Domain of validity of Szegö quadrature formