Orthogonal rational functions and quadrature on the real half line

In this paper we generalize the notion of orthogonal Laurent polynomials to orthogonal rational functions. Orthogonality is considered with respect to a measure on the positive real line. From this, Gauss-type quadrature formulas are derived and multipoint Padé approximants for the Stieltjes transform of the measure. Convergence of both the quadrature formula and the multipoint Padé approximants is discussed.     

Computation of special functions by Padé approximants with orthogonal polynomial denominators

This paper deals with the computation of special functions of mathematics and physics in the complex domain using continued fraction (one-point or two-point Padé) approximants. We consider three families of continued fractions (Stieltjes fractions, real J-fractions and non-negative T-fractions) whose denominators are orthogonal polynomials or Laurent polynomials. Orthogonality of these denominators plays an important role in the analysis of errors due to numerical roundoff and truncation of infinite sequences of approximants. From the rigorous error bounds described one can determine the exact number of significant decimal digits contained in the approximation of a given function value. Results from computational experiments are given to illustrate the methods.Research supported in part by the National Science Foudation under Grant No. DMS-9302584.     

Gaussian quadrature formulae on the unit circle

Let μ be a probability measure on [0,2π]. In this paper we shall be concerned with the estimation of integrals of the form

Full-size image

For this purpose we will construct quadrature formulae which are exact in a certain linear subspace of Laurent polynomials. The zeros of Szegö polynomials are chosen as nodes of the corresponding quadratures. We will study this quadrature formula in terms of error expressions and convergence, as well as, its relation with certain two-point Padé approximants for the Herglotz–Riesz transform of μ. Furthermore, a comparison with the so-called Szegö quadrature formulae is presented through some illustrative numerical examples.     

Natural solutions of rational Stieltjes moment problems

In the strong or two-point Stieltjes moment problem, one has to find a positive measure on [0,∞) for which infinitely many moments are prescribed at the origin and at infinity. Here we consider a multipoint version in which the origin and the point at infinity are replaced by sequences of points that may or may not coincide. In the indeterminate case, two natural solutions μ₀ and μ_∞ exist that can be constructed by a limiting process of approximating quadrature formulas. The supports of these natural solutions are disjoint (with possible exception of the origin). The support points are accumulation points of sequences of zeros of even and odd indexed orthogonal rational functions. These functions are recursively computed and appear as denominators in approximants of continued fractions. They replace the orthogonal Laurent polynomials that appear in the two-point case. In this paper we consider the properties of these natural solutions and analyze the precise behavior of which zero sequences converge to which support points.     

Chebyshev series method for computing weighted quadrature formulas

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

Sequences of orthogonal Laurent polynomials,bi-orthogonality and quadrature formulas on the unit circle

In this paper, the construction of orthogonal bases in the space of Laurent polynomials on the unit circle is considered. As an application, a connection with the so-called bi-orthogonal systems of trigonometric polynomials is established and quadrature formulas on the unit circle based on Laurent polynomials are studied.     

Sequences of orthogonal Laurent polynomials,bi-orthogonality and quadrature formulas on the unit circle

In this paper, the construction of orthogonal bases in the space of Laurent polynomials on the unit circle is considered. As an application, a connection with the so-called bi-orthogonal systems of trigonometric polynomials is established and quadrature formulas on the unit circle based on Laurent polynomials are studied.     

Talbot quadratures and rational approximations

Many computational problems can be solved with the aid of contour integrals containing e ^z in the integrand: examples include inverse Laplace transforms, special functions, functions of matrices and operators, parabolic PDEs, and reaction-diffusion equations. One approach to the numerical quadrature of such integrals is to apply the trapezoid rule on a Hankel contour defined by a suitable change of variables. Optimal parameters for three classes of such contours have recently been derived: (a) parabolas, (b) hyperbolas, and (c) cotangent contours, following Talbot in 1979. The convergence rates for these optimized quadrature formulas are very fast: roughly O(3^-N ), where N is the number of sample points or function evaluations. On the other hand, convergence at a rate apparently about twice as fast, O(9.28903^-N ), can be achieved by using a different approach: best supremum-norm rational approximants to e ^z for z∈(–∞,0], following Cody, Meinardus and Varga in 1969. (All these rates are doubled in the case of self-adjoint operators or real integrands.) It is shown that the quadrature formulas can be interpreted as rational approximations and the rational approximations as quadrature formulas, and the strengths and weaknesses of the different approaches are discussed in the light of these connections. A MATLAB function is provided for computing Cody–Meinardus–Varga approximants by the method of Carathéodory–Fejér approximation. In memory of Germund Dahlquist (1925–2005).AMS subject classification (2000) 65D30, 41A20     

Rational approximation to harmonic functions

Padé-type approximation is the rational function analogue of Taylor’s polynomial approximation to a power series. A general method for obtaining Padé-type approximants to Fourier series expansions of harmonic functions is defined. This method is based on the Newton-Cotes and Gauss quadrature formulas. Several concrete examples are given and the convergence behavior of a sequence of such approximants is studied. This revised version was published online in June 2006 with corrections to the Cover Date.     

