Christoffel functions for doubling measures on quasismooth curves and arcs

Matching two-sided estimates are given for Christoffel functions associated with a doubling measure ν over a quasismooth curve or arc. The size of the the n-th Christoffel function at a point z is given by the ν-measure of the largest disk about z which lies within the 1/n-level line of the Green’s function. The main theorem contains as special case all previously known weak asymptotics for Christoffel functions, and it also gives their size in explicit form about smooth corners. Applications are given for estimating the size of orthonormal polynomials and for Nikolskii-type inequalities.     

Matrix Christoffel Functions

We introduce and study matrix Christoffel functions for a matrix weight W. We find an explicit expression of the matrix Christoffel functions in terms of any sequence of orthonormal matrix polynomials with respect to W. An extremal property related to the matrix moment problem defined by W is established for the matrix Christoffel functions. We finally find the relative asymptotic behavior of the matrix Christoffel functions associated to matrix weights in the matrix Nevai class.     

Asymptotics of orthogonal polynomials and the numerical solution of ill-posed problems

The paper reviews the impact of modern orthogonal polynomial theory on the analysis of numerical algorithms for ill-posed problems. Of major importance are uniform bounds for orthogonal polynomials on the support of the weight function, the growth of the extremal zeros, and asymptotics of the Christoffel functions.     

Generalized Christoffel functions and error of positive quadrature

The author gives some upper and lower bounds for the generalized Christoffel functions related to a Ditzian-Totik generalized weight. As an application, an error estimate of Gauss quadrature formula inL ₁-weighted norm is derived.Dedicated to Prof. Luigi Gatteschi on the occasion of his 70th birthdayWork sponsored by MURST 40%.     

On three-term recurrence and Christoffel–Darboux identity for orthogonal rational functions on the real line

First, we give an algebraic proof to the Christoffel–Darboux identity of formal orthogonal rational functions on the real line by exposing some underlying algebraic properties. This proof does not involve the three-term recurrence relationship. Besides, it is shown that if a family of rational functions satisfies the Christoffel–Darboux relation, then it also admits a three-term recurrence relationship. Thus, the equivalence between both relations is revealed.     

Perturbations of Christoffel–Darboux Kernels: Detection of Outliers

Foundations of Computational Mathematics - Two central objects in constructive approximation, the Christoffel–Darboux kernel and the Christoffel function, encode ample information about the...     

Asymptotics for Christoffel functions for general measures on the real line

We consider asymptotics of Christoffel functions for measures ν with compact support on the real line. It is shown that under some natural conditionsn times thenth Christoffel function has a limit asn→∞ almost everywhere on the support, and the limit is the Radon-Nikodym derivative of ν with respect to the equilibrium measure of the support of ν. The case in which the support is an interval was settled previously by A. Máté, P. Nevai and the author. The present paper solves the general problem. Work was supported by the National Science Foundation, DMS 9801435 and by the Hungarian National Science Foundation for Research, T/022983.     

Generalized Christoffel functions for power orthogonal polynomials

In this paper we extend the Christoffel functions to the case of power orthogonal polynomials. The existence and uniqueness as well as some properties are given.     

Projective equivalence of Finsler and Riemannian surfaces

In this paper we ask when a Finsler surface is projectively equivalent to a given Riemannian surface and when is a Finsler surface projectively equivalent to some Riemannian surface in general. We obtain a necessary and sufficient condition for projective equivalence in both cases. We then consider the latter condition in terms of the Christoffel symbols of the Riemannian metric and investigate when six functions of two variables are the Christoffel symbols of a Riemannian metric. We employ an exterior differential system to analyze when four functions of two variables are the four projective quantities of a Riemannian metric. We end the paper with a theorem which applies the necessary and sufficient condition to 2-dimensional Randers metrics.     

Orthogonal polynomials for exponential weights on

Let I=[0,d), where d is finite or infinite. Let , where and Q is continuous and increasing on I, with limit ∞ at d. We study the orthonormal polynomials associated with the weight , obtaining bounds on the orthonormal polynomials, zeros, and Christoffel functions. In addition, we obtain restricted range inequalities.     

Asymptotics for Christoffel Functions with Varying Weights

We prove sharp asymptotics for Christoffel functions λ_n(w²ⁿ, · ) with respect to varying weights w²ⁿ. The result has an interpretation in the theory of statistical-mechanical models of quantum systems, and it is sufficiently strong to yield asymptotics for Christoffel functions for weights defined on several intervals.     

