interlacing of zeros of gegenbauer polynomials of non consecutive degree from different sequences a theorem due to stieltjes states that if p n n 0 infty is any orthogonal sequence then between any two consecutive zeros of p k there is at least one zero of p n whenever k frac 12 if 0 t k 1 and also to the zeros of c n 1 lambda and c n 2 lambda k if k in 1 2 3 more generally we prove that stieltjes interlacing holds between the zeros of the kth derivative of c n lambda and the zeros of c n 1 lambda k in 1 2 dots n 1 and we derive associated polynomials that play an analogous role to the de boor saff polynomials in completing the interlacing process of the zeros

orthogonal polynomials associated polynomials and functions of the second kind abstract a survey is given of the interaction between orthogonal polynomials associated polynomials and functions of the second kind with an emphasis on asymptotic results various formulas are presented in a unified way in terms of wronskians of solutions of linear recurrence relations some of these formulas are classical and go back to the previous century but usually they are hard to locate in the literature some new formulas are also given in particular a formula expressing the derivative of an orthogonal polynomial in terms of the orthogonal polynomials and the associated polynomials

approximations to bayesian clustering rules summary the amount of computation required for implementing the bayesian cluster analysis suggested by binder 1978 is often too large for exact results to be feasible a general algorithm is proposed for approximating the similarity matrix and the resulting optimal partition this algorithm is applied to artificial and to real data for the real data it appears that the algorithm is successful at identifying the optimal partitions as well as those units whose group membership is doubtful

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

stieltjes polynomials and functions of the second kind abstract in this paper we discuss known and new facts on the asymptotic behaviour of the stieltjes polynomials and the close relation to the asymptotic behaviour of the boundary values of the functions of the second kind on 1 1 most of the results are derived by new methods of proof where mainly the theory of real orthogonal polynomials is used

anti gaussian quadrature formulas an anti gaussian quadrature formula is an n 1 point formula of degree 2n 1 which integrates polynomials of degree up to 2n 1 with an error equal in magnitude but of opposite sign to that of the n point gaussian formula its intended application is to estimate the error incurred in gaussian integration by halving the difference between the results obtained from the two formulas we show that an anti gaussian formula has positive weights and that its nodes are in the integration interval and are interlaced by those of the corresponding gaussian formula similar results for gaussian formulas with respect to a positive weight are given except that for some weight functions at most two of the nodes may be outside the integration interval the anti gaussian formula has only interior nodes in many cases when the kronrod extension does not and is as easy to compute as the n 1 point gaussian formula

bayesian estimation for nonstandard loss functions using a parametric family of estimators bayesian estimation with other loss functions than the standard hit or miss loss or the quadratic loss often yields optimal bayesian estimators obes that can only be formulated as optimization problems and which have to be solved for each new observation the contribution of this paper is to introduce a new parametric family of estimators to circumvent this problem by restricting the estimator to lie in this family we split the estimation problem into two parts in a first step we have to find the best estimator with respect to the bayes risk for a given nonstandard loss function which has to be done only once the second step then calculates the estimate for an observation using importance sampling the computational complexity of this second step is therefore comparable to that of an mmse estimator if the mmse estimator also uses monte carlo integration we study the proposed parametric family using two examples and show that the estimator family gives for both a good approximation of the obe

minimal solutions of three term recurrence relations and orthogonal polynomials we observe that the well known recurrence relation pn x z z a pn z bnp l z for orthogonal polynomials admits a minimal solution if z is outside the spectrum of the mass distribution ds t with respect to which the polynomials are orthogonal and if the moment problem for this distribution is determined the minimal solution indeed is f z m ds t z t and can be computed accurately by means of the author s continued fraction algorithm an application is made to special gauss type quadrature formulas 1 introduction minimal solutions of three term recurrence relations and their computational implications are discussed systematically in 7 effective algorithms for computing minimal solutions have been developed in 7 16 17 and continue to be the subject of further study see e g 14 26 5 2 24 25 3 with these powerful algorithms at hand it seems desirable to delineate large classes of recurrence relations for which the presence of minimal solutions can be ascertained and the minimal solution itself identified even more desirable is an understanding of the deeper reasons for the occurrence of minimal solutions in terms of intrinsic features of the subject area in which they arise few attempts have been made along these lines the work of thacher 22 23 on power series solution of linear differential equations with polynomial coeffi cients may be considered a beginning inasmuch as the presence of minimal solutions of the recurrence relation satisfied by the expansion coefficients is conjectured to be related in a specified way to the type of singularities in the differential equation here we wish to consider the recurrence relation

