spectral methods for weakly singular volterra integral equations with smooth solutions we propose and analyze a spectral jacobi collocation approximation for the linear volterra integral equations vies of the second kind with weakly singular kernels in this work we consider the case when the underlying solutions of the vies are sufficiently smooth in this case we provide a rigorous error analysis for the proposed method which shows that the numerical errors decay exponentially in the infinity norm and weighted sobolev space norms numerical results are presented to confirm the theoretical prediction of the exponential rate of convergence

data free inference of the joint distribution of uncertain model parameters a critical problem in accurately estimating uncertainty in model predictions is the lack of details in the literature on the correlation or full joint distribution of uncertain model parameters in this paper we describe a framework and a class of algorithms for analyzing such missing data problems in the setting of bayesian statistics the analysis focuses on the family of posterior distributions consistent with given statistics e g nominal values confidence intervals the combining of consistent distributions is addressed via techniques from the opinion pooling literature the developed approach allows subsequent propagation of uncertainty in model inputs consistent with reported statistics in the absence of data

error estimates in the numerical evaluation of some bem singular integrals in some applications of galerkin boundary element methods one has to compute integrals which after proper normalization are of the form b 2 x y dxdy a 1 x y y where a b 1 1 or a b a 1 or a b 1 b and f x y is a smooth function in this paper we derive error estimates for a numerical approach recently proposed to evaluate the above integral when a p or h p formulation of a galerkin method is used this approach suggests approximating the inner integral by a quadrature formula of interpolatory type that exactly integrates the cauchy kernel and the outer integral by a rule which takes into account the log endpoint singularities of its integrand some numerical examples are also given

gaussian quadrature formulae of the third kind for cauchy principal value integrals basic properties and error estimates abstract let n 1 f be the polynomial of degree n 1 interpolating the function f at the points x1 x2 xn with pn xi 0 i e at the nodes of the classical gaussian quadrature formula for the numerical approximation of the cauchy principal value integral 1 1 x x 1 d x with 1 1 and f c1 1 1 we present the quadrature formula qn 1g3 given by q n 1 g3 1 n 1 x n 1 x d x 1 n 1 1 we show that this quadrature formula does not have the disadvantages of the other two well known quadrature formulae based on the same set of nodes in particular we prove that the sequence based on the same set of nodes in particular we prove that the sequence qn 1g3 f converges to the true value of the integral uniformly for all 1 1 we give estimates for the error term furthermore we state some relations connecting the present quadrature formula to the previously introduced formulae

product integration in the presence of a singularity this paper considers the approximate evaluation of int a b k x f x dx where k is a fixed lebesgue integrable function by quadrature rules of the form sigma i 0 m n w in f x in normally rules of this kind are used only for smooth functions f and any singularities are incorporated into k here however we allow f to have a singularity either at an endpoint or in the interior general convergence properties are established for a wide class of product integration rules more detailed results are established for the class of rules which are exact if f belongs to a specified family of piecewise polynomials

maximum entropy and bayesian data analysis entropic prior distributions the problem of assigning probability distributions which reflect the prior information available about experiments is one of the major stumbling blocks in the use of bayesian methods of data analysis in this paper the method of maximum relative entropy me is used to translate the information contained in the known form of the likelihood into a prior distribution for bayesian inference the argument is inspired and guided by intuition gained from the successful use of me methods in statistical mechanics for experiments that cannot be repeated the resulting entropic prior is formally identical with the einstein fluctuation formula for repeatable experiments however the expected value of the entropy of the likelihood turns out to be relevant information that must be included in the analysis the important case of a gaussian likelihood is treated in detail

multi element generalized polynomial chaos for arbitrary probability measures we develop a multi element generalized polynomial chaos me gpc method for arbitrary probability measures and apply it to solve ordinary and partial differential equations with stochastic inputs given a stochastic input with an arbitrary probability measure its random space is decomposed into smaller elements subsequently in each element a new random variable with respect to a conditional probability density function pdf is defined and a set of orthogonal polynomials in terms of this random variable is constructed numerically then the generalized polynomial chaos gpc method is implemented element by element numerical experiments show that the cost for the construction of orthogonal polynomials is negligible compared to the total time cost efficiency and convergence of me gpc are studied numerically by considering some commonly used random variables me gpc provides an efficient and flexible approach to solving differential equations with random inputs especially for problems related to long term integration large perturbation and stochastic discontinuities

