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.     

Potentiometric response of a commercial (Ag2S+PbS)-membrane selective electrode to Hg(II) in aqueous solution

Summary The preparation and performance of an Hg(II)-ISE are described. The analytical behaviour of this electrode, made by chemical treatment of a commercial silver sulphide type Pb(II)-ISE, is discussed in terms of potentialvs. concentration curves, potentialvs. pH curves, and selectivity. The response to fixed c_Hg(II) as the pH is increased in the alkaline region corresponds to decrease in the free Hg²⁺concentration down to pHg 17. The estimated limit of detection of total Hg(II) concentration is 10^–5 M.

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Potentiometrisches Verhalten einer handelsüblichen (Ag₂S+PbS)-Membran-Elektrode gegenüber Hg(tt) in wäßriger Lösung

Zusammenfassung Herstellung und Wirkungsweise einer gegen Hg(II) ionenselektiven Elektrode wurde beschrieben. Das analytische Verhalten dieser Elektrode, die sich durch chemische Behandlung einer handelsüblichen Silbersulfidelektrode, die für Pb(II)-Ionen selektiv ist, herstellen läßt, wurde an Hand von Potential-Konzentrationskurven, Potential-pH-Kurven und ihrer Selektivität diskutiert. Das Verhalten gegenüber fixemc _Hg(ii)bei Anstieg des pH ins alkalische Gebiet korrespondiert mit der Abnahme der Konzentration an freien Hg²⁺-ionen bis herab zu pHg 17. Die Nachweisgrenze liegt schätzungsweise bei einer totalen Hg(II)-Konzentration von 10^–5 M.

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Presented at the Fifth European Conference on Analytical Chemistry (EUROANALYSIS-V), Cracow, Poland, 26–31 August, 1984.     

A moment problem associated to rational Szegő functions

Leta ₁,...,a _p be distinct points in the finite complex plane ?, such that |a _j|>1,j=1,..., p and let \(b_j = 1/\bar \alpha _j ,\) j=1,..., p. Let μ₀, μ _π ^(j) , ν _π ^(j) j=1,..., p;n=1, 2,... be given complex numbers. We consider the following moment problem. Find a distribution ψ on [?π, π], with infinitely many points of increase, such that $$\begin{array}{l} \int_{ - \pi }^\pi {d\psi (\theta ) = \mu _0 ,} \\ \int_{ - \pi }^\pi {\frac{{d\psi (\theta )}}{{(e^{i\theta } - a_j )^n }} = \mu _n^{(j)} ,} \int_{ - \pi }^\pi {\frac{{d\psi (\theta )}}{{(e^{i\theta } - b_j )^n }} = v_n^{(j)} ,} j = 1,...,p;n = 1,2,.... \\ \end{array}$$ It will be shown that this problem has a unique solution if the moments generate a positive-definite Hermitian inner product on the linear space of rational functions with no poles in the extended complex plane ?^* outside {a ₁,...,a _p,b ₁,...,b _p}.     

Preparation of an ion-selective electrode by chemical treatment of copper wire for the measurement of copper(II) and iodide by batch and flow-injection potentiometry

The preparation of an ion-selective electrode by chemical treatment of copper wire and its application for the measurements of copper (II) and iodide ions is described. The proposed reaction mechanism at the sensing surface, which explains the response of the electrode to Cu²⁺ and iodide ions, is discussed. The prepared electrode was suitable for direct potentiometric measurements of iodide and copper (II) in batch experiments down to concentrations of 1 × 10^–5 mol L^–1. A tubular electrode, prepared in the same way, may be used as a potentiometric sensor in a flow-injection analysis for Cu (II) and/or iodide determinations. Received: 4 December 1998 / Revised: 31 March 1999 / Accepted: 6 April 1999     

Preparation of an ion-selective electrode by chemical treatment of copper wire for the measurement of copper(II) and iodide by batch and flow-injection potentiometry

The preparation of an ion-selective electrode by chemical treatment of copper wire and its application for the measurements of copper (II) and iodide ions is described. The proposed reaction mechanism at the sensing surface, which explains the response of the electrode to Cu²⁺ and iodide ions, is discussed. The prepared electrode was suitable for direct potentiometric measurements of iodide and copper (II) in batch experiments down to concentrations of 1 × 10^–5 mol L^–1. A tubular electrode, prepared in the same way, may be used as a potentiometric sensor in a flow-injection analysis for Cu (II) and/or iodide determinations.     

Wall Rational Functions and Khrushchev’s Formula for Orthogonal Rational Functions

The Nevalinna–Pick algorithm yields a continued fraction expansion of every Schur function, whose approximants are identified. These approximants are quotients of rational functions which can be understood as the rational analogs of the Wall polynomials. The properties of these Wall rational functions and the corresponding approximants permit us to obtain a Khrushchev’s formula for orthogonal rational functions. An introduction to the convergence of the Wall approximants in the indeterminate case is presented. This work was partially realized during two stays of the second author at the Norwegian University of Science and Technology (NTNU) financed respectively by Secretaría de Estado de Universidades e Investigación from the Ministry of Education and Science of Spain and by the Department of Mathematical Sciences of NTNU. The work of the second author was also partially supported by the Spanish grants from the Ministry of Education and Science, project code MTM2005-08648-C02-01, and the Ministry of Science and Innovation, project code MTM2008-06689-C02-01, and by Project E-64 of Diputación General de Aragón (Spain).     

The computation of orthogonal rational functions and their interpolating properties

We shall consider nested spacesl _n ,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,l _n = _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.The work of the first author is partially supported by a research grant from the Belgian National Fund for Scientific Research     

Asymptotics for Szegö polynomial zeros

The Wiener-Levinson method and algorithm, formulated here in terms of Szegö polynomials _n ( _N,I ;z) orthogonal on the unit circle, is used to find unknown frequencies _j from anN-sample of a discrete time signal consisting of the superposition of sinusoidal waves with frequencies ₁,...,₁. In a recent paper the authors (and W.J. Thron) have shown that zerosz(j, n, N, I) of _n ( _N,I ;z) converge asN to the critical points ,j=1, 2,...,I, providednn ₀ (I)=2I+L, whereL is 0 or 1. The present paper gives results on the convergence of zerosz(j, n, N, I) to some of the for the case in whichnn ₀ (I), wheren is the degree of _n ( _N,I ;z).Research supported in part by the United States Educational Foundation in Norway (Fulbright Grant), the Norwegian Research Council (NAVF) and the US National Science Foundation under Grant No. DMS-9103141.     

Orthogonal Laurent polynomials and the strong Hamburger moment problem

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)ⁿdψ(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.