stopping functionals for gaussian quadrature formulas abstract gaussian formulas are among the most often used quadrature formulas in practice in this survey an overview is given on stopping functionals for gaussian formulas which are of the same type as quadrature formulas i e linear combinations of function evaluations in particular methods based on extended formulas like the important gauss kronrod and patterson schemes and methods which are based on gaussian nodes are presented and compared

finite perturbations of orthogonal polynomials in this paper we study orthogonal polynomials pn which arise from a given system of orthogonal polynomials pn by a finite perturbation of the recurrence coefficients i e the recurrence coefficients n n 1 of pn respectively n n 1 of pn are related to each other by n l n m and n 1 l n 1 m for n n where l and m are fixed nonnegative integers closed expressions for the polynomial pn in terms of pn and its associated polynomial pn 1 1 as well as for the orthogonality measure of pn in terms of the orthogonality measure of pn are given in fact the results are presented for the most general case when is an arbitrary not necessarily positive measure

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

bayes risk reduction of estimators using artificial observation noise this paper studies noise enhanced ne estimators which are constructed from an original estimator by artificially adding noise to the observation and computing the expected estimator output by this expectation operation we take into account the neighbourhood of an observation and the resulting ne estimator often has a smaller bayes risk than the original one we derive some general properties of this estimator and also present a method to obtain a suitable approximation of the optimal ne estimator which can be computed numerically by solving a constrained optimization problem finally we study two examples to show the bayes risk improvement that we can obtain from the ne estimator and we compare it to the stochastic resonance estimator

gauss kronrod integration rules for cauchy principal value integrals kronrod extensions to two classes of gauss and lobatto integration rules for the evaluation of cauchy principal value integrals are derived since in one frequently occurring case the kronrod extension involves evaluating the derivative of the integrand a new extension is introduced using n 2 points which requires only values of the integrand however this new rule does not exist for all n and when it does several significant figures are lost in its use 1 introduction in this paper we shall consider kronrod extensions ke to integration rules based on gauss and lobatto points for the evaluation of cauchy principal value cpv integrals of the form fix vyx 1 since the existence theory of ke s for regular integrals insures such extensions for only certain classes of weight functions w x we shall restrict our attention here to the most important class namely w x 1 x2y x 2 where 0 p 2 in the gauss case and p 1 in the lobatto case 9 the only other relevant weight functions are the jacobi weight functions w x 1 x a l x with a i s or s y a f and w x x2 i rx2 oo r 1 5 for our case w x 1 x2 m 1 2 the corresponding orthogonal polynomials are the gegenbauer polynomials c x usually written c x which have the following normalization 11 p 174 2 w x c x cmil x dx 8 mhnil where 3 hn i ft n 2p r p n u n r p r 2p

asymptotic behavior of varying discrete jacobi sobolev orthogonal polynomials in this contribution we deal with a varying discrete sobolev inner product involving the jacobi weight our aim is to study the asymptotic properties of the corresponding orthogonal polynomials and the behavior of their zeros we are interested in mehler heine type formulae because they describe the essential differences from the point of view of the asymptotic behavior between these sobolev orthogonal polynomials and the jacobi ones moreover this asymptotic behavior provides an approximation of the zeros of the sobolev polynomials in terms of the zeros of other well known special functions we generalize some results appeared in the literature very recently

stieltjes polynomials and related quadrature rules in this paper we present a survey of theoretical results some of them new and numerical evidence of others concerning stieltjes polynomials kronrod schemes and their generalizations applied to integrals with a classical nonnegative weight function the use of these schemes in automatic integration routines is briefly outlined

separation theorems for the zeros of certain hypergeometric polynomials we study interlacing properties of the zeros of two contiguous 2f1 hypergeometric polynomials we use the connection between hypergeometric 2f1 and jacobi polynomials as well as a monotonicity property of zeros of orthogonal polynomials due to markoff to prove that the zeros of contiguous hypergeometric polynomials separate each other we also discuss interlacing results for the zeros of 2f1 and those of the polynomial obtained by shifting one of the parameters of 2f1 by t where 0 t 1

gauss kronrod quadrature formulae for weight functions of bernstein szego type ii we study the kronrod extensions of gaussian quadrature rules whose weight functions on 1 l consist of any one of the four chebyshev weights divided by an arbitrary quadratic polynomial that remains positive on 1 11 we show that in almost all cases these extended gauss kronrod quadrature rules have all the desirable properties kronrod nodes interlacing with gauss nodes all nodes contained in 1 11 and all weights positive and representable by semiexplicit formulas exceptions to these properties occur only for small values of n the number of gauss nodes namely n 3 and are carefully identified the precise degree of exactness of each of these gauss kronrod formulae is determined and shown to grow like 4n rather than 3n as is normally the case our findings are the result of a detailed analysis of the underlying orthogonal polynomials and stieltjes polynomials the paper concludes with a study of the limit case of a linear divisor polynomial in the weight function

necessary and sufficient conditions for boundedness of some commutators with weighted lipschitz functions abstract in this paper we show that b lip if and only if the commutator b t of the multiplication operator by b and the singular integral operator t is bounded from l p to l q 1 q where 1 p q 0 1 and 1 q 1 p n also we will obtain that b lip if and only if the commutator b i of the multiplication operator by b and the fractional integral operator i is bounded from l p to l r 1 1 n r where 1 p 0 1 and 1 r 1 p n with 1 p n

modified optimal quadrature extensions optimal extensions of quadrature rules are of importance in the construction of automatic integrators but many sequences fail to exist in usable form the paper considers some techniques for overcoming the problem of inextensibility with a minimal effect on integrating efficiency a modification to the extension procedure proposed recently by begumisa and robinson is shown to be just a special case of the standard theory for quadrature extension some illustrative examples are included

corrigendum generalized chebyshev interpolation and its application to automatic quadrature math comp 41 1983 no 164 537 553 mr0717701 84m 65037 by hasegawa t torii and i ninomiya

speech enhancement using a minimum mean square error short time spectral amplitude estimator this paper focuses on the class of speech enhancement systems which capitalize on the major importance of the short time spectral amplitude stsa of the speech signal in its perception a system which utilizes a minimum mean square error mmse stsa estimator is proposed and then compared with other widely used systems which are based on wiener filtering and the spectral subtraction algorithm in this paper we derive the mmse stsa estimator based on modeling speech and noise spectral components as statistically independent gaussian random variables we analyze the performance of the proposed stsa estimator and compare it with a stsa estimator derived from the wiener estimator we also examine the mmse stsa estimator under uncertainty of signal presence in the noisy observations in constructing the enhanced signal the mmse stsa estimator is combined with the complex exponential of the noisy phase it is shown here that the latter is the mmse estimator of the complex exponential of the original phase which does not affect the stsa estimation the proposed approach results in a significant reduction of the noise and provides enhanced speech with colorless residual noise the complexity of the proposed algorithm is approximately that of other systems in the discussed class

a characterization of positive quadrature formulae a positive quadrature formula with n nodes which is exact for polynomials of degree in r 1 0 r is based on the zeros of certain quasi orthogonal polynomials of degree n we show that the quasi orthogonal polynomials that lead to the positive quadrature formulae can all be expressed as characteristic polynomials of a symmetric tridiagonal matrix with positive subdiagonal entries as a consequence for a fixed n every positive quadrature formula is a gaussian quadrature formula for some nonnegative measure

internality of generalized averaged gaussian quadrature rules and truncated variants for modified chebyshev measures of the second kind abstract generalized averaged gaussian quadrature rules associated with some measure and truncated variants of these rules can be used to estimate the error in gaussian quadrature rules however the former quadrature rules may have nodes outside the interval of integration and therefore it may not be possible to apply them when the integrand is defined on the interval of integration only this paper investigates whether generalized averaged gaussian quadrature rules associated with modified chebyshev measures of the second kind and truncated variants of these rules are internal i e if all nodes of these quadrature rules are in the interval of integration

inverses of generalized vandermonde matrices on demontre plusieurs equations matricielles comprenant des matrices de vandermonde generalisees qui donnent une information algebrique explicite sur les matrices inverses