a lagrange type projector on the real line we introduce an interpolation process based on some of the zeros of the mth generalized freud polynomial convergence results and error estimates are given in particular we show that in some important function spaces the interpolating polynomial behaves like the best approximation moreover the stability and the convergence of some quadrature rules are proved

polynomial approximations of functions with endpoint singularities and product integration formulas several problems of mathematical physics lead to fredholm integral equations of the second kind where the kernels are either weakly or strongly sin gular and the known terms are smooth these equations have solutions which are smooth in the whole interval of integration except at the endpoints where they have mild singularities in this paper we derive new pointwise and uniform polynomial approximation error estimates for that type of function these esti mates are then used to obtain bounds for the remainder terms of interpolatory product rules based on the zeros of classical jacobi orthogonal polynomials that have been proposed for the discretization of integrals of the form k x y f x dx appearing in the integral equations mentioned above

optimal systems of nodes for lagrange interpolation on bounded intervals a survey in this brief survey special attention is paid to some recent procedures for constructing optimal interpolation processes i e with lebesgue constant having logarithmic behaviour a new result on lagrange interpolation based on the zeros of the associated jacobi polynomials and on suitable additional nodes is given

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

on minimax robustness a general approach and applications the minimax approach to the design of systems that are robust with respect to modeling uncertainties is studied using a game theoretic formulation in which the performance functional and the sets of modeling uncertainties and admissible design policies are arbitrary the existence and characterization of minimax robust solutions that form saddle points are discussed through various methods that take into account several common features of the games encountered in applications in particular it is shown that if the performance functional and the uncertainty set are convex then a certain type of regularity condition on the functional is sufficient to ensure that the optimal strategy for a least favorable element of the uncertainty set is minimax robust the efficacy of the methods proposed for a general game is tested in the problems of matched filtering wiener filtering quadratic detection and output energy filtering in which uncertainties in their respective signal and noise models are assumed to exist these problems are analyzed in a common hilbert space framework and they serve to point out the advantages and limitations of the proposed techniques

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

associated generalized jacobi functions and polynomials abstract pointwise estimates are obtained for associated functions and polynomials corresponding to generalized jacobi weights these estimates hold uniformly in the interval of orthogonality

multi resolution analysis of wiener type uncertainty propagation schemes a multi resolution analysis mra is applied to an uncertainty propagation scheme based on a generalized polynomial chaos pc representation the mra relies on an orthogonal projection of uncertain data and solution variables onto a multi wavelet basis consisting of compact piecewise smooth polynomial functions the coefficients of the expansion are computed through a galerkin procedure the mra scheme is applied to the simulation of the lorenz system having a single random parameter the convergence of the solution with respect to the resolution level and expansion order is investigated in particular results are compared to two monte carlo sampling strategies demonstrating the superiority of the mra for more complex problems however the mra approach may require excessive cpu times adaptive methods are consequently developed in order to overcome this drawback two approaches are explored the first is based on adaptive refinement of the multi wavelet basis while the second is based on adaptive block partitioning of the space of random variables computational tests indicate that the latter approach is better suited for large problems leading to a more efficient flexible and parallelizable scheme

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

on the rate distortion function of sources with incomplete statistics maximization of the rate distortion function rdf of a class of memoryless sources constrained by moments is transformed to a minimax problem kuhn tucker conditions for determining channel capacity and an rdf are solved simultaneously by a linear program special cases include the maximum entropy method gaussian densities and spherical invariance

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

remarks on two integral operators and numerical methods for csie in this paper the author extends the mapping properties of some singular integral operators in zygmund spaces equipped with uniform norm as a by product quadrature methods for solving csie having indices 0 and 1 are proposed their stability and convergence are proved and error estimates in zygmund norm are given some numerical tests are also shown

convergence of product integration rules over 0 for functions with weak singularities at the origin in this paper we consider integrals of the form 0 e x k x y f x dx with f c p 0 c q 0 q p 0 and x i f p 1 x c 0 i 1 q p when q p they appear for instance in certain wiener hopf integral equations and are of interest if one wants to solve these by a nystrom method to discretize the integral above we propose to use a product rule of interpolatory type based on the zeros of laguerre polynomials for this rule we derive weighted uniform convergence estimates and present some numerical examples

