sampling and reconstruction of signals in a reproducing kernel subspace of lp rd in this paper we consider sampling and reconstruction of signals in a reproducing kernel subspace of lp rd 1 p associated with an idempotent integral operator whose kernel has certain off diagonal decay and regularity the space of p integrable non uniform splines and the shift invariant spaces generated by finitely many localized functions are our model examples of such reproducing kernel subspaces of lp rd we show that a signal in such reproducing kernel subspaces can be reconstructed in a stable way from its samples taken on a relatively separated set with sufficiently small gap we also study the exponential convergence consistency and the asymptotic pointwise error estimate of the iterative approximation projection algorithm and the iterative frame algorithm for reconstructing a signal in those reproducing kernel spaces from its samples with sufficiently small gap

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

analysis and design of minimax optimal interpolators we consider a class of interpolation algorithms including the least squares optimal yen 1956 interpolator and we derive a closed form expression for the interpolation error for interpolators of this type the error depends on the eigenvalue distribution of a matrix that is specified for each set of sampling points the error expression can be used to prove that the yen interpolator is optimal the implementation of the yen algorithm suffers from numerical ill conditioning forcing the use of a regularized approximate solution we suggest a new approximate solution consisting of a sinc kernel interpolator with specially chosen weighting coefficients the newly designed sinc kernel interpolator is compared with the usual sinc interpolator using jacobian area weighting through numerical simulations we show that the sinc interpolator with jacobian weighting works well only when the sampling is nearly uniform the newly designed sinc kernel interpolator is shown to perform better than the sinc interpolator with jacobian weighting

direct inversion of the nonequispaced fast fourier transform abstract various applications such as mri solution of pdes etc need to perform an inverse nonequispaced fast fourier transform nfft i e compute m fourier coefficients from given n nonequispaced data in the present paper we consider direct methods for the inversion of the nfft we introduce algorithms for the setting m n as well as for the underdetermined and overdetermined cases for the setting m n a direct method of complexity o n log n is presented which utilizes lagrange interpolation and the fast summation for the remaining cases we use the matrix representation of the nfft to deduce our algorithms thereby we are able to compute an inverse nfft up to a certain accuracy by means of a modified adjoint nfft in o m log m n arithmetic operations finally we show that these approaches can also be explained by means of frame approximation

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

a more powerful method for triangularizing input output matrices and the similarity of production structures studies in the production structure of any national economy as revealed in its inputoutput table have been closely related to the question of the interindustrial dependence or hierarchical structures of productive sectors leading from primary to final production in particular the notion of hierarchy provides a useful tool when one wants to draw inferences on the structural change or international difference of industrial structures the standard technique for studying this notion is to triangularize the input output table by interchanging sectors in order to maximize the entries below the main diagonal in a perfect triangularized table the entries above the main diagonal should be zero for example such a strong one way interdependence relation as cotton textiles clothing can easily establish the hierarchy due to the existence of circular relations like coal steel mining equipment coal it is not possible to triangularize the input output table perfectly this paper extends and revises the previous triangulation method which is based on a permutation theorem deriving from the interchange of adjoining two industrial groups into that among three industrial groups theorem 2 the new algorithm based on theorem 2 is demonstrated by actually computing the suboptimal orderings for the four input output tables for such more developed countries mdc s as the united states italy norway japan and the two tables for such less developed countries ldc s as india and korea the empirical results suggest that sectors can be arranged in a similar hierarchical order among mdc s and ldc s transport equipment machinery apparel leather and products grain mill products and processed foods which link directly to the final demand are recorded as higher order sectors trade transport and services are lower order ones and energy sectors such as electric power coal products coal mining petroleum petroleum products and natural gas are the lowest consequently this paper provides some more evidence in support of the similarity of hierarchical structures of production among these countries

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

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

sinc interpolation of nonuniform samples it is well known that a bandlimited signal can be uniquely recovered from nonuniformly spaced samples under certain conditions on the nonuniform grid and provided that the average sampling rate meets or exceeds the nyquist rate however reconstruction of the continuous time signal from nonuniform samples is typically more difficult to implement than from uniform samples motivated by the fact that sinc interpolation results in perfect reconstruction for uniform sampling we develop a class of approximate reconstruction methods from nonuniform samples based on the use of time invariant lowpass filtering i e sinc interpolation the methods discussed consist of four cases incorporated in a single framework the case of sub nyquist sampling is also discussed and nonuniform sampling is shown as a possible approach to mitigating the impact of aliasing