design of linear equalizers optimized for the structural similarity index we propose an algorithm for designing linear equalizers that maximize the structural similarity ssim index between the reference and restored signals the ssim index has enjoyed considerable application in the evaluation of image processing algorithms algorithms however have not been designed yet to explicitly optimize for this measure the design of such an algorithm is nontrivial due to the nonconvex nature of the distortion measure in this paper we reformulate the nonconvex problem as a quasi convex optimization problem which admits a tractable solution we compute the optimal solution in near closed form with complexity of the resulting algorithm comparable to complexity of the linear minimum mean squared error mmse solution independent of the number of filter taps to demonstrate the usefulness of the proposed algorithm it is applied to restore images that have been blurred and corrupted with additive white gaussian noise as a special case we consider blur free image denoising in each case its performance is compared to a locally adaptive linear mse optimal filter we show that the images denoised and restored using the ssim optimal filter have higher ssim index and superior perceptual quality than those restored using the mse optimal adaptive linear filter through these results we demonstrate that a designing image processing algorithms and in particular denoising and restoration type algorithms can yield significant gains over existing in particular linear mmse based algorithms by optimizing them for perceptual distortion measures and b these gains may be obtained without significant increase in the computational complexity of the algorithm

orthogonal polynomials on sobolev spaces old and new directions abstract during the last years orthogonal polynomials on sobolev spaces have attracted considerable attention algebraic properties distribution of their zeros and fourier expansions as well as their relevance in the analysis of spectral methods for partial differential equations provide a very large field to explore and to compare with the standard case in this paper we present an introductory survey about the subject the origin of the problems and their development show the interest and the promising future of this field

gauss turan quadrature formulas asymptotics of weights first we derive asymptotics for the derivatives of s orthogonal polynomials and the derivatives of cauchy principal value integrals associated with s orthogonal polynomials with the help of these results asymptotics for the quadrature weights of gauss turan quadrature formulas are obtained so far asymptotics have been known with respect to the chebyshev weight function only

quasi orthogonal polynomials quadrature and interpolation abstract we investigate the quasi orthogonal polynomials by expressing them as characteristic polynomials of symmetric matrices many properties of zeros of quasi orthogonal polynomials are derived from this formulation we derive new formulae for quadrature and interpolation based on the zeros of quasi orthogonal polynomials and prove the mean convergence of lagrange interpolation

endpoint estimates for commutators of singular integral operators abstract we prove endpoint estimates for commutatotrs of singular integrals with bmo functions we first show that they satisfy l log l type inequalities and then that there is a hardy space type estimate where the usual atomic hardy space is replaced by an appropriate subspace

on generalized averaged gaussian formulas we present a simple numerical method for constructing the optimal generalized averaged gaussian quadrature formulas which are the optimal stratified extensions of gauss quadrature formulas these extensions exist in many cases in which real positive kronrod formulas do not exist for the jacobi weight functions w x w x 1 x 1 x 1 we give a necessary and sufficient condition on the parameters a and such that the optimal averaged gaussian quadrature formulas are internal

asymptotic behaviour of stieltjes polynomials for ultraspherical weight functions abstract for the ultraspherical weight functions w x 1 x 2 1 2 an asymptotic representation of the stieltjes polynomials is proved for 1 n n the main result is applied to prove positivity results for kronrod extensions of gauss and lobatto quadrature formulae

extended lagrange interpolation in weighted uniform norm the author studies the uniform convergence of extended lagrange interpolation processes based on the zeros of generalized laguerre polynomials

generalized associated polynomials and functions of second kind let pv v n0 pv v v 1 be a sequence of polynomials generated by a three term recurrence relation shifting the recurrence coefficients of the elements of pv v n0 we get a sequence of so called associated polynomials which play an important role in the theory of orthogonal polynomials we generalize this concept of associating for arbitrary polynomials vn n especially if vn is expanded in terms of pv v 0 n their associated polynomials are clenshaw polynomials which are used in numerical mathematics as a consequence of it we present some results from the viewpoint of associated polynomials and from the viewpoint of clenshaw polynomials analogously as for orthogonal polynomials we define functions of second kind for vn we prove some properties of them which depend on the generalized associated polynomials and functions of second kind for orthogonal polynomials

zeros of linear combinations of laguerre polynomials from different sequences we study interlacing properties of the zeros of two types of linear combinations of laguerre polynomials with different parameters namely r n l n a al n a and s n l n a bl n 1 a proofs and numerical counterexamples are given in situations where the zeros of r n and s n respectively interlace or do not in general with the zeros of l k a l k a k n or n 1 the results we prove hold for continuous as well as integral shifts of the parameter a

