be described by the analysis part of a uniform discrete fourier transform dft modulated filterbank from which the k uniformly distributed and down sampled frequency bands are mixed in a very specific way this description gives a clear relationship between uniform and recurrent nonuniform discrete time sampling schemes a side benefit of this model is an efficient structure with which one can reconstruct uniform discrete time nyquist signal samples from recurrent nonuniform samples with known mutual delays between the nonuniform distributed samples this reconstruction structure can be viewed as a natural extension of the synthesis part of an uniform dft modulated filterbank

reconstruction of multidimensional signals from irregular noisy samples we focus on a multidimensional field with uncorrelated spectrum and study the quality of the reconstructed signal when the field samples are irregularly spaced and affected by independent and identically distributed noise more specifically we apply linear reconstruction techniques and take the mean square error mse of the field estimate as a metric to evaluate the signal reconstruction quality we find that the mse analysis could be carried out by using the closed form expression of the eigenvalue distribution of the matrix representing the sampling system unfortunately such distribution is still unknown thus we first derive a closed form expression of the distribution moments and we find that the eigenvalue distribution tends to the marcenko pastur distribution as the field dimension goes to infinity finally by using our approach we derive a tight approximation to the mse of the reconstructed field

random sampling in shift invariant spaces abstract the set of sampling in a shift invariant space plays an important role in signal processing and has many applications this paper addresses the problem when some randomly chosen samples x x j j j form a set of sampling in a shift invariant space that is when the inequality of the form c p f l p r d p x j x f x j p c p f l p r d p holds uniformly for all functions f in a shift invariant space where c p and c p are positive constants 1 p we prove that with overwhelming probability the above sampling inequality holds for certain compact subsets of the shift invariant space when the sampling size is sufficiently large

polynomial spline signal approximations filter design and asymptotic equivalence with shannon s sampling theorem the least squares polynomial spline approximation of a signal g t in l sub 2 r is obtained by projecting g t on s sup n r the space of polynomial splines of order n it is shown that this process can be linked to the classical problem of cardinal spline interpolation by first convolving g t with a b spline of order n more specifically the coefficients of the b spline interpolation of order 2n 1 of the sampled filtered sequence are identical to the coefficients of the least squares approximation of g t of order n it is shown that this approximation can be obtained from a succession of three basic operations prefiltering sampling and postfiltering which confirms the parallel with the classical sampling reconstruction procedure for bandlimited signals the frequency responses of these filters are determined for three equivalent spline representations using alternative sets of shift invariant basis functions of s sup n r the standard expansion in terms of b spline coefficients a representation in terms of sampled signal values and a representation using orthogonal basis functions

on the connection between exponential bases and certain related sequences in l2 π π a sequence of vectors fn in a separable hilbert space h is a frame if there are positive constants a b such that a f 2 n f fn 2 b f 2 for all f h and fn is a riesz sequence if it is a riesz basis in the closure of the space spanned by the vectors fn the latter of the following two questions has been raised by khrushchev nikolskii and pavlov can every frame of complex exponentials ei nx in l2 be made into a riesz basis by removing from ei nx a suitable collection of the functions ei nx can every riesz sequence ei nx in l2 be made into a riesz basis by adjoining to ei nx a suitable collection of exponentials ei nx ei nx we show that this is indeed so for all frames and riesz sequences ei nx studied so far and we prove that we can always solve both problems in a certain weak sense however our main conclusion is that the answer to both questions is no

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

on shrinking and boundedly complete schauder frames of banach spaces this paper studies schauder frames in banach spaces a concept which is a natural generalization of frames in hilbert spaces and schauder bases in banach spaces the associated minimal and maximal spaces are introduced as are shrinking and boundedly complete schauder frames our main results extend the classical duality theorems on bases to the situation of schauder frames in particular we will generalize james results on shrinking and boundedly complete bases to frames secondly we will extend his characterization of the reflexivity of spaces with unconditional bases to spaces with unconditional frames