generalized average sampling in shift invariant spaces average sampling is motivated by realistic needs e g physical limitation of acquisition devices or characteristics of sampling procedure as an extension of the average sampling we study generalized average sampling in shift invariant spaces with frame generators in which averages are taken from suitable channeled version of signals illustrative examples are given

vector sampling expansions in shift invariant subspaces multi input multi output mimo sampling scheme which is motivated by applications in multi channel deconvolution and multi source separation has been investigated in many aspects common for most of results on mimo systems is that the input signals are supposed to be band limited in this paper we study vector sampling expansions on general finitely generated shift invariant subspaces necessary and sufficient conditions for a vector sampling theorem to hold are given we also give several examples to illustrate the main result

estimation of time limited channel spectra from nonuniform samples this paper deals with the estimation of a time invariant channel spectrum from its own nonuniform samples assuming there is a bound on the channel s delay spread except for this last assumption this is the basic estimation problem in systems providing channel spectral samples however as shown in the paper the delay spread bound leads us to view the spectrum as a band limited signal rather than the fourier transform of a tapped delay line tdl using this alternative model a linear estimator is presented that approximately minimizes the expected root mean square rms error for a deterministic channel its main advantage over the tdl is that it takes into account the spectrum s smoothness time width thus providing a performance improvement the proposed estimator is compared numerically with the maximum likelihood ml estimator based on a tdl model in pilot assisted channel estimation pace for ofdm

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

a master identity for homotopy gerstenhaber algebras we produce a master identity for a certain type of homotopy gerstenhaber algebras in particular suitable for the prototype namely the hochschild complex of an associative algebra this algebraic master identity was inspired by the work of getzler jones and kimura voronov zuckerman in the context of topological conformal field theories to this end we introduce the notion of a partitioned multilinear map and explain the mechanics of composing such maps in addition many new examples of pre lie algebras and homotopy gerstenhaber algebras are given

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

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

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

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

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

efficient algorithms for burst error recovery using fft and other transform kernels we show that the problem of signal reconstruction from missing samples can be handled by using reconstruction algorithms similar to the reed solomon rs decoding techniques usually the rs algorithm is used for error detection and correction of samples in finite fields for the case of missing samples of a speech signal we work with samples in the field of real or complex numbers and we can use fft or some new transforms in the reconstruction algorithm dsp implementation and simulation results show that the proposed methods are better than the ones previously published in terms of the quality of recovered speech signal for a given complexity the burst error recovery method using the fft kernel is sensitive to quantization and additive noise like the other techniques however other proposed transform kernels are very robust in correcting bursts of errors with the presence of quantization and additive noise

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

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

on discrete wirtinger northcott problems abstract some discrete wirtinger northcott inequalities are established in this paper let dx and x be the periodic difference and non periodic difference of a vector x r n respectively for 1 p we give the maximum and the minimum of d x x d x and x in the l p ball b p 0 n in particular we prove that for p the result min b 0 n d x 4 n n is even 4 n n 2 1 n is odd is equivalent to the well known discrete first northcott inequality established by fan taussky and todd

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

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

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

an average sampling theorem for bandlimited stochastic processes in this correspondence we give an average sampling theorem for bandlimited stochastic processes which shows that many average sampling theorems have their counterparts for stochastic signals

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

sampling high dimensional bandlimited fields on low dimensional manifolds consider the task of sampling and reconstructing a bandlimited spatial field in r2 using moving sensors that take measurements along their path it is inexpensive to increase the sampling rate along the paths of the sensors but more expensive to increase the total distance traveled by the sensors per unit area which we call the path density in this paper we introduce the problem of designing sensor trajectories that are minimal in path density subject to the condition that the measurements of the field on these trajectories admit perfect reconstruction of bandlimited fields we study various possible designs of sampling trajectories generalizing some ideas from the classical theory of sampling on lattices we obtain necessary and sufficient conditions on the trajectories for perfect reconstruction we show that a single set of equispaced parallel lines has the lowest path density from certain restricted classes of trajectories that admit perfect reconstruction we then generalize some of our results to higher dimensions we first obtain results on designing sampling trajectories in higher dimensional fields further interpreting trajectories as 1 d manifolds we extend some of our ideas to higher dimensional sampling manifolds we formulate the problem of designing dimensional sampling manifolds for d dimensional spatial fields that are minimal in manifold density a natural generalization of the path density we show that our results on sampling trajectories for fields in r2 can be generalized to analogous results on d dimensional sampling manifolds for d dimensional spatial fields

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