interlacing properties of the zeros of the orthogonal polynomials and approximation of the hilbert transform abstract in this paper starting from interlacing properties of the zeros of the orthogonal polynomials the authors propose a new method to approximate the finite hilbert transform for this method they give error estimates in uniform norm

an algebraic study of gauss kronrod quadrature formulae for jacobi weight functions we study gauss kronrod quadrature formulae for the jacobi weight function w d t 1 t a 1 t and its special case a f a of the gegenbauer weight function we are interested in delineating regions in the a f plane resp intervals in a for which the quadrature rule has a the interlacing property i e the gauss nodes and the kronrod nodes interlace b all nodes contained in 1 1 c all weights positive d only real nodes not necessarily satisfying a and or b we determine the respective regions numerically for n 1 1 20 4 40 in the gegenbauer case and for n 1 1 10 in the jacobi case where n is the number of gauss nodes algebraic criteria in particular the vanishing of appropriate resultants and discriminants are used to determine the boundaries of the regions identifying properties a and d the regions for properties b and c are found more directly a number of conjectures are suggested by the numerical results finally the gauss kronrod formula for the weight w al 2 is obtained from the one for the weight w ca and similarly the gauss kronrod formula with an odd number of gauss nodes for the weight function w t itlv 1 t2 c is derived from the gauss kronrod formula for the weight w a l y 2

positive interpolatory quadrature rules generated by some biorthogonal polynomials interpolatory quadrature rules whose abscissas are zeros of a biorthogonal polynomial have proved to be useful especially in numerical integration of singular integrands however the positivity of their weights has remained an open question in some cases since 1980 we present a general criterion for this positivity as a consequence we establish positivity of the weights in a quadrature rule introduced by the second author in 1980 generated by a polynomial that is biorthogonal to log x j 0 j n 1

zeros of orthogonal polynomials generated by canonical perturbations of measures abstract in this paper we analyze the behaviour of the zeros of polynomials orthogonal with respect to the uvarov perturbation of a positive borel measure d x when the measure is semiclassical then its electrostatic interpretation is given

positive quadrature formulas iii asymptotics of weights first we discuss briefly our former characterization theorem for positive interpolation quadrature formulas abbreviated qf provide an equivalent characterization in terms of jacobi matrices and give links and applications to other qf in particular to gauss kronrod quadratures and recent rediscoveries then for any polynomial t n which generates a positive qf a weight function depending on n is given with respect to which t n is orthogonal to p n 1 with the help of this result an asymptotic representation of the quadrature weights is derived in general the asymptotic behaviour is different from that of the gaussian weights only under additional conditions do the quadrature weights satisfy the so called circle law corresponding results are obtained for positive qf of radau and lobatto type

positivity of the weights of extended clenshaw curtis quadrature rules we prove that some extended clenshaw curtis quadrature rules have all weights positive we also present extended filippi rules of open type having all weights positive conjectures on the possibility of other positive quadrature rules embedded in clenshaw curtis or filippi rule are suggested

symmetrical orthogonal polynomials for sobolev type inner products abstract in this paper families of symmetric orthogonal polynomials qn with respect to the sobolev type inner product f g ifgd rj 0mj j 0 g j 0 where i is a symmetric interval and is a symmetric positive borel measure with infinite support on i and whose moments are all finite are considered if q2n x un x2 and q2n 1 x xvn x2 we deduce that un and vn are sobolev type orthogonal polynomials and in several particular cases standard orthogonal polynomials we study the zeros of qn showing that in some cases qn has two complex conjugate zeros moreover a partial result about separation of the zeros is given we also discuss the symmetrization problem for this kind of inner products finally some sobolev type inner products with two symmetric mass points are considered

stieltjes polynomials and gauss kronrod quadrature for jacobi weight functions let w beta t 1 t 1 t 1 denote the jacobi weight function for 0 5 2 the stieltjes polynomial has only few real zeros and that gauss kronrod quadrature is not possible so far results of the above type have been known only for ultraspherical weight functions w lambda t big 1 t 2 big lambda 1 2 for 0 2 for 3 the impossibility of gauss kronrod quadrature has been proved by the authors recently