Quadrature formulas for Bessel polynomials

A quadrature formula is a formula computing a definite integration by evaluation at finite points. The existence of certain quadrature formulas for orthogonal polynomials is related to interesting problems such as Waring’s problem in number theory and spherical designs in algebraic combinatorics. Sawa and Uchida proved the existence and the non-existence of certain rational quadrature formulas for the weight functions of certain classical orthogonal polynomials. Classical orthogonal polynomials belong to the Askey-scheme, which is a hierarchy of hypergeometric orthogonal polynomials. Thus, it is natural to extend the work of Sawa and Uchida to other polynomials in the Askey-scheme. In this article, we extend the work of Sawa and Uchida to the weight function of the Bessel polynomials. In the proofs, we use the Riesz–Shohat theorem and Newton polygons. It is also of number theoretic interest that proofs of some results are reduced to determining the sets of rational points on elliptic curves.     

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.     

Recurrence relations for orthogonal rational functions

It is well known that members of families of polynomials, that are orthogonal with respect to an inner product determined by a nonnegative measure on the real axis, satisfy a three-term recursion relation. Analogous recursion formulas are available for orthogonal Laurent polynomials with a pole at the origin. This paper investigates recursion relations for orthogonal rational functions with arbitrary prescribed real or complex conjugate poles. The number of terms in the recursion relation is shown to be related to the structure of the orthogonal rational functions.     

Sharp quadrature formulas and inequalities between various metrics for rational functions

We obtain the sharp quadrature formulas for integrals of complex rational functions over circles, segments of the real axis, and the real axis itself. Among them there are formulas for calculating the L₂-norms of rational functions. Using the quadrature formulas for rational functions, in particular, for simple partial fractions and polynomials, we derive some sharp inequalities between various metrics (Nikol’ski?-type inequalities).     

The Strong Chebyshev Distribution and Orthogonal Laurent Polynomials

The strong Chebyshev distribution and the Chebyshev orthogonal Laurent polynomials are examined in detail. Explicit formulas are derived for the orthogonal Laurent polynomials, uniform convergence of the associated continued fraction is established, and the zeros of the Chebyshev L-polynomials are given. This provides another well-developed example of a sequence of orthogonal L-polynomials     

Hermite‐Padé Approximants for a Pair of Cauchy Transforms with Overlapping Symmetric Supports

Hermite‐Padé approximants of type II are vectors of rational functions with a common denominator that interpolate a given vector of power series at infinity with maximal order. We are interested in the situation when the approximated vector is given by a pair of Cauchy transforms of smooth complex measures supported on the real line. The convergence properties of the approximants are rather well understood when the supports consist of two disjoint intervals (Angelesco systems) or two intervals that coincide under the condition that the ratio of the measures is a restriction of the Cauchy transform of a third measure (Nikishin systems). In this work we consider the case where the supports form two overlapping intervals (in a symmetric way) and the ratio of the measures extends to a holomorphic function in a region that depends on the size of the overlap. We derive Szeg?‐type formulae for the asymptotics of the approximants, identify the convergence and divergence domains (the divergence domains appear for Angelesco systems but are not present for Nikishin systems), and show the presence of overinterpolation (a feature peculiar for Nikishin systems but not for Angelesco systems). Our analysis is based on a Riemann‐Hilbert problem for multiple orthogonal polynomials (the common denominator).© 2016 Wiley Periodicals, Inc.     

Generalized Stieltjes Polynomials and Rational Gauss–Kronrod Quadrature

Generalized Stieltjes polynomials are introduced and their asymptotic properties outside the support of the measure are studied. As applications, we prove the convergence of sequences of interpolating rational functions, whose poles are partially fixed, to Markov functions and give an asymptotic estimate of the error of rational Gauss–Kronrod quadrature formulas when functions which are analytic on some neighborhood of the set of integration are considered.     

Numerical Quadratures for Hadamard Hypersingular Integrals

In this paper,we develop Gaussian quadrature formulas for the Hadamard fi- nite part integrals.In our formulas,the classical orthogonal polynomials such as Legendre and Chebyshev polynomials are used to approximate the density function f(x)so that the Gaussian quadrature formulas have degree n-1.The error estimates of the formulas are obtained.It is found from the numerical examples that the convergence rate and the accu- racy of the approximation results are satisfactory.Moreover,the rate and the accuracy can be improved by choosing appropriate weight functions.     

Anti-Gaussian Padé Approximants

This paper gives a synthesis of Padé approximants and anti-Gaussian quadratures. New rational approximants for Stieltjes series have been constructed. In addition, a three term recurrence relation is given for the numerator and denominator, which is useful when the given functional is not defin ite positive.We give the different algebraic properties of these new polynomials, which are similar to those obtained with the Gaussian quadrature formula. We find an easy definition and several relations with Padé approximants. Finally, some numerical results are given in the last section.     

On the convergence of multipoint Padé-type approximants and quadrature formulas associated with the unit circle

We study the convergence of rational interpolants with prescribed poles on the unit circle to the Herglotz-Riesz transform of a complex measure supported on [–, ]. As a consequence, quadrature formulas arise which integrate exactly certain rational functions. Estimates of the rate of convergence of these quadrature formulas are also included.This research was performed as part of the European project ROLLS under contract CHRX-CT93-0416.     

Rational Approximation to the Exponential Function with Complex Conjugate Interpolation Points

In this paper, we study asymptotic properties of rational functions that interpolate the exponential function. The interpolation is performed with respect to a triangular scheme of complex conjugate points lying in bounded rectangular domains included in the horizontal strip |Im z|<2π. Moreover, the height of these domains cannot exceed some upper bound which depends on the type of rational functions. We obtain different convergence results and precise estimates for the error function in compact sets of that generalize the classical properties of Padé approximants to the exponential function. The proofs rely on, among others, Walsh's theorem on the location of the zeros of linear combinations of derivatives of a polynomial and on Rolle's theorem for real exponential polynomials in the complex domain.