comparison results for parallel multisplitting methods with applications to aor methods abstract in this paper we present some comparison theorems between two general parallel multisplittings these comparison theorems can be applied to many classical splittings such as jacobi gauss seidel and aor splittings some significant improvements and generalizations of the existing comparison results are obtained

convergence theory for the general gaor type iterative method and the msor iterative method applied to h matrices abstract we discuss the general gaor type ggaor iterative method of which special cases are the aor the gaor and the msor iterative methods to solve the linear system ax b where a i l u with l and u being general matrices the ggaor iteration matrix is expressed by l r i rl 1 i r l u where r and are diagonal matrices and the msor iteration matrix l i 1 i u some basic results on the upper bounds of the spectral radii l r and l are given when a is strictly or irreducibly diagonally dominant by rows based upon these results we obtain new results on the convergence regions of the ggaor and the msor iterative methods when a is an h matrix or a has property a and recover and improve previous ones

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

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

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

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

convergence for a general form of the gaor method and its application to the msor method abstract two types of gaor iterative methods were respectively given by a hadjidimos and hu jia gan to solve the linear system ax b where a d e f with d to be nonsingular and diagonal in the former method the parameters involved are two numbers and and in the latter method two diagonal matrices r and the msor iterative method given by d m young is a special case of the latter gaor method for uniformity of the results the paper considers the convergence for a general form of the aor type iteration matrix l r i rl 1 i r l u where l d 1e and u d 1f and gives some important results for the convergence domain of lr when a d e f is an h matrix applying the results to the gaor aor gsor and msor methods we recover certain recent main results and greatly extend certain other previous results

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

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

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

comparisons of spectral radii and the theorem of stein rosenberg abstract the theorem of stein rosenberg is generalized to the case of two m splittings a m 1 n 1 m 2 n 2 where the m i are m matrices and n 1 n 2 0 n 1 n 2 n 2 0

forced oscillations of a massive point on a compact surface with a boundary abstract we present sufficient conditions for the existence of a periodic solution for a class of systems describing the periodically forced motion of a massive point on a compact surface with a boundary

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

on forced oscillations in groups of interacting nonlinear systems abstract consider a periodically forced nonlinear system which can be presented as a collection of smaller subsystems with pairwise interactions between them each subsystem is assumed to be a massive point moving with friction on a compact surface possibly with a boundary in an external periodic field we present sufficient conditions for the existence of a periodic solution for the whole system the result is illustrated by a series of examples including a chain of strongly coupled pendulums in a periodic field

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

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

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

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

periodic solutions for systems of forced coupled pendulum like equations abstract the existence of periodic solutions for systems of forced pendulum like equations was studied in the papers by j a marlin internat j nonlinear mech 3 1968 439 447 and j mawhin internat j nonlinear mech 5 1970 335 339 in both works some symmetry hypotheses on the forcing terms were considered this paper discusses the existence and multiplicity of periodic solutions of systems under consideration without any requirement on the symmetry of the forcing terms note that as a model example it is possible to consider the motion of n coupled pendulums see the already mentioned paper by j a marlin or the oscillations of an n coupled point josephson junction with external time dependent disturbances studied in the autonomous case by m levi f c hoppensteadt and w l miranker quart appl math 36 1978 167 198

generalized stein rosenberg theorems for the regular splittings and convergence of some generalized iterative methods we study the convergence properties of the aor and gaor iterative methods for the solution of the linear system ax b and we give more general forms of the stein rosenberg theorem under the assumptions that 1 l u 0 in a i b i l u are not necessarily strictly lower and strictly upper triangular matrices respectively and 2 l 1 we derive new results regarding the spectral radii of the gaor or aor and the gj or jacobi iteration matrices and give in the plane sufficient conditions domains for the gaor and aor method to converge

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

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

comparison theorems for a class of parallel multisplitting aor type iterative methods abstract we establish some comparison theorems for parallel multisplitting aor type iteration matrices with multiset parameters as a consequence we obtain a stein rosenberg type theorem for the parallel multisplitting aor iterative method and its special cases

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

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

generalized extrapolation principle and convergence of some generalized iterative methods abstract to solve the linear system ax b this paper presents a generalized extrapolated method by replacing the extrapolation parameter with the diagonal matrix and systematically gives the basic results for its convergence based upon these results the paper considers the convergence of the gj and gaor iterative methods and using the set of the equimodularized diagonally similar matrices defined here gives some new further convergence results for h matrices and their subclasses strictly or irreducibly diagonally dominant matrices which unify improve and extend previously given various results finally conditions equivalent to the statement that a is a nonsingular h matrix or a strictly or an irreducibly diagonally dominant matrix are given in connection with the gj and gaor methods

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 stein rosenberg type theorems for nonnegative and perron frobenius splittings abstract the stein rosenberg theorem is extended and generalized to the class of nonnegative splittings a m 1 n 1 m 2 n 2 as well as to the most generalized class of perron frobenius splittings two types of the stein rosenberg theorem are stated and proved for both classes these theorems allow us to obtain comparison results for the rate of convergence of the associated iterative methods specific assumptions are given under which the inequalities of the spectral radii become equalities or strict inequalities the theoretical results are confirmed by numerical examples

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