interpolatory quadrature formulas on the unit circle for chebyshev weight functions in this paper interpolatory quadrature formulas based upon the roots of unity are studied for certain weight functions positivity of the coefficients in these formulas is deduced along with computable error estimations for analytic integrands a comparison is made with szego quadrature formulas finally an application to the interval 1 1 is also carried out

chebyshev series method for computing weighted quadrature formulas abstract in this paper we study convergence and computation of interpolatory quadrature formulas with respect to a wide variety of weight functions the main goal is to evaluate accurately a definite integral whose mass is highly concentrated near some points the numerical implementation of this approach is based on the calculation of chebyshev series and some integration formulas which are exact for polynomials in terms of accuracy the proposed method can be compared with rational gauss quadrature formula

product rules on the unit circle with uniformly distributed nodes error bounds for analytic functions this paper deals with the numerical calculation of integrals over the unit circle in the complex plane by means of quadrature formulas with uniformly distributed nodes on such region several properties of the coefficients as well as error bounds for analytic functions are established

on the numerical resolution of the generalized airfoil equation with possio kernel in this paper we consider the so called generalized airfoil singular integral equation with a smooth input function and present ad hoc quadrature rules to compute efficiently the elements of collocation and galerkin matrices

asymptotic completeness of short range many body systems on annonce une demonstration de la completude asymptotique pour des systemes de mecanique quantique constitues de nombres arbitraires de particules interagissant par des forces a courte portee

on the convergence of an interpolatory product rule for evaluating cauchy principal value integrals etablissement des theoremes de convergence des regles de produits interpolatoires pour l evaluation d integrales singulieres de cauchy avec estimation asymptotique du reste amelioration et generalisation de resultats anterieurs

parameter estimation for a stationary process on a d dimensional lattice summary we study asymptotic properties of various estimation procedures for a general stationary regular process on a d dimensional lattice differences between d 1 time series and d 2 spatial processes are pointed out we suppose that the process is observed on a set pn with cn points which tends to infinity with the same speed in all directions the relative edge effect is of order cn lid increasing with d this effect is without statistical importance if d 1 but is important if d 2 we give a lcn consistent asymptotically normal estimator of the underlying parameter the procedure being constructed by a modification of whittle s approximation to the log likelihood in the gaussian case this procedure is asymptotically efficient

approximation of the hilbert transform on the real line using hermite zeros the authors study the hilbert transform on the real line they introduce some polynomial approximations and some algorithms for its numerical evaluation error estimates in uniform norm are given

spectral properties of vaguely elliptic pseudodifferential operators with momentum dependent long range potentials using time dependent scattering theory a absence of singular continuous spectrum for h and unitary equivalence of h0 p with the absolutely continuous part of h is proved for the self adjoint operator h h0 p ws q p wl q p on l2 rn where i h0 is a smooth nonnegative real valued function on rn of at most polynomial growth with h0 ii the critical values of h0 has a countable closure iii the symbol ws x is a smooth function of at most polynomial growth in and is of short range in x and iv wl x is a smooth real valued function of x with at most a polynomial growth in and o x at for some 0 further we have b essential spectra of h and h0 p are equal and c eigenvalues of h can accumulate only at the critical values of h0

on the convergence of product formulas for the numerical evaluation of derivatives of cauchy principal value integrals product rules of interpolatory type on the zeros of generalized smooth jacobi polynomials for the numerical approximation of cauchy principal value integrals and their derivatives are considered convergence results for these rules are given and some estimates of the remainder are established in relation to the regularity of the density function

orthogonal laurent polynomials corresponding to certain strong stieltjes distributions with applications to numerical quadratures in this paper we shall be mainly concerned with sequences of orthogonal laurent polynomials associated with a class of strong stieltjes distributions introduced by a s ranga algebraic properties of certain quadratures formulae exactly integrating laurent polynomials along with an application to estimate weighted integrals on 1 1 with nearby singularities are given finally numerical examples involving interpolatory rules whose nodes are zeros of orthogonal laurent polynomials are also presented

interpolary product integration in the presence of singularities l 2 theory we study interpolatory product integration rules based on gauss radau and lobatto points with respect to a generalized jacobi weight function as applied to the integration of functions with singularities we give sufficient conditions for convergence of such rules for both endpoint and interior singularities

an algorithm based on the fft for a generalized chebyshev interpolation an algorithm for a generalized chebyshev interpolation procedure increasing the number of sample points more moderately than doubling is pre sented the fft for a real sequence is incorporated into the algorithm to enhance its efficiency numerical comparison with other existing algorithms is given