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

super resolution from unregistered and totally aliased signals using subspace methods in many applications the sampling frequency is limited by the physical characteristics of the components the pixel pitch the rate of the analog to digital aid converter etc a low pass filter is usually applied before the sampling operation to avoid aliasing however when multiple copies are available it is possible to use the information that is inherently present in the aliasing to reconstruct a higher resolution signal if the different copies have unknown relative offsets this is a nonlinear problem in the offsets and the signal coefficients they are not easily separable in the set of equations describing the super resolution problem thus we perform joint registration and reconstruction from multiple unregistered sets of samples we give a mathematical formulation for the problem when there are m sets of n samples of a signal that is described by l expansion coefficients we prove that the solution of the registration and reconstruction problem is generically unique if mn ges l m 1 we describe two subspace based methods to compute this solution their complexity is analyzed and some heuristic methods are proposed finally some numerical simulation results on one and two dimensional signals are given to show the performance of these methods

some exact solutions of the vibration of non homogeneous membranes the purpose of this note is to show that the linear density variation case studied by masad has a closed form exact solution also we shall present another exact solution for the vibration of a continuously non homogeneous annular membrane

vibration of non homogeneous rectangular membranes with arbitrary interfaces abstract the study deals with the generalized solution of the title problem the free vibration problem of a rectangular membrane with partial domains each of uniform density and arbitrary interface is tackled previous studies by other researchers include straight parallel to the borders and oblique interfaces and bent ones the solution is found by means of a direct variational method with a series composed with a complete set of functions two alternative sets are explored trigonometric and power series such series are uniformly convergent to the exact solution the approach is straightforward and very efficient from the computational viewpoint a determinant factorization method is employed to automatically eliminate eventual spurious frequencies the well known analogy between plates and membranes does not hold in this problem and a demonstration is included diverse illustrations are worked out as the cases of an oblique straight line interface and an open curve line which divides the membrane in two domains each of different density are first presented also a rectangular membrane with an interior closed domain is stated and the numerical example of a circular interior zone is included comparison between the two alternatives and with other authors results show excellent agreement in all cases the computational cost is very low

a theory for multiresolution signal decomposition the wavelet representation multiresolution representations are effective for analyzing the information content of images the properties of the operator which approximates a signal at a given resolution were studied it is shown that the difference of information between the approximation of a signal at the resolutions 2 sup j 1 and 2 sup j where j is an integer can be extracted by decomposing this signal on a wavelet orthonormal basis of l sup 2 r sup n the vector space of measurable square integrable n dimensional functions in l sup 2 r a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function psi x this decomposition defines an orthogonal multiresolution representation called a wavelet representation it is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters wavelet representation lies between the spatial and fourier domains for images the wavelet representation differentiates several spatial orientations the application of this representation to data compression in image coding texture discrimination and fractal analysis is discussed

multichannel interpolation of nonuniform samples with application to image recovery abstract the multichannel trigonometric reconstruction from uniform samples was proposed recently it not only makes use of multichannel information about the signal but is also capable to generate various kinds of interpolation formulas according to the types and amounts of the collected samples the paper presents the theory of multichannel interpolation from nonuniform samples two distinct models of nonuniform sampling patterns are considered namely recurrent and generic nonuniform sampling each model involves two types of samples nonuniform samples of the observed signal and its derivatives numerical examples and quantitative error analysis are provided to demonstrate the effectiveness of the proposed algorithms additionally the proposed algorithms for recovering highly corrupted images are also investigated in comparison with the median filter and the shepard interpolation the proposed algorithms produce superior results with lower errors

exponential bases paley wiener spaces and applications abstract we investigate the connection between translation bases for paley wiener spaces and exponential fourier bases for a domain we apply these results to the characterization of vector valued time frequency translations of a paley wiener window signal

generalized sampling expansion on lattices the papoulis generalized sampling expansion gse can be combined with petersen middleton lattice sampling to obtain a gse for n dimensional bandlimited functions such expansions will be discussed for nonperiodic periodic hybrids the periodic and nonperiodic cases are incorporated as special cases for a restrictive special case a parseval like formula is proved for n dimensional bandlimited functions sampled on periodic nonuniform sample sets