on mean convergence of hermite fejer and hermite interpolation for erdos weights abstract we investigate convergence of hermite fejer and hermite interpolation polynomials in l p 0 for erdos weights

stieltjes polynomials and lagrange interpolation bounds are proved for the stieltjes polynomial e n 1 and lower bounds are proved for the distances of consecutive zeros of the stieltjes polynomials and the legendre polynomials p n this sharpens a known interlacing result of szego as a byproduct bounds are obtained for the geronimus polynomials g n applying these results convergence theorems are proved for the lagrange interpolation process with respect to the zeros of e n 1 and for the extended lagrange interpolation process with respect to the zeros of p n e n 1 in the uniform and weighted l p norms the corresponding lebesgue constants are of optimal order

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

product integration for the linear transport equation in slab geometry in this paper we present a product quadrature rule for the discretization of the well known linear transport equation in slab geometry in particular we give an algorithm for constructing the weights of the rule and prove that the order of convergence iso n 3 0 small as we like numerical examples are given and our formula is also compared with product simpson rules finally we examine a nystrom method based on our quadrature

robust memoryless quantization for minimum signal distortion robust quantizers are designed for situations where there is only an incomplete statistical description of the quantizer input the goal of the design is to closely approximate quantizer inputs by quantizer outputs without using more than a specified number of quantization levels the exact probability distribution of the input is unknown but this distribution is known to belong to some set c the primary set c considered is the set of all unimodal probability distributions which satisfy generalized moment constraint e g mean square value less than or equal to a constant a quantizer is derived which minimizes over all quantizers the maximum distortion over all distributions in c this robust quantizer guarantees a significanfiy lower worst case distortion than the classical gaussian optimal quantizer while performing nearly as well as the gaussian optimal quantizer when the input is in fact gaussian

asymptotic error bounds of product formulas for functions with interior singularity this paper is concerned with the convergence of product quadrature formulas of interpolatory type based on the zeros of jacobi polynomials for the approximation of integrals of the type f x is supposed to be of the form vert x x 0 vert sigma sigma 0 not an integer x 0 in 1 1 the kernel k x y can be a smooth one or it can contain an algebraic or a logarithmic singularity

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

convergence analysis of the jacobi collocation method for nonlinear weakly singular volterra integral equations in this work we present an efficient spectral collocation method for numerical solution of a class of nonlinear weakly singular volterra integral equations this type of equations typically has a singular behavior at the left endpoint of the interval of integration for overcoming this non smooth behavior we apply the jacobi collocation method the convergence analysis of the proposed method is investigated in the l and the weighted l2 norms and the results of several numerical experiments are presented which support the theoretical results the computed results are compared wherever possible with those already available in the literature

product integration for volterra integral equations of the second kind with weakly singular kernels we introduce a new numerical approach for solving volterra integral equations of the second kind when the kernel contains a mild singularity we give a convergence result we also present numerical examples which show the performance and efficiency of our method

uncertainty quantification in chemical systems summary we demonstrate the use of multiwavelet spectral polynomial chaos techniques for uncertainty quantification in non isothermal ignition of a methane air system we employ bayesian inference for identifying the probabilistic representation of the uncertain parameters and propagate this uncertainty through the ignition process we analyze the time evolution of moments and probability density functions of the solution we also examine the role and significance of dependence among the uncertain parameters we finish with a discussion of the role of non linearity and the performance of the algorithm copyright q 2009 john wiley sons ltd

rates of convergence of gaussian quadrature for singular integrands the authors obtain the rates of convergence or divergence of gaussian quadrature on functions with an algebraic or logarithmic singularity inside or at an endpoint of the interval of integration a typical result is the following for a bounded smooth weight function on 1 1 the error in n point gaussian quadrature of f x ix y1 1 is o n 2 28 if y 1 and o n if y e 1 1 provided we avoid the singularity if we ignore the singularityy the error is o n1 23 log n 8 loglog n 8t 1 for almost all choices of y these assertions are sharp with respect to order

