Olav Njåstad

Orthogonal Laurent polynomials and the strong Hamburger moment problem

Abstract This paper is concerned with the strong Hamburger moment problem (SHMP): For a given double sequence of real numbers C = { c n } ∞ −∞ , does there exist a real-valued, bounded, non-decreasing function ψ on (−∞, ∞) with infinitely many points of increase such that for every integer n , c n = ∝ ∞ −∞ (− t ) n dψ ( t )? Necessary and sufficient conditions for the existence of such a function ψ are given in terms of the positivity of certain Hankel determinants associated with C. Our approach is made through the study of orthogonal (and quasi-orthogonal) Laurent polynomials (referred to here as L-polynomials) and closely related Gaussian-type quadrature formulas. In the proof of sufficiency an inner product for L-polynomials is defined in terms of the given double sequence C . Since orthogonal L-polynomials are believed to be of interest in themselves, some examples of specific systems are considered.

Two-point Padé expansions for a family of analytic functions☆

Abstract Each member G(z) of a family of analytic functions defined by Stieltjes transforms is shown to be represented by a positive T-fraction, the approximants of which form the main diagonal in the two-point Pade table of G(z). The positive T-fraction is shown to converge to G(z) throughout a domain D(a, b) = [z: z∋[−b, −a]], uniformly on compact subsets. In addition, truncation error bounds are given for the approximants of the continued function; these bounds supplement previously known bounds and apply in part of the domain of G(z) not covered by other bounds. The proofs of our results employ properties of orthogonal L -polynomials (Laurent polynomials) and L -Gaussian quadrature which are of some interest in themselves. A number of examples are considered.

Some results about numerical quadrature on the unit circle

In this paper, quadrature formulas on the unit circle are considered. Algebraic properties are given and results concerning error and convergence established.Finally, numerical experiments are carried out.

The computation of orthogonal rational functions and their interpolating properties

We shall consider nested spacesln,n = 0, 1, 21... of rational functions withn prescribed poles outside the unit disk of the complex plane. We study orthogonal basis functions of these spaces for a general positive measure on the unit circle. In the special case where all poles are placed at infinity,ln =∏n, the polynomials of degree at mostn. Thus the present paper is a study of orthogonal rational functions, which generalize the orthogonal Szegö polynomials. In this paper we shall concentrate on the functions of the second kind which are natural generalizations of the corresponding polynomials.

Orthogonal Laurent polynomials and strong moment theory: a survey

These topics are found in many parts of 20th century mathematics and its applications in mathematical physics, chemistry, statistics and engineering. Historically, the analytic theory of continued fractions has played a central role in both the origin and the development of the other topics. Continued fractions are intimately related to Pad e approximants and special functions. Emphasis is given to the development of strong moment theory and orthogonal Laurent polynomials and to the related continued fractions, quadrature formulas, integral transforms and linear functionals. By a strong moment problem we mean the following: For a given bisequence { n}n=−∞ of real numbers, does there exist a distribution function such that

Szego¨ polynomials associated with Wiener-Levinson filters

Abstract Szego polynomials are studied in connection with Wiener–Levinson filters formed from discrete signals xN={xN(k)}N−1k=0. Our main interest is in the frequency analysis problem of finding the unknown frequencies ωj, when the signal is a trigonometric polynomial x N (k)= ∑ j=−I I α j e i ω j k . Associated with this signal is the sequence of monic Szego polynomials {ρn(ψN; z)}∞n=0 orthogonal on the unit circle with respect to a distribution function ψN(θ). Explicit expressions for the weight function ψ′N(θ) and associated Szego function DN(z) are given in terms of the Z-transform XN(z) of the signal xN. Several theorems are given to support the following conjecture which was suggested by numerical experiments: As N and n increase, the 2I + 1 zeros of ρn(ψN; z) of largest modulus approach the points eiωj. We conclude by showing that the reciprocal polynomials ρ ∗ n (ψ N ; z)≔z n ρ n (ψ N ; 1 z ) are Pade numerators for Pade approximants (of fixed denominator degree) to a meromorphic function related to DN(z).

Continued fractions associated with trigonometric and other strong moment problems

General T-fractions and M-fractions whose approximants form diagonals in two-point Padé tables are subsumed here under the study of Perron-Carathéodory continued fractions (PC-fractions) whose approximants form diagonals in weak two-point Padé tables. The correspondence of PC-fractions with pairs of formal power series is characterized in terms of Toeplitz determinants. For the subclass of positive PC-fractions, it is shown that even ordered approximants converge to Carathéodory functions. This result is used to establish sufficient conditions for the existence of a solution to the trigonometric moment problem and to provide a new starting point for the study of Szegö polynomials orthogonal on the unit circle. Szegö polynomials are shown to be the odd ordered denominators of positive PC-fractions. Positive PC-fractions are also related to Wiener filters used in digital signal processing [3], [25].

Convergence of two-point Pade´ approximants to series of Stieltjes

Abstract In this paper we study the convergence of two-point Pade approximants to Stieltjes series. A Carleman-type condition given by Gragg (1980) in the context of convergence theory for continued fractions is here established in the framework of orthogonal Laurent polynomials (Jones, Njastad and Thron (1984), Jones and Thron (1981), Njastad and Thron (1983, 1986)). A bound for the error is also given.

Orthogonal rational functions and quadrature on the unit circle

In this paper we shall be concerned with the problem of approximating the integralIμ{f}=∫−ππf(eiθ) dμ(θ), by means of the formulaIn{f}=Σj=1nAj(n)f(xj(n)) where μ is some finite positive measure. We want the approximation to be so thatIn{f}=Iμ{f} forf belonging to certain classes of rational functions with prescribed poles which generalize in a certain sense the space of polynomials. In order to get nodes {xj(n)} of modulus 1 and positive weightsAj(n), it will be fundamental to use rational functions orthogonal on the unit circle analogous to Szegő polynomials.

Orthogonal rational functions and interpolatory product rules on the unit circle. III.

Let R be the space of rational functions with poles among f k ; 1= k g 1 k=0 with 0 = 0 and j k j < 1, k 1. We consider a sequence fR n g 1 n=0 of nested subspaces with 1 n=0 R n = R. First we recall from part I how to nd orthogonal bases for R for a positive measure on the unit circle. These are used in the construction of interpolatory quadrature rules for integrals with respect to a complex measure on the unit circle. Integration for the (2n + 1)-point rule is exact for all f 2 R n. Also their convergence is discussed as n ! 1. Finally we discuss the convergence of multipoint rational approximants to the Riesz-Herglotz transform associated with such a complex measure.