on long range scattering even if special cases are not well understood it pays sometimes to consider a problem at hand in the most general framework thus addressing the existence of wave operators for the most general potentials hormander ho1 has discovered a natural asymptotic evolution for the long range scattering it is determined by a solution s k t of a certain hamilton jacobi equation

asymptotic completeness for long range many particle systems with stark effect ii we prove the existence and the asymptotic completeness of the dollard type modified wave operators for many particle stark hamiltonians with long range potentials

convergence and computation of simultaneous rational quadrature formulas we discuss the convergence and numerical evaluation of simultaneous quadrature formulas which are exact for rational functions the problem consists in integrating a single function with respect to different measures using a common set of quadrature nodes given a multi index n the nodes of the integration rule are the zeros of the multi orthogonal hermite pade polynomial with respect to s n where s is a collection of measures and is a polynomial which modifies the measures in s the theory is based on the connection between gauss type simultaneous quadrature formulas of rational type and multipoint hermite pade approximation the numerical treatment relies on the technique of modifying the integrand by means of a change of variable when it has real poles close to the integration interval the output of some tests show the power of this approach in comparison with other ones in use

properties of interpolatory product integration rules this paper is concerned with the approximate evaluation of int 1 1 k x f x d x where k is lebesgue integrable and f is at least riemann integrable and preferably smooth the integral is approximated by a rule of the form sum nolimits i 1 n w ni k f x ni where the points x ni are chosen in some prescribed way and the weights w ni k are such that the rule is exact if f is any polynomial of degree n for suitable choices of the points and suitable functions k the rule is shown to converge to the exact integral and the companion rule sum w ni k f x ni to int k x f x dx for all riemann integrable functions f the companion property ensures for example that the weights have the asymptotic positivity property if k is positive error bounds are given that guarantee rapid convergence of the rule if f is smooth finally by establishing a connection with mean convergence of lagrangian interpolation some negative theorems are proved in pa

bayesian clustering for unsupervised estimation of surface and texture models a method of calculating the maximum likelihood clustering for the unsupervised estimation of polynomial models for the data in images of smooth surfaces or for range data for such surfaces is presented an image or a depth map of a region of smooth 3 d surface is modeled as a polynomial plus white noise a region of physically meaningful textured image such as the image of foliage grass or road in outdoor scenes or conductor or lintburn on a thick film substrate is modeled as a colored gaussian markov random field mrf with a polynomial mean value function unsupervised model parameter estimation is accomplished by determining the segmentation and model parameter values that maximize the likelihood of the data or a more general bayesian performance functional agglomerative clustering is used for this purpose

high order methods for weakly singular integral equations with nonsmooth input functions to solve one dimensional linear weakly singular integral equations on bounded intervals with input functions which may be smooth or not we propose to introduce first a simple smoothing change of variable and then to apply classical numerical methods such as product integration and collocation based on global polynomial approximations the advantage of this approach is that the order of the methods can be arbitrarily high and that the associated linear systems one has to solve are very well conditioned

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

rational quadrature formulae on the unit circle with arbitrary poles interpolatory quadrature rules exactly integrating rational functions on the unit circle are considered the poles are prescribed under the only restriction of not lying on the unit circle a computable upper bound of the error is obtained which is valid for any choice of poles arbitrary weight functions and any degree of exactness provided that the integrand is analytic on a neighborhood of the unit circle a number of numerical examples are given which show the advantages of using such rules as well as the sharpness of the error bound also a comparison is made with other error bounds appearing in the literature

on the dense divergence of the product quadrature formulas of interpolatory type this paper concerns divergent aspects regarding interpolatory product quadrature formulas generated by arbitrary projection operators associated to the banach space c of all real continuous functions defined on the interval 1 1 1 1 of r r in supremum norm the main result is obtained by using principles of functional analysis and it highlights the phenomenon of double condensation of singularities for the considered quadrature formulas meaning unbounded divergence on large subsets in topological sense belonging to c and to the banach space of absolute integrable functions on 1 1 1 1

product integration of piecewise continuous integrands based on cubic splineinterpolation at equally spaced nodes in this paper we consider the approximate evaluation of int limits a b k x f x dx wherek x is a fixed lebesgue integrable function by product formulas of the form sum limits i 0 n w i f x i based on cubic spline interpolation of the functionf generally whenever it is possible product quadratures incorporate the bad behaviour of the integrand in the kernelk here however we allowf to have a finite number of jump discontinuities in a b convergence results are established and some numerical applications are given for a logarithmic singularity structure in the kernel