robust quantization of e contaminated data the problem of robust quantization of data with uncertain statistical properties is considered uncertainty in the statistics of the data is modeled by assuming that the data have a probability density function of the e contaminated form and a minimax approach to robust design is adopted an approximation is developed for the asymptotic worst case distortion over the e contaminated class produced by an arbitrary companded quantizer and the quantizer design which minimizes this worst case distortion is derived the robustness of the resulting design is verified numerically for the particular problem of quantizing e contaminated gaussian data

solution of stochastic partial differential equations using galerkin finite element techniques this paper presents a framework for the construction of galerkin approximations of elliptic boundary value problems with stochastic input data a variational formulation is developed which allows among others numerical treatment by the finite element method a theory of a posteriori error estimation and corresponding adaptive approaches based on practical experience can be utilized the paper develops a foundation for treating stochastic partial differential equations pdes which can be further developed in many directions

robust quantization of memoryless sources using dispersive fir filters an approach to quantizing discrete time memoryless sources is presented an important feature is that its performance is largely insensitive to errors in modeling the input pdf the method involves changing the amplitude distribution of the source to be approximately gaussian by all pass filtering then applying a lloyd max quantizer designed for a gaussian source after quantization the samples are passed through another all pass filter which is an approximate inverse of the first filter the mean square error mse for the overall process is roughly equal to the quantization mse for the intermediate gaussian signal independent of the source statistics for some sources this is actually an improvement over direct correct model lloyd max quantization the cost of this technique is some delay due to filtering

projection methods and condition numbers in uniform norm for fredholm and cauchy singular integral equations in this paper the authors propose a numerical method for the approximate solution of some classes of fredholm and cauchy integral equations including the discrete collocation and collocation methods

noninterpolatory integration rules for cauchy principal value integrals etude de la convergence des regles d integration interpolat pour l approximation des integrales de cauchy

design of quantizers from histograms the design of signal quantizers when the statistical description of the source is a histogram on a finite domain is considered minimum mean and minimax error criteria are discussed both leading to the design of piecewise linear compressor functions the problem of allocating the histogram intervals in the symmetric source density case is addressed through the use of chebyshev probability inequalities finally an adaptive implementation of the proposed system is described

the numerical computation of some integrals on the real line abstract we discuss the approximation of integrals of type i f t r f x k x t e x 2 x d x 1 k is the weakly singular algebraic kernel x t 1 for large value of the parameter t moreover we consider strongly oscillatory kernels of type k 1 x t sin tx 2 k 2 x t cos tx 2 weighted error estimates in uniform and l1 norm are stated and numerical examples to confirm the efficiency of the proposed procedures are given

some weighted polynomial inequalities abstract the author considers a weight function of sub exponential type with zeros and singularities in 1 1 some inequalities of remez bernstein markov and nikolsky type are proved the results hold for polynomials and generalized polynomials moreover an upper and a lower bound for the christoffel functions are derived

random matrices nonbacktracking walks and orthogonal polynomials several well known results from the random matrix theory such as wigner s law and the marchenko pastur law can be interpreted and proved in terms of nonbacktracking walks on a certain graph orthogonal polynomials with respect to the limiting spectral measure play a role in this approach

interpolatory product quadratures for cauchy principal value integrals with freud weights we prove convergence results and error estimates for interpolatory product quadrature formulas for cauchy principal value integrals on the real line with freud type weight functions the formulas are based on polynomial interpolation at the zeros of orthogonal polynomials associated with the weight function under consideration as a by product we obtain new bounds for the derivative of the functions of the second kind for these weight functions

numerical solutions for fredholm integral equations of the second kind with weakly singular kernel using spectral collocation method abstract we investigate the spectral collocation for fredholm integral equations of the second kind with weakly singular kernel the jacobi gauss quadrature formula is used to approximate the integral operator in the numerical implementation we obtain the convergence rates for the approximated solution of weakly singular fredholm integral equations which show that the errors of the approximate solution decay exponentially in l norm and weighted l2 norm some numerical examples are given to illustrate the theoretical results

numerical methods for fredholm integral equations on the square abstract in this paper we shall investigate the numerical solution of two dimensional fredholm integral equations by nystrom and collocation methods based on the zeros of jacobi orthogonal polynomials the convergence stability and well conditioning of the method are proved in suitable weighted spaces of functions some numerical examples illustrate the efficiency of the methods

