William B. Jones

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.

Orthogonal Laurent Polynomials and Gaussian Quadrature

The purpose of this paper is to introduce a class of orthogonal functions similar in many respects to the classical orthogonal polynomials [6]. These functions are linear combinations of integral (positive, negative and zero) powers of a single complex variable z. Hence they are called Laurent 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].

Szego polynomials applied to frequency analysis

Abstract This paper is concerned with the problem of determining unknown frequencies ω1,…,ωI, using an observed discrete time signal xN={xN(m)} arising from a continuous waveform that is the superposition of a finite number of sinusoidal waves with well-defined frequencies ωj ( periods p j =2πn/ω j , n ϵ Z ) , j=1,…,I. We investigate the Wiener-Levinson method formulated here in terms of Szegő polynomials ϱn(ψN; z) orthogonal on the unit circle with respect to a distribution function ψN(θ) defined by the N observed values of the signal. We prove that if n0 denotes the number of critical points eiwj, then for every n ⩾ n0 and N ⩾ 1, the zeros z(j, n, N) of ϱn(ψN; z) can be arranged so that limN → ∞z(j, n, N) = eiwj for each of the frequencies wj. This result confirms one of the main parts of the conjecture given by Jones, Njastad and Saff (this journal (1990)) on the convergence of zeros of the Szegő polynomials. A related result on the convergence of corresponding two-point Pade approximants is also given.

Accelerating convergence of trigonometric approximations

Lanczos has recently developed a method for accelerating the convergence of trigonometric approximations for smooth, nonperiodic functions by modifying their boundary behavior. The method is reformulated here in terms of interpolation theory and is shown to be related to the theory of Lidstone interpolation. Extensions given include a new type of modifying function and the establishment of criteria for the convergence of associated interpolation series. Applications are given for the error function and its derivative.

Numerical Stability in Evaluating Continued Fractions

A careful analysis of the backward recurrence algorithm for evaluating approxi- mants of continued fractions provides rigorous bounds for the accumulated relative error due to rounding. Such errors are produced by machine operations which carry only a fixed number v of significant digits in the computations. The resulting error bounds are expressed in terms of the machine parameter v. The derivation uses a basic assumption about continued fractions, which has played a fundamental role in developing convergence criteria. Hence, its appear- ance in the present context is quite natural. For illustration, the new error bounds are applied to two large classes of continued fractions, which subsume many expansions of special functions of physics and engineering, including those represented by Stieltjes fractions. In many cases, the results insure numerical stability of the backward recurrence algorithm. 1. Introduction. The analytic theory of continued fractions provides a useful means for representation and continuation of special functions of mathematical physics (1), (2), (10). Many applications of continued fractions and the closely related Pade approximants have recently been made in various areas of numerical analysis and of theoretical physics, chemistry and engineering (4), (5), (7). Thus, it is important to establish a sound understanding of the basic computational problems associated with continued fractions. The present paper is written to help fulfill that aim.

Blaschke product interpolation and its application to the design of digital filters

A new proof is given thatn distinct points on the unit circle can be mapped inton arbitrary points on the unit circle of the complex plane by a finite Blaschke product. A result of this proof is that the mapping can be done with at mostn−1 factors in the product. The problem is studied in the context of its application to frequency transformations used to design digital filters.