quadrature on the half line and two point pade approximants to stieltjes functions ii convergence abstract let be a distribution function on a b 0 a c k b a d x exist for all the integers k the main course of the paper is to give convergence results both for sequences of two point pade approximants to the cauchy transform of the distribution stieltjes function and for quadrature formulas based on exactly integrating laurent polynomials the case when d x k x dx k x being a possibly complex function is also considered and estimates of the rate of convergence are given

propagation of states in dilation analytic potentials and asymptotic completeness we estimate the space time behavior of scattering states for two body schrodinger operators with smooth dilation analytic potentials we use our estimates to give a simple proof of asymptotic completeness for a class of long range potentials including the coulomb potential plus a fairly general short range perturbation

an approximate method of evaluating the joint likelihood for first order gmrfs a highly accurate approximation is proposed for computing the joint likelihood for first order gauss markov random fields gmrfs defined on irregularly shaped lattices the problem in computing the likelihood lies in evaluating the determinant of a very large matrix b its exact evaluation is limited to either very small irregular regions or a few regularly shaped regions the approximation proposed here is based on an eigenanalysis of b

unsupervised image restoration and edge location using compound gauss markov random fields and the mdl principle discontinuity preserving bayesian image restoration typically involves two markov random fields one representing the image intensities gray levels to be recovered and another one signaling discontinuities edges to be preserved the usual strategy is to perform joint maximum a posterori map estimation of the image and its edges which requires the specification of priors for both fields instead of taking an edge prior we interpret discontinuities in fact their locations as deterministic unknown parameters of the compound gauss markov random field cgmrf which is assumed to model the intensities this strategy should allow inferring the discontinuity locations directly from the image with no further assumptions however an additional problem emerges the number of parameters edges is unknown to deal with it we invoke the minimum description length mdl principle according to mdl the best edge configuration is the one that allows the shortest description of the image and its edges taking the other model parameters noise and cgmrf variances also as unknown we propose a new unsupervised discontinuity preserving image restoration criterion implementation is carried out by a continuation type iterative algorithm which provides estimates of the number of discontinuities their locations the noise variance the original image variance and the original image itself restored image experimental results with real and synthetic images are reported

numerical evaluation of cauchy principal value integrals with singular integrands convergence results are proved for sequences of interpolatory integration rules for cauchy principal value integrals of the form fk x f x x a dx 1 a 1 when f x is singular at a point a 7 i and the singularity is ignored or avoided

uniform convergence of derivatives of lagrange interpolation abstract uniform convergence of derivatives of lagrange interpolation at the union of zeros of jacobi polynomials and some additional points is investigated

product integration based on hermite feje r interpolation abstract the use of generalized hermite fejer interpolation polynomials for product integration is studied algorithms for implementing this method are developed and convergence results are proved comparisons are made between this new approach and the classical approach based on lagrange interpolation polynomials it turns out that integration rules based on hermite fejer interpolation converge in more cases but in general more slowly than those based on lagrange interpolation when the latter converge

rational interpolation ii quadrature and convergence abstract consider an n th rational interpolatory quadrature rule j n f j 1 n j f x j to approximate integrals of the form j f 1 1 f x d x where is a possibly complex bounded measure with infinite support on the interval 1 1 first we discuss the connection of j n f with certain rational interpolatory quadratures on the complex unit circle to approximate integrals of the form f e i d c where c is a possibly complex bounded measure with infinite support on next we provide conditions to ensure the convergence of j n f to j f for n tending to infinity finally an upper bound for the error on the n th approximation and an estimate for the rate of convergence is provided

texture synthesis using 2 d noncausal autoregressive models the purpose of this paper is to illustrate the usefulness of two dimensional noncausal autoregressive ncar models for the synthesis of textures these models characterize the gray level at a pixel as a linear combination of gray levels at nearby locations in all directions and an additive white noise variable we first show that the class of ncar models is capable of generating a wide variety of image patterns posessing the local replication attribute an essential ingredient of many natural textures it is also shown that the theoretical variograms of many ncar models possess an oscillatory behavior a characteristic of the variograms of many natural textures next we give experimental results of synthesis of 64 64 textures resembling several real textures in the brodatz album the synthetic textures generated by 16 parameter ncar models retain most the visual characteristics of the original textures the variograms of the original and synthetic textures are also similar