on weight functions which admit explicit gauss tura n quadrature formulas the main purpose of this paper is the construction of explicit gauss turan quadrature formulas they are relative to some classes of weight functions which have the peculiarity that the corresponding s orthogonal polynomials of the same degree are independent of s these weights too are introduced and discussed here moreover highest precision quadratures for evaluating fourier chebyshev coefficients are given

spectral collocation method for the time fractional diffusion wave equation and convergence analysis in this paper we consider the numerical solution of the time fractional diffusion wave equation essentially the time fractional diffusion wave equation differs from the standard diffusion wave equation in the time derivative term we propose a spectral collocation method in both temporal and spatial discretizations with a spectral expansion of jacobi interpolation polynomial for this equation the convergence of the method is rigorously established numerical tests are carried out to confirm the theoretical results

the wiener askey polynomial chaos for stochastic differential equations we present a new method for solving stochastic differential equations based on galerkin projections and extensions of wiener s polynomial chaos specifically we represent the stochastic processes with an optimum trial basis from the askey family of orthogonal polynomials that reduces the dimensionality of the system and leads to exponential convergence of the error several continuous and discrete processes are treated and numerical examples show substantial speed up compared to monte carlo simulations for low dimensional stochastic inputs

chebyshev spectral solution of nonlinear volterra hammerstein integral equations abstract in this paper the chebyshev spectral cs method for the approximate solution of nonlinear volterra hammerstein integral equations f o k r g r y r d r 0 t is investigated the method is applied to approximate the solution not to the equation in its original form but rather to an equivalent equation z t g t y t t 1 1 the function z is approximated by the nth degree interpolating polynomial zn with coefficients determined by discretizing g t y t at the chebyshev gauss labatto nodes we then define the approximation to y to be of the form y n t f t 1 1 k t s z n s d s 1 1 and establish that under suitable conditions limn yn t y t uniformly in t finally a numerical experiment for a nonlinear volterra hammerstein integral equation is presented which confirms the convergence demonstrates the applicability and the accuracy of the chebyshev spectral cs method

hermite and hermite fejer interpolation for stieltjes polynomials let x 1 x 2 1 2 and p n be the ultraspherical polynomials with respect to x then we denote by e n 1 the stieltjes polynomials with respect to x satisfying in this paper we show uniform convergence of the hermite fejer interpolation polynomials h n 1 and h 2n 1 based on the zeros of the stieltjes polynomials e n 1 and the product e n 1 p n for 0 1 and 0 a 1 2 respectively to prove these results we prove that the lebesgue constants of hermite fejer interpolation operators for the stieltjes polynomials e n 1 and the product e n 1 p n are optimal that is the lebesgue constants h n 1 0 a 1 and h 2n 1 0 a 1 2 have optimal order o 1 in the case of the hermite fejer interpolation polynomials h 2n 1 for 1 2 1 we prove weighted uniform convergence moreover we give some convergence theorems of hermite fejer and hermite interpolation polynomials for 0 1 in weighted lp norms

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

mean convergence of hermite fejer interpolation abstract weighted l p convergence of hermite fejer interpolation based on the zeros of generalized jacobi polynomials is investigated the main result of the paper gives necessary and sufficient conditions for such convergence for all continuous functions

numerical treatment of second kind fredholm integral equations systems on bounded intervals in this paper the authors propose numerical methods to approximate the solutions of systems of second kind fredholm integral equations they prove that such methods are stable and convergent error estimates in weighted l p norm 1

remez type inequalities and their applications abstract the remez inequality gives a sharp uniform bound on 1 1 for real algebraic polynomials p of degree at most n if the lebesgue measure of the subset of 1 1 where p is at most 1 is known remez type inequalities give bounds for classes of functions on a line segment on a curve or on a region of the complex plane given that the modulus of the functions is bounded by 1 on some subset of prescribed measure this paper offers a survey of the extensive recent research on remez type inequalities for polynomials generalized nonnegative polynomials exponentials of logarithmic potentials and muntz polynomials remez type inequalities play a central role in proving other important inequalities for the above classes the paper illustrates the power of remez type inequalities by giving a number of applications

state control spectral chebyshev parameterization for linearly constrained quadratic optimal control problems in this paper we propose a computationally attractive numerical method for determining the optimal control of constrained linear dynamic systems with a quadratic performance the method is based upon constructing the mth degree interpolating polynomials using chebyshev nodes to approximate the control and the state vectors the system dynamics are collocated at chebyshev nodes the performance index is discretized by a cell averaging method the state and control inequality constraints are converted into algebraic inequalities through collocation at the nodes the linear quadratic optimal control problem is thereby transformed into a quadratic programming one simulation studies demonstrate computational advantages relative to a standard riccati method a classical chebyshev based method fourier based method and other methods in the literature