sampling theorems for shift invariant spaces gabor frames and totally positive functions we study nonuniform sampling in shift invariant spaces and the construction of gabor frames with respect to the class of totally positive functions whose fourier transform factors as hat g xi prod j 1 n 1 2 pi i delta j xi 1 e c xi 2 for delta 1 ldots delta n in mathbb r c 0 in which case g is called totally positive of gaussian type in analogy to beurling s sampling theorem for the paley wiener space of entire functions we prove that every separated set with lower beurling density 1 is a sampling set for the shift invariant space generated by such a g in view of the known necessary density conditions this result is optimal and validates the heuristic reasonings in the engineering literature using a subtle connection between sampling in shift invariant spaces and the theory of gabor frames we show that the set of phase space shifts of g with respect to a rectangular lattice alpha mathbb z times beta mathbb z forms a frame if and only if alpha beta 1 this solves an open problem going back to daubechies in 1990 for the class of totally positive functions of gaussian type the proof strategy involves the connection between sampling in shift invariant spaces and gabor frames a new characterization of sampling sets without inequalities in the style of beurling new properties of totally positive functions and the interplay between zero sets of functions in a shift invariant space and functions in the bargmann fock space

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

variational image reconstruction from arbitrarily spaced samples a fast multiresolution spline solution we propose a novel method for image reconstruction from nonuniform samples with no constraints on their locations we adopt a variational approach where the reconstruction is formulated as the minimizer of a cost that is a weighted sum of two terms 1 the sum of squared errors at the specified points and 2 a quadratic functional that penalizes the lack of smoothness we search for a solution that is a uniform spline and show how it can be determined by solving a large sparse system of linear equations we interpret the solution of our approach as an approximation of the analytical solution that involves radial basis functions and demonstrate the computational advantages of our approach using the two scale relation for b splines we derive an algebraic relation that links together the linear systems of equations specifying reconstructions at different levels of resolution we use this relation to develop a fast multigrid algorithm we demonstrate the effectiveness of our approach on some image reconstruction examples

finite sampling in multiple generated u invariant subspaces the relevance in a sampling theory of u invariant subspaces of a hilbert space mathcal h where u denotes a unitary operator on mathcal h is nowadays a recognized fact indeed shift invariant subspaces of l 2 mathbb r become a particular example periodic extensions of finite signals also provide a remarkable example as a consequence the availability of an abstract u sampling theory becomes a useful tool to handle these problems in this paper we derive a sampling theory for finite dimensional multiple generated u invariant subspaces of a hilbert space mathcal h as the involved samples are identified as frame coefficients in a suitable euclidean space the relevant mathematical technique is that of the finite frame theory since finite frames are nothing but spanning sets of vectors the used technique naturally meets matrix analysis

error analysis of frame reconstruction from noisy samples this paper addresses the problem of reconstructing a continuous function defined on rd from a countable collection of samples corrupted by noise the additive noise is assumed to be i i d with mean zero and variance sigma2 we sample the continuous function on the uniform lattice 1 m zd and show for large enough m that the variance of the error between the frame reconstruction fepsiv m from noisy samples of f and the function f satisfy var fepsiv m x f x ap sigma2 md cx where cx is the best constant for every x isin rd we also prove a similar result in the case that our data are weighted average samples of corrupted by additive noise

recovering signals from lowpass data the problem of recovering a signal from its low frequency components occurs often in practical applications due to the lowpass behavior of many physical systems here we study in detail conditions under which a signal can be determined from its low frequency content we focus on signals in shift invariant spaces generated by multiple generators for these signals we derive necessary conditions on the cutoff frequency of the lowpass filter as well as necessary and sufficient conditions on the generators such that signal recovery is possible when the lowpass content is not sufficient to determine the signal we propose appropriate pre processing that can improve the reconstruction ability in particular we show that modulating the signal with one or more mixing functions prior to lowpass filtering can ensure the recovery of the signal in many cases and reduces the necessary bandwidth of the filter

perfect reconstruction formulas and bounds on aliasing error in sub nyquist nonuniform sampling of multiband signals we examine the problem of periodic nonuniform sampling of a multiband signal and its reconstruction from the samples this sampling scheme which has been studied previously has an interesting optimality property that uniform sampling lacks one can sample and reconstruct the class spl bscr spl fscr of multiband signals with spectral support spl fscr at rates arbitrarily close to the landau 1969 minimum rate equal to the lebesgue measure of spl fscr even when spl fscr does not tile r under translation using the conditions for exact reconstruction we derive an explicit reconstruction formula we compute bounds on the peak value and the energy of the aliasing error in the event that the input signal is band limited to the span of spl fscr the smallest interval containing spl fscr which is a bigger class than the valid signals spl bscr spl fscr band limited to spl fscr we also examine the performance of the reconstruction system when the input contains additive sample noise