computation of gauss kronrod of quadrature rules recently laurie presented a new algorithm for the computation of 2n 1 point gauss kronrod quadrature rules with real nodes and positive weights this algorithm first determines a symmetric tridiagonal matrix of order 2n 1 from certain mixed moments and then computes a partial spectral factorization we describe a new algorithm that does not require the entries of the tridiagonal matrix to be determined and thereby avoids computations that can be sensitive to perturbations our algorithm uses the consolidation phase of a divide and conquer algorithm for the symmetric tridiagonal eigenproblem we also discuss how the algorithm can be applied to compute kronrod extensions of gauss radau and gauss lobatto quadrature rules throughout the paper we emphasize how the structure of the algorithm makes efficient implementation on parallel computers possible numerical examples illustrate the performance of the algorithm

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

on the associated orthogonal polynomials abstract by using the second order recurrence relation this paper gives some new results on associated orthogonal polynomials without referring to the continued fractions tool some results are very useful for obtaining the second order differential equation satisfied by the semi classical orthogonal polynomials hendriksen and van rossum 1985 maroni 1987 cf section 3 also the main formula derives from proposition 2 6 by which the fourth order differential equation satisfied by some laguerre hahn polynomials magnus 1984 is obtained cf belmehdi and ronveaux 1989 dini 1989 ronveaux 1990

global properties of zeros for sobolev type orthogonal polynomials perez t e and m a pifiar global properties of zeros for sobolev type orthogonal polynomials journal of computational and applied mathematics 49 1993 22 5 232 in this paper we analyze some properties concerning the zeros of orthogonal polynomials q x associated to the inner product f g f xlg x dp x1 t mf c g c nf c g c where i is a not necessarily bounded real interval p is a positive measure on i c e w and m n 2 0 in particular some properties concerning the localization and separation for the roots of q x are obtained

a tool for locating zeros of orthogonal polynomials in sobolev inner product spaces abstract in the theory of polynomials orthogonal with respect to an inner product of the form f 0f x g x d x mk 1akf ik 0 g ik 0 one is confronted with the following situation for certain values of the parameters the orthogonal polynomial of degree n does not have all its zeros inside the support of the distribution function d this paper gives a method to investigate the zero distribution by looking at a type of limiting polynomial for the case m 2 it is shown that there are exactly two zeros outside the true interval of orthogonality for a1 a2 large moreover it is proved that these zeros are nonreal complex conjugates in the case i1 1 i2 also several examples are given

fundamentals of statistical signal processing estimation theory minimum variance unbiased estimation cramer rao lower bound linear models general minimum variance unbiased estimation best linear unbiased estimators maximum likelihood estimation least squares method of moments the bayesian philosophy general bayesian estimators linear bayesian estimators kalman filters summary of estimators extension for complex data and parameters

generalized chebyshev interpolation and its application to automatic quadrature a generalized chebyshev interpolation procedure increasing a fixed number of sample points at a time is developed and analvzed it is incorporated into an efficient automatic quadrature scheme of clenshaw curtis tvpe numerical examples indicate that the present method is efficient not onlv for well behaved functions but for those w ith discontinuous low order derivatives by virtue of adequate error estimation as wvell as saving of sample points

polynomial evaluation and associated polynomials we show a connection between the clenshaw algorithm for evaluating a polynomial q n expanded in terms of a system of orthogonal polynomials and special linear combinations of associated polynomials these results enable us to get the derivatives of q n z analogously to the horner algorithm for evaluating polynomials in monomial representations furthermore we show how a polynomial hat q n z given in monomial representation can be evaluated for z in bbb c using the clenshaw algorithm without complex arithmetic from this we get a connection between zeros of polynomials expanded in terms of chebyshev polynomials and the corresponding polynomials in monomial representation with the same coefficients

asymptotics for varying discrete sobolev orthogonal polynomials abstract we consider a varying discrete sobolev inner product such as f g s f x g x d m n f j c g j c where is a finite positive borel measure supported on an infinite subset of the real line c is adequately located on the real axis j 0 and mn n 0 is a sequence of nonnegative real numbers satisfying a very general condition our aim is to study asymptotic properties of the sequence of orthonormal polynomials with respect to this sobolev inner product in this way we focus our attention on mehler heine type formulae as they describe in detail the asymptotic behavior of these polynomials around c just the point where we have located the perturbation of the standard inner product moreover we pay attention to the asymptotic behavior of the scaled zeros of these varying sobolev polynomials and some numerical experiments are shown finally we provide other asymptotic results which strengthen the idea that mehler heine asymptotics describe in a precise way the differences between sobolev orthogonal polynomials and standard ones