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

uniform convergence of derivatives of extended lagrange interpolation the authors construct some extended interpolation formulae to approximate the derivatives of a function in uniform norm they prove theorems on uniform convergence and give estimates of pointwise type and of simultaneous approximation

mean convergence of derivatives of lagrange interpolation abstract weighted lp convergence of derivatives of lagrange interpolation at the union of zeros of generalized jacobi polynomials and some additional points is investigated

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

a spectral collocation method for multidimensional nonlinear weakly singular volterra integral equation abstract this paper is concerned with the convergence properties of chebyshev spectral collocation method when used to approximate the solution of multidimensional nonlinear volterra integral equation of the second kind with a weakly singular kernel we consider the case that the underlying solution is sufficiently smooth the chebyshev collocation discretization is proposed for this equation in the present paper we provide a rigorous error analysis which justifies that the errors of approximate solution decay exponentially in weighted l 2 norm and l norm numerical results are presented to demonstrate the effectiveness of the spectral method

piecewise uniform vector quantizers the companding model for quantizer design and analysis has been widely applied in the scalar quantization case however if the signal to be quantized is a vector then the optimum companding system can be designed for only a limited number of distributions on the other hand multidimensional piecewise linear companders can be designed for any signal density generating quantizers that are uniform on each region of the compander these systems while not optimal can have asymptotic performance arbitrarily close to the optimum their analysis and implementation can be simpler than those of optimal systems piecewise linear companders for asymptotic multidimensional quantization are analyzed and a method for their design is suggested

finite elements for stochastic media problems we consider discretization methods for stochastic partial differential equations which are used to model media with stochastic properties the wick product formulation of the stochastic pde is used and it is shown how it may be discretized both spatially and in the stochastic part a stochastic variational principle serves as the guideline for the discretization procedure it is shown that both the projection onto the wiener chaos and the hermite transform lead to the same fully discrete equations the application examples show that there are other advantageous bases besides the karhunen loeve expansion and that the methods may be used in a variety of settings

a survey on mean convergence of interpolatory processes abstract a detailed account of what happened in the theory of mean convergence of lagrange and hermite fejer type interpolation in the last fifty five years is given starting from the classical result of erd s and turan hopefully all important developments are considered in some cases with an indication of the method of proof even some yet unpublished results are included a list of references helps to orientate those interested in the details

legendre polynomials of the second kind fourier series and lagrange interpolation abstract the distribution of the zeros of jacobi polynomials of the second kind p m w is studied and lower and upper bounds for the corresponding christoffel functions are given fourier expansions in the system p m w are also considered moreover some theorems on the convergence of lagrange interpolating polynomials based on the zeros of p m w are proved

rates of convergence of gauss lobattom and radau integration rules for singular integrands rates of convergence or divergence are obtained in the application of gauss lobatto and radau integration rules to functions with an algebraic or logarithmic singularity inside or at an endpoint of the interval of integration a typical result is the following for a generalized jacobi weight function on 1 1 the error in applying an n point rule to f x ix vi 6 is o n 2 28 if y 1 and o n 1 6 if y e 1 1 providedwe avoid the singularity if we ignore the singularity y the error is o n 1 28 log n 6 log log n 1 e for almost all choices of y these assertions are sharp with respect to order 1 introduction this paper is a sequel to that of lubinsky and myself 2 on the rates of convergence of gauss integration rules for singular integrands in this paper we extend the results in 2 in several directions on the one hand we extend the results for gauss integration rules to results for lobatto and radau rules this extension follows easily from the generalized markov stieltjes inequality for lobatto and radau rules given in lemma 3 2 and from a representation of the coefficients or weights or cotes numbers in the lobatto and radau rules in terms of the coefficients in the gauss rule with respect to a related weight function a second extension is to the special case of gauss jacobi integration rules as well as lobatto jacobi and radau jacobi rules for certain values of the parameters defining the jacobi weight function