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

design of prefilters for discrete multiwavelet transforms the pyramid algorithm for computing single wavelet transform coefficients is well known the pyramid algorithm can be implemented by using tree structured multirate filter banks the authors propose a general algorithm to compute multiwavelet transform coefficients by adding proper premultirate filter banks before the vector filter banks that generate multiwavelets the proposed algorithm can be thought of as a discrete vector valued wavelet transform for certain discrete time vector valued signals the proposed algorithm can be also thought of as a discrete multiwavelet transform for discrete time signals the authors then present some numerical experiments to illustrate the performance of the algorithm which indicates that the energy compaction for discrete multiwavelet transforms may be better than the one for conventional discrete wavelet transforms

exact correspondence between eigenvalues of membranes and functionally graded simply supported polygonal plates we use reddy s third order plate theory to study buckling and steady state vibrations of a simply supported functionally gradient isotropic polygonal plate resting on a winkler pasternak elastic foundation and subjected to uniform in plane hydrostatic loads young s modulus and the poisson ratio for the material of the plate are assumed to vary only in the thickness direction effects of rotary inertia are considered the problem of determining the critical buckling load or the vibration frequency of the plate is found to be analogous to that of ascertaining the frequency of a membrane clamped at the edges and whose shape coincides with that of the plate the critical buckling load and the vibration frequency are shown to be positive some available results for plates symmetric about the mid plane can be retrieved from the present analysis

oversampling and reconstruction functions with compact support assume that a sequence of samples of a filtered version of a function in a shift invariant space is given this paper deals with the existence of a sampling formula involving these samples and having reconstruction functions with compact support this is done in the light of the generalized sampling theory by using the oversampling technique a necessary and sufficient condition is given in terms of the smith canonical form of a polynomial matrix finally we prove that the aforesaid oversampled formulas provide nice approximation schemes with respect to the uniform norm

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

wavelets and dilation equations a brief introduction wavelets are new families of basis functions that yield the representation f x sum b jk w 2 j x k their construction begins with the solution phi x to a dilation equation with coefficients c k then w comes from phi and the basis comes by translation and dilation of w it is shown in part 1 how conditions on the c k lead to approximation properties and orthogonality properties of the wavelets part 2 describes the recursive algorithms also based on the c k that decompose and reconstruct f the object of wavelets is to localize as far as possible in both time and frequency with efficient algorithms

quantitative fourier analysis of approximation techniques i interpolators and projectors we present a general fourier based method that provides an accurate prediction of the approximation error as a function of the sampling step t our formalism applies to an extended class of convolution based signal approximation techniques which includes interpolation generalized sampling with prefiltering and the projectors encountered in wavelet theory we claim that we can predict the l sup 2 approximation error by integrating the spectrum of the function to approximate not necessarily bandlimited against a frequency kernel e spl omega that characterizes the approximation operator this prediction is easier yet more precise than was previously available our approach has the remarkable property of providing a global error estimate that is the average of the true approximation error over all possible shifts of the input function our error prediction is exact for stationary processes as well as for bandlimited signals we apply this method to the comparison of standard interpolation and approximation techniques our method has interesting implications for approximation theory in particular we use our results to obtain some new asymptotic expansions of the error as t spl rarr 0 as well as to derive improved upper bounds of the kind found in the strang fix 1971 theory we finally show how we can design quasi interpolators that are near optimal in the least squares sense

u invariant sampling and reconstruction in atomic spaces with multiple generators given a u invariant sampling scheme on a hilbert space h this paper characterizes atomic subspaces a of h such that every signal x a can be reconstructed from its generalized samples acquired with the u invariant sampling scheme if signal recovery is possible a linear filter is derived which reconstructs the signal from the samples moreover the paper gives necessary and sufficient conditions on a sequence of sampling functions such that it forms a pseudoframe or pseudo riesz basis for a given atomic subspace a

regular multivariate sampling and approximation in lp shift invariant spaces abstract in this paper some results about regular multivariate generalized sampling in the l p setting 1 p are proven thus stable multivariate regular sampling formulas are derived for l p shift invariant spaces v p i e the l p closure of the linear span of the shifts of a finite set of generators these sampling formulas include regular samples of the function its derivatives and or some filtered versions of the function itself taken at a lattice of r d approximation schemes using these generalized sampling formulas are also included