analysis of general quadrature methods for integral equations of the second kind this paper is concerned with a class of approximation methods for integral equations of the form y t f t int limits a b k t s y s ds wherea andb are finite f andy are continuous and the kernelk may be weakly singular the methods are characterized by approximate equations of the form y n t f t sum limits i 1 n w ni t y n s ni such methods include the nystrom method and a variety of product integration methods a general convergence theory is developed for methods of this type in suitable cases it has the feature that its application to a specific method depends only on a knowledge of convergence properties of the underlying quadrature rule the theory is used to deduce convergence results some of them new for a number of specific methods

quadrature formulas for functions with poles near the interval of integration on examine la construction des regles de quadrature de type interpolatoire utilisant uniquement des valeurs de fonctions reelles pour des fonctions ayant des paires complesex conjuguees de poles pres de l intervalle d integration

convergence of product integration rules for functions with interior and end point singularities over bounded and unbounded intervals the convergence of product integration rules based on gaussian quadrature points is investigated for functions with interior and endpoint singularities over bounded and unbounded intervals the investigation is based on a new convergence result for lagrangian interpolation and gaussian quadrature of singular integrands

product integration of singular integrands based on cubic spline interpolation at equally spaced nodes in this paper the convergence of product integration rules based on cubic spline interpolation at equally spaced nodes with not a knot end condition is investigated for integrand functions with a interior or endpoint singularity in the integration interval

numerical calculation of integrals involving oscillatory and singular kernels and some applications of quadratures numerical methods for strongly oscillatory and singular functions are given in this paper beside a summary of standard methods and product integration rules we consider a class of complex integration methods several applications of quadrature processes in problems in telecom munications and physics are also presented 1998 elsevier science ltd all rights reserved integration oscillatory kernel singular kernel orthogonal polynomials product rules gaussian quaclratures error function bessel functions legendre functions

application of a modified fft to product type integration abstract an automatic integration scheme is proposed for evaluating the so called product type indefinite integral q k y x k t t dt 1 x y where t is assumed to be a smooth function and k t are some singular or badly behaved functions typical examples for k t are 1n t c t c 1 cauchy principal value 1 t c and ei t 1 the function t is approximated by a truncated chebyshev series pn t of degree n whose coefficients are efficiently computed using the fft the approximation qn k to the integral q k is given by q k pn the sequence pn t is recursively generated until the required tolerance for the integral is satisfied to enhance the efficiency of the automatic quadrature the degree n is increased more slowly than doubling which is usually the case the evaluations of qn k q k pn for a set of x y c can be efficiently made by using recurrence relations for the singular kernels k t above numerical examples for the algebraic singular kernel k t t c 1 are included

on product integration with gauss kronrod nodes gauss kronrod product quadrature formulas for the numerical approximation of i k x f x dx are shown to converge for every riemann integrable f and to possess optimal stability similar results are proved for the product formulas based on the kronrod nodes only an application to the uniform convergence of approximate solutions of integral equations is given

product integration over infinite intervals i rules based on the zeros of hermite polynomials the paper discusses both theoretical properties and practical implementation of product integration rules of the form n j k x f x dx wnlj xnj o i l where f is continuous k is absolutely integrable the nodes xn are roots of the hermite polynomials hn x and the weights wn are chosen so that the rule is exact if f is any polynomial of degree n convergence of the rule to the exact integral as n oo is proved for a wide class of functions f and k including singular or oscillatory functions k and rates of convergence are estimated the rules are shown to have the property of asymptotic positivity and as a consequence exhibit good numerical stability numerical calculations for some practical cases are presented which show the method to be computationally effective for integrands including highly oscillatory ones that decay suitably at infinity applications of the method to integration over 0 oo are also discussed

asymptotic completeness of long range n body quantum systems where x is a euclidean space and lra a e a is a finite family of projections onto certain subspaces xa of x we will always assume that the family xa a e a is closed with respect to the algebraic sum and contains xamin 0 the class of operators of the form 0 2 is a generalization of 0 1 up to a linear change of coordinates in the configuration space x they usually go under the name of generalized n body schrbdinger operators and were first considered in a in what follows we will drop the word generalized we refer the reader to rsi vol iii for a general introduction to n body schr6dinger operators defined as in equation 0 1 we will consistently use the definition 0 2 and assume that the reader is familiar with the physical meaning of various objects under study