numerical evaluation of integrals containing a spherical bessel function by product integration a method is developed for numerical evaluation of integrals with k integration range from 0 to that contain a spherical bessel function jl kr explicitly the required quadrature weights are easily calculated and the rate of convergence is rapid only a relatively small number of quadrature points is needed for an accurate evaluation even when r is large the quadrature rule is obtained by the method of product integration with the abscissas chosen to be those of clenshaw curtis and the chebyshev polynomials as the interpolating polynomials quadrature weights are obtained that depend on the spherical bessel function an inhomogenous recurrence relation is derived from which the weights can be calculated without accumulation of roundoff error the procedure is summarized as an easily implementable algorithm questions of convergence are discussed and the rate of convergence demonstrated for several test integrals alternative procedures are given for generating the integration weights and an error analy

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

nystrom method for cauchy singular integral equations with negative index in this paper the authors propose a nystrom method to approximate the solutions of cauchy singular integral equations with constant coefficients having a negative index they consider the equations in spaces of continuous functions with weighted uniform norm they prove the stability and the convergence of the method and show some numerical tests that confirm the error estimates

convergence properties of a class of product formulas for weakly singular integral equations we examine the convergence of product quadrature formulas of interpolatory type based on the zeros of certain generalized jacobi polynomials for the discretization of integrals of the type f k x y f x dx l 1 and f x has algebraic singularities of the form 1 x a a 1 we prove that the uniform rate of convergence of the rules is o m log2 m in the case of the first kernel and o m 2 a2v logim if v 0 for the second where m is the number of points in the quadrature rule

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

spectral collocation methods for nonlinear coupled time fractional nernst planck equations in two dimensions and its convergence analysis abstract in this paper a nonlinear coupled time fractional nernst planck equation in two dimensions space is considered using properties of the caputo fractional derivative and riemann liouville fractional integral the time fractional nernst planck problem is transformed into a volterra integral model with singular kernel equivalently we propose a novelty spectral collocation method in time and space with a spectral expansion of jacobi interpolation polynomial to discretize the model we also derive the optimal rate of convergence of this method numerical results are presented to verify efficiency of collocation methods and spectral accuracy theory

a stochastic projection method for fluid flow i basic formulation we describe the construction and implementation of a stochastic navier stokes solver the solver combines a spectral stochastic uncertainty representation scheme with a finite difference projection method for flow simulation the uncertainty quantification scheme is adapted from the spectral stochastic finite element method ssfem which is based on regarding uncertainty as generating a new dimension and the solution as being dependent on this dimension in the ssfem formalism the stochastic dependence is represented in terms of the polynomial chaos system and the coefficients in the corresponding spectral representation are obtained using a galerkin approach it is shown that incorporation of the spectral uncertainty representation scheme into the projection method results in a coupled system of advection diffusion equations for the various uncertainty fields and in a decoupled system of pressure projection steps this leads to a very efficient stochastic solver whose advantages are illustrated using steady and transient simulations of transport and mixing in a microchannel

convergence analysis of the jacobi spectral collocation methods for volterra integral equations with a weakly singular kernel in this paper a jacobi collocation spectral method is developed for volterra integral equations of second kind with a weakly singular kernel we use some function transformation and variable transformations to change the equation into a new volterra integral equation deflned on the standard interval i1 1 so that the solution of the new equation possesses better regularity and the jacobi orthogonal polynomial theory can be applied conveniently in order to obtain high order accuracy for the approx imation the integral term in the resulting equation is approximated by using jacobi spectral quadrature rules the convergence analysis of this novel method is based on the lebesgue constants corresponding to the lagrange interpolation polynomials poly nomials approximation theory for orthogonal polynomials and the operator theory the spectral rate of convergence for the proposed method is established in the l 1 norm and weighted l 2 norm numerical results are presented to demonstrate the efiectiveness of the proposed method

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

numerical solutions for solving time fractional fokker planck equations based on spectral collocation methods abstract in this paper we consider the numerical solution of the time fractional fokker planck equation we propose a spectral collocation method in both temporal and spatial discretizations with a spectral expansion of jacobi interpolation polynomial for this equation the convergence of the method is rigorously established numerical tests are carried out to confirm the theoretical results

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