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

an irregular sampling theorem for functions bandlimited in a generalized sense it is shown that a well known irregular sampling formula for ordinary bandlimited functions is also valid for functions bandlimited in a generalized sense except for some necessary modifications

the exponentially convergent trapezoidal rule it is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line the mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity it is linked with computational methods all across scientific computing including algorithms related to inverse laplace transforms special functions complex analysis rational approximation integral equations and the computation of functions and eigenvalues of matrices and operators

smooth approximation of potential fields from noisy scattered data geophysical potential field data generally are observed at scattered sampling points and are contaminated by measurement and preprocessing errors analysis and further treatment require a representation of the underlying potential fields on a regular grid in the presence of noise an approximation of the data is more appropriate than an exact interpolation analysis of the smoothness of potential fields indicates an approximation by band limited functions the resulting least squares problem can be formulated as a linear system of equations in which the system matrix is of block toeplitz type this special structure allows a computationally attractive and robust solution via the fast fourier transform and the conjugate gradient algorithm on this basis we present a gridding method that incorporates additional knowledge about the physical properties of potential fields and the statistics of the data experimental results for synthetic and real world gravity data demonstrate good performance for highly sc

wavelets in wandering subspaces mallat s construction via a multiresolution approximation of orthonormal wavelets generated by a single function is extended to wavelets generated by a finite set of functions the connection between multiresolution approximation and the concept of wandering subspaces of unitary operators in hilbert space is exploited in the general setting an example of multiresolution approximation generated by cardinal hermite b splines is constructed

sampling theorem and irregular sampling theorem for multiwavelet subspaces we generalize the multiwavelet sampling theorem by reproducing a kernel that is easy to use multiscaling functions that have orthogonality regularity short compact support symmetry and high approximation order satisfy the conditions in the sampling theorem which is not possible in the case of the scalar wavelet we consider the general cases of the uniform noninteger and irregular sampling a reconstruction from more general sets of points is necessary since the measurements may not be made at uniform points finally we establish a general irregular sampling theorem for multiwavelet subspaces and derive an estimate for the perturbations of uniform noninteger sampling in shift invariant spaces

remarks on the voronoi method in paley wiener space abstract in voronoi method the reconstruction functions n are not well behaved i e not continuous which affect the convergence rate of this algorithm in this paper first we obtain well behaved reconstruction functions in place of n second when the sampling set x n n z is denser we obtain a faster convergent reconstruction iterative algorithm using analog to the mth order b spline reconstruction functions n f x

granger causality analysis with nonuniform sampling and its application to pulse coupled nonlinear dynamics the granger causality gc analysis is an effective approach to infer causal relations for time series however for data obtained by uniform sampling i e with an equal sampling time interval it is known that gc can yield unreliable causal inference due to aliasing if the sampling rate is not sufficiently high to solve this unreliability issue we consider the nonuniform sampling scheme as it can mitigate against aliasing by developing an unbiased estimation of power spectral density of nonuniformly sampled time series we establish a framework of spectrum based nonparametric gc analysis applying this framework to a general class of pulse coupled nonlinear networks and utilizing some particular spectral structure possessed by these nonlinear network data we demonstrate that for such nonlinear networks with nonuniformly sampled data reliable gc inference can be achieved at a low nonuniform mean sampling rate at which the traditional uniform sampling gc may lead to spurious causal inference

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

time sampling and reconstruction in weighted reproducing kernel subspaces abstract this paper mainly studies the time sampling and reconstruction of signals in weighted reproducing kernel subspaces of l p firstly a class of examples for weighted reproducing kernel subspaces are given then using samples produced by a crossing time encoding machine or an integrate and fire time encoding machine satisfying some density constraints iterative approximation algorithms are established for perfect recovery of signals in weighted reproducing kernel subspaces finally an integrate and fire sampling scheme is discussed and an approximate reconstruction procedure is proposed for signals in weighted reproducing kernel subspaces moreover we give detailed analysis for the approximate reconstruction method in the setting of shift invariant subspaces

a new method for studying the vibration of non homogeneous membranes abstract we present a method to solve the helmholtz equation for a non homogeneous membrane with dirichlet boundary conditions at the border of arbitrary two dimensional domains the method uses a collocation approach based on a set of localized functions called little sinc functions which are used to discretize two dimensional regions we have performed extensive numerical tests and we have compared the results obtained with the present method with the ones available from the literature our results show that the present method is very accurate and that its implementation for general problems is straightforward