unsupervised contour representation and estimation using b splines and a minimum description length criterion this paper describes a new approach to adaptive estimation of parametric deformable contours based on b spline representations the problem is formulated in a statistical framework with the likelihood function being derived from a region based image model the parameters of the image model the contour parameters and the b spline parameterization order i e the number of control points are all considered unknown the parameterization order is estimated via a minimum description length mdl type criterion a deterministic iterative algorithm is developed to implement the derived contour estimation criterion the result is an unsupervised parametric deformable contour it adapts its degree of smoothness complexity number of control points and it also estimates the observation image model parameters the experiments reported in the paper performed on synthetic and real medical images confirm the adequate and good performance of the approach

maximum likelihood estimation of models for residual covariance in spatial regression we describe the maximum likelihood method for fitting the linear model when residuals are correlated and when the covariance among the residuals is determined by a parametric model containing unknown parameters observations are assumed to be gaussian we give conditions which ensure consistency and asymptotic normality of the estimators our main concern is with the analysis of spatial data and in this context we describe some simulation experiments to assess the small sample behaviour of estimators we also discuss an application of the spectral approximation to the likelihood for processes on a lattice

stieltjes type polynomials on the unit circle stieltjes type polynomials corresponding to measures supported on the unit circle t are introduced and their asymptotic properties away from t are studied for general classes of measures as an application we prove the convergence of an associated sequence of interpolating rational functions to the corresponding caratheodory function in turn this is used to give an estimate of the rate of convergence of certain quadrature formulae that resemble the gauss kronrod rule provided that the integrand is analytic in a neighborhood of t

ultraspherical stieltjes polynomials and gauss kronrod quadrature behave nicely for lambda 0 we show that the zeros of the stieltjes polynomials associated with the ultraspherical weight function with parameter lambda 0 are real and simple and except for two of them belong to 1 1 we also prove that they strictly interlace with the zeros of the corresponding ultraspherical polynomials consequently a gauss kronrod quadrature formula with two nodes outside the interval 1 1 is obtained we prove that the coefficients of such quadrature rules are positive finally an asymptotic representation of stieltjes polynomials which converges uniformly on the whole interval 1 1 is provided

orthogonality interpolation and quadratures on the unit circle and the interval 1 1 given a weight function s x on 1 1 or more generally a positive borel measure the erdos turan theorem assures the convergence in l 2 s norm to a function f of its sequence of interpolating polynomials at the zeros of the orthogonal polynomials or equivalently at the nodes of the gauss christoffel quadrature formulas associated with s in this paper we will extend this result to the nodes of the gauss radau and gauss lobatto quadrature formulas by passing to the unit circle and taking advantage of the results on interpolation by means of laurent polynomials at the zeros of certain para orthogonal polynomials with respect to the weight function w q s cos q sin q on p p as a consequence an application to the construction of certain product integration rules on finite intervals of the real line will be given several numerical experiments are finally carried out

product integration of logarithmic singular integrands based on cubic splines abstract this paper is concerned with the practical evaluation of the product integral 1 1 f x k x d x for the case when k x in x 1 1 and f is bounded in 1 1 the approximation is a quadrature rule where the weights w n n i are chosen to be exact when f is given by a linear combination of a chosen set of functions n j in this paper the functions n j are chosen to be cubic b splines an error bound for product quadrature rules based on cubic splines is provided examples that test the performance of the product quadrature rules for different choices of the function are given a comparison is made with product quadrature rules based on first kind chebyshev polynomials

mean convergence of lagrange interpolation for freud s weights with application to product integration rules the connection between convergence of product integration rules and mean convergence of lagrange interpolation in l 1 p 00 has been thoroughly analysed by sloan and smith 37 motivated by this connection we investigate mean convergence of lagrange interpolation at the zeros of orthogonal polynomials associated with freud weights on r our results apply to the weights exp x 2 m 2 4 6 and for the hermite weight m 2 extend results of nevai 28 and bonan 2 in at least one direction the results are sharp in l 1 p g 2 as a consequence we can improve results of smith sloan and opie 38 on convergence of product integration rules based on the zeros of the orthogonal polynomials associated with the hermite weight in the process we prove a new markov stieltjes inequality for gauss quadrature sums and solve a problem of nevai on how to estimate certain quadrature sums

likelihood analysis of spatial point patterns summary the likelihood procedure for estimating the pairwise interaction potential function is developed for statistical analysis of homogeneous spatial point patterns approximation methods of the normalizing factor of gibbs canonical distribution are discussed both to estimate a scale parameter and to measure the softness or hardness of repulsive interactions the approximations are useful up to a considerably high density the validity of our procedure is demonstrated by some computer experi ments some real data are analysed