CMV Biorthogonal Laurent Polynomials: Perturbations and Christoffel Formulas

Quasidefinite sesquilinear forms for Laurent polynomials in the complex plane and corresponding CMV biorthogonal Laurent polynomial families are studied. Bivariate linear functionals encompass large families of orthogonalities such as Sobolev and discrete Sobolev types. Two possible Christoffel transformations of these linear functionals are discussed. Either the linear functionals are multiplied by a Laurent polynomial, or are multiplied by the complex conjugate of a Laurent polynomial. For the Geronimus transformation, the linear functional is perturbed in two possible manners as well, by a division by a Laurent polynomial or by a complex conjugate of a Laurent polynomial, in both cases the addition of appropriate masses (linear functionals supported on the zeros of the perturbing Laurent polynomial) is considered. The connection formulas for the CMV biorthogonal Laurent polynomials, its norms, and Christoffel–Darboux kernels, in all the four cases, are given. For the Geronimus transformation, the connection formulas for the second kind functions and mixed Christoffel–Darboux kernels are also given in the two possible cases. For prepared Laurent polynomials, i.e., of the form , , these connection formulas lead to quasideterminantal (quotient of determinants) Christoffel formulas for all the four transformations, expressing an arbitrary degree perturbed biorthogonal Laurent polynomial in terms of 2n unperturbed biorthogonal Laurent polynomials, their second kind functions or Christoffel–Darboux kernels and its mixed versions. Different curves are presented as examples, such as the real line, the circle, the Cassini oval, and the cardioid. The unit circle case, given its exceptional properties, is discussed in more detail. In this case, a particularly relevant role is played by the reciprocal polynomial, and the Christoffel formulas provide now with two possible ways of expressing the same perturbed quantities in terms of the original ones, one using only the nonperturbed biorthogonal family of Laurent polynomials, and the other using the Christoffel–Darboux kernels and its mixed versions, as well. Two examples are discussed in detail.     

<Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">m</Emphasis></Subscript> Extemal Polynomials Associated with Generalized Jacobi Weights

Asymptotic estimations of the Christoffel type functions for Lm extremal polynomials with an even integer m associated with generalized Jacobi weights are established. Also, asymptotic behavior of the zeros of the Lm extremal polynomials and the Cotes numbers of the corresponding Turan quadrature formula is given.     

Christoffel functions and universality on the boundary of the ball

We establish asymptotics for Christoffel functions, and universality limits, associated with multivariate orthogonal polynomials, on the boundary of the unit ball in ? ^d .     

Asymptotics of Christoffel functions on arcs and curves

For a system of smooth Jordan curves and arcs asymptotics for Christoffel functions is established. A separate new method is developed to handle the upper and lower estimates. In the course to the upper bound a theorem of Widom on the norm of Chebyshev polynomials is generalized.     

How poles of orthogonal rational functions affect their Christoffel functions

We show that even a relatively small number of poles of a sequence of orthogonal rational functions approaching the interval of orthogonality, can prevent their Christoffel functions from having the expected asymptotics. We also establish a sufficient condition on the rate for such asymptotics, provided the rate of approach of the poles is sufficiently slow. This provides a supplement to recent results of the authors where poles were assumed to stay away from the interval of orthogonality.     

Universality for locally Szegő measures

In this paper, we use asymptotic estimates of the Christoffel functions associated with regular measures satisfying Szegő’s condition locally to extend a recent universality result by D.S. Lubinsky. As a consequence, we obtain under the same conditions an extension of a very precise zero-spacing result of Levin and Lubinsky.     

The Christoffel function for the Hermite weight is bell-shaped

We show that the Christoffel function λ_n associated with the Hermite weight function w_H(x)=exp(−x²) is bell-shaped. As a consequence, we describe completely how the weights in a Gauss-type quadrature formula associated with w_H(x) are arranged in magnitude.     

Turán Quadrature Formulas and Christoffel Type Functions for the Chebyshev Polynomials of the Second Kind

An explicit representation for the Cotes numbers of Turán quadrature formulas based on the zeros of the Chebyshev polynomials of the second kind and its asymptotic behavior are given. The asymptotic formula for the corresponding Christoffel type functions is also provided. This revised version was published online in June 2006 with corrections to the Cover Date.     

On sensitivity of Gauss–Christoffel quadrature

In numerical computations the question how much does a function change under perturbations of its arguments is of central importance. In this work, we investigate sensitivity of Gauss–Christoffel quadrature with respect to small perturbations of the distribution function. In numerical quadrature, a definite integral is approximated by a finite sum of functional values evaluated at given quadrature nodes and multiplied by given weights. Consider a sufficiently smooth integrated function uncorrelated with the perturbation of the distribution function. Then it seems natural that given the same number of function evaluations, the difference between the quadrature approximations is of the same order as the difference between the (original and perturbed) approximated integrals. That is perhaps one of the reasons why, to our knowledge, the sensitivity question has not been formulated and addressed in the literature, though several other sensitivity problems, motivated, in particular, by computation of the quadrature nodes and weights from moments, have been thoroughly studied by many authors. We survey existing particular results and show that even a small perturbation of a distribution function can cause large differences in Gauss–Christoffel quadrature estimates. We then discuss conditions under which the Gauss–Christoffel quadrature is insensitive under perturbation of the distribution function, present illustrative examples, and relate our observations to known conjectures on some sensitivity problems. The work of the first author was supported by the National Science Foundation under Grants CCR-0204084 and CCF-0514213. The work of the other two authors was supported by the Program Information Society under project 1ET400300415 and by the Institutional Research Plan AV0Z100300504. P. Tichy in the years 2003–2006 on leave at the Institute of Mathematics, TU Berlin, Germany.