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

a fourier domain formula for the least squares projection of a function onto a repetitive basis in n dimensional space a theorem concerning the least squares projection of an arbitrary function onto an infinite basis of translated function is given the theorem provides an explicit formula for the fourier transform of the projected function the formula has the advantage of being valid for least squares projections in any n dimensional space the expression for the projected function can be approximately inverted using the discrete fourier transform to find the actual basis coefficients

generalized sampling theorems in multiresolution subspaces it is well known that under very mild conditions on the scaling function multiresolution subspaces are reproducing kernel hilbert spaces rkhss this allows for the development of a sampling theory in this paper we extend the existing sampling theory for wavelet subspaces in several directions we consider periodically nonuniform sampling sampling of a function and its derivatives oversampling multiband sampling and reconstruction from local averages all these problems are treated in a unified way using the perfect reconstruction pr filter bank theory we give conditions for stable reconstructions in each of these cases sampling theorems developed in the past do not allow the scaling function and the synthesizing function to be both compactly supported except in trivial cases this restriction no longer applies for the generalizations we study here due to the existence of fir pr banks in fact with nonuniform sampling oversampling and reconstruction from local averages we can guarantee compactly supported synthesizing functions moreover local averaging schemes have additional nice properties robustness to the input noise and compression capabilities we also show that some of the proposed methods can be used for efficient computation of inner products in multiresolution analysis after this we extend the sampling theory to random processes we require autocorrelation functions to belong to some subspace related to wavelet subspaces it turns out that we cannot recover random processes themselves unless they are bandlimited but only their power spectral density functions we consider both uniform and nonuniform sampling

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

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

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

natural frequencies of rectangular membranes with partial intermediate supports abstract the natural vibrations of rectangular membranes with partial intermediate supports are solved by a direct variational method known as whole element method wem it is based on the use of extended trigonometrical series of uniform convergence fortunately for the case of membranes supported on the perimeter which is the case that interests us the simplest series that we will use is reduced in the unitary domain to a fourier series of sines in both coordinate axes the characteristic that the supports are internal and partial instead of complete in the membrane gives the work one of its conditions of singularity to the authors knowledge the analysis of the aforementioned case is not reported elsewhere in the literature the proposed methodology guarantees that the frequencies found are only those related to the problem eliminating spurious frequencies it is demonstrated how depending on the characteristic algorithm it is possible to identify in an unmistakable way spurious parameters that result when adopting this approach it is proved that in general the frequency parameter of polygonal membranes does not match the square root of the parameter for frequency simple supported plates of the same shape evidently this is due to the addition of intermediate supports as has been known for the last century without the presence of the analogy of the quadratic ratio between corresponding parameters is verified

optimal tight frames and quantum measurement tight frames and rank one quantum measurements are shown to be intimately related in fact the family of normalized tight frames for the space in which a quantum mechanical system lies is precisely the family of rank one generalized quantum measurements on that space using this relationship frame theoretical analogs of various quantum mechanical concepts and results are developed the analog of a least squares quantum measurement is a tight frame that is closest in a least squares sense to a given set of vectors the least squares tight frame is found for both the case in which the scaling of the frame is specified constrained least squares frame clsf and the case in which the scaling is chosen to minimize the least squares error unconstrained least squares frame ulsf the well known canonical frame is shown to be proportional to the ulsf and to coincide with the clsf with a certain scaling

stability of localized operators let lp 1 p be the space of all p summable sequences and ca be the convolution operator associated with a summable sequence a it is known that the lp stability of the convolution operator ca for different 1 p are equivalent to each other i e if ca has lp stability for some 1 p then ca has lq stability for all 1 q in the study of spline approximation wavelet analysis time frequency analysis and sampling there are many localized operators of non convolution type whose stability is one of the basic assumptions in this paper we consider the stability of those localized operators including infinite matrices in the sjostrand class synthesis operators with generating functions enveloped by shifts of a function in the wiener amalgam space and integral operators with kernels having certain regularity and decay at infinity we show that the lp stability or lp stability of those three classes of localized operators are equivalent to each other and we also prove that the left inverse of those localized operators are well localized

nonuniform sequential sampling for signal analysis new electronic technologies for signal analysis raise the possibility of sampling very rapidly with a time varying density and determining empirically both the sampling rate and the window width as the signal evolves in time these opportunities also point to the possibility of sequentially sampling in a time varying way in more traditional problems motivated by these ideas we establish a sampling formula valid in cases where both sampling rate and window width may be varied the formula states that in terms of the ways in which these quantities should alter with time optimal performance is achieved when the window width is inversely proportional to squared sampling rate and sampling rate is directly proportional to squared bias