some properties of zeros of sobolev type orthogonal polynomials abstract for polynomials orthogonal with respect to a discrete sobolev product we prove that for each n q n has at least n m zeros on the convex hull of the support of the measure where m denotes the number of terms in the discrete part interlacing properties of zeros are also described

eine fehlerabschatzung fur positive quadraturformeln some well known error estimates for the quadrature formulas of lobatto and radau are shown to be applicable to all positive quadrature formulae of the same degree

error estimates of anti gaussian quadrature formulae anti gauss quadrature formulae associated with four classical chebyshev weight functions are considered complex variable methods are used to obtain expansions of the error in anti gaussian quadrature formulae over the interval 1 1 the kernel of the remainder term in anti gaussian quadrature formulae is analyzed the location on the elliptic contours where the modulus of the kernel attains its maximum value is investigated this leads to effective l error bounds of anti gauss quadratures moreover the effective l 1 error estimates are also derived the results obtained here are an analogue of some results of gautschi and varga 1983 11 gautschi et al 1990 9 and hunter 1995 10 concerning gaussian quadratures

linear combinations of orthogonal polynomials generating positive quadrature formulas let pk x x k e n0 be the polynomials orthogonal on 1 1 with respect to the positive measure da we give sufficient conditions on the real numbers p j 0 m such that the linear combination of orthogonal polynomials yflo jpn j has n simple zeros in 1 1 1 and that the interpolatory quadrature formula whose nodes are the zeros of yfj osjpn j has positive weights

calculation of gauss kronrod quadrature rules the jacobi matrix of the 2n 1 point gauss kronrod quadrature rule for a given measure is calculated efficiently by a five term recurrence relation the algorithm uses only rational operations and is therefore also useful for obtaining the jacobi kronrod matrix analytically the nodes and weights can then be computed directly by standard software for gaussian quadrature formulas

bayesian estimation and prediction using asymmetric loss functions abstract estimators and predictors that are optimal relative to varian s asymmetric linex loss function are derived for a number of well known models their risk functions and bayes risks are derived and compared with those of usual estimators and predictors it is shown that some usual estimators for example a scalar sample mean or a scalar least squares regression coefficient estimator are inadmissible relative to asymmetric linex loss by providing alternative estimators that dominate them uniformly in terms of risk

interlacing of zeros of shifted sequences of one parameter orthogonal polynomials we study the interlacing property of zeros of laguerre polynomials of adjacent degree where the free parameters differ by an integer and of the same degree where the free parameter is shifted continuously similar interlacing results are proven for the positive zeros of gegenbauer polynomials

a fast algorithm for gaussian type quadrature formulae with mixed boundary conditions and some lumped mass spectral approximations after studying gaussian type quadrature formulae with mixed boundary conditions we suggest a fast algorithm for computing their nodes and weights it is shown that the latter are computed in the same manner as in the theory of the classical gauss quadrature formulae in fact all nodes and weights are again computed as eigenvalues and eigenvectors of a real symmetric tridiagonal matrix hence we can adapt existing procedures for generating such quadrature formulae comparative results with various methods now in use are given in the second part of this paper new algorithms for spectral approximations for second order elliptic problems are derived the key to the efficiency of our algorithms is to find an appropriate spectral approximation by using the most accurate quadrature formula which takes the boundary conditions into account in such a way that the resulting discrete system has a diagonal mass matrix hence our algorithms can be used to introduce explicit resolutions for the time dependent problems this is the so called lumped mass method the performance of the approach is illustrated with several numerical examples in one and two space dimensions

new loss functions in bayesian imaging abstract unlike the development of more accurate prior distributions for use in bayesian imaging the design of more sensible estimators through loss functions has been neglected in the literature we discuss the design of loss functions with a local structure that depend only on a binary misclassification vector the proposed approach is similar to modeling with a markov random field the bayes estimate is calculated in a two step algorithm using markov chain monte carlo and simulated annealing algorithms we present simulation experiments with the ising model where the observations are corrupted with gaussian and flip noise

a family of gauss kronrod quadrature formulae etude d une famille de formules de quadrature de gauss kronrad formules explicites des poids