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

on the stability of multivariate trigonometric systems abstract kadec s 1 4 theorem says that if n n z is a sequence of real numbers for which n n l 1 4 then ei n n z forms a riesz basis for l2 s favier r zalik c chui and x shi extended this result to the multivariate case but their results lead to very small stability bounds in this paper we give an optimal stability bound for the multivariate trigonometric systems moreover for the case of fourier frames in l2 d we also give the stability bounds

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

shannon sampling and function reconstruction from point values i first met rene at the well known 1956 meeting on topology in mexico city he then came to the university of chicago where i was starting my job as instructor for the fall of 1956 he suzanne clara and i became good friends and saw much of each other for many decades especially at ihes in paris thom s encouragement and support were important for me especially in my first years after my ph d i studied his work in cobordism singularities of maps and transversality gaining many insights i also enjoyed listening to his provocations for example his disparaging remarks on complex analysis 19th century mathematics and bourbaki there was also a stormy side in our relationship neither of us could hide the pain that our public conflicts over catastrophe theory caused rene thom was a great mathematician leaving his impact on a wide part of mathematics i will always treasure my memories of him steve smale

the recovery of irregularly sampled band limited functions via tempered splines abstract we show that band limited functions can be recovered from their values on certain irregularly distributed discrete sampling sets as the limits of the piecewise polynomial spline interpolants when the order of the splines goes to infinity this is an extension of the classical case when the sampling set is a lattice which was considered by collatz quade schoenberg and others

continuous and discrete wavelet transforms this paper is an expository survey of results on integral representations and discrete sum expansions of functions in l 2 bf r in terms of coherent states two types of coherent states are considered weyl heisenberg coherent states which arise from translations and modulations of a single function and affine coherent states called wavelets which arise as translations and dilations of a single function in each case it is shown how to represent any function in l 2 bf r as a sum or integral of these states most of the paper is a survey of literature most notably the work of i daubechies a grossmann and j morlet a few results of the authors are included

a generalization of the membrane plate analogy to non homogeneous polygonal domains consisting of homogeneous subdomains the well known membrane plate analogy that relates the natural frequencies when dealing with polygonal homogeneous domains is herein extended to non homogeneous systems comprised of homogeneous subdomains the analogy is generalized and demonstrated and it is shown that certain restrictions among the frequency parameters of the membranes and plates arise several examples of membranes and plates with interfaces separating areas with different material properties are numerically solved with different approaches the subdomains are separated by straight curved and closed line interfaces it is shown that the analogy is verified provided that the restrictions are satisfied the analogy is first demonstrated and presented as a practical methodology to find the natural frequencies of membranes knowing the corresponding ones of the plates or vice versa second the plate and membrane vibration problems governed by the bi laplacian and laplacian differential operators respectively can be solved without distinction though under certain conditions i e solve one of them and deduce the other using the analogy various numerical examples validate the analogy

a minimum squared error framework for generalized sampling we treat the problem of reconstructing a signal from its nonideal samples where the sampling and reconstruction spaces as well as the class of input signals can be arbitrary subspaces of a hilbert space our formulation is general and includes as special cases reconstruction from finitely many samples as well as uniform sampling of continuous time signals which are not necessarily bandlimited to obtain a good approximation of the signal in the reconstruction space from its samples we suggest two design strategies that attempt to minimize the squared norm error between the signal and its reconstruction the approaches we propose differ in their assumptions on the input signal if the signal is known to lie in an appropriately chosen subspace then we propose a method that achieves the minimal squared error on the other hand when the signal is not restricted we show that the minimal norm reconstruction cannot generally be obtained instead we suggest minimizing the worst case squared error between the reconstructed signal and the best possible but usually unattainable approximation of the signal within the reconstruction space we demonstrate both theoretically and through simulations that the suggested methods can outperform the consistent reconstruction approach previously proposed for this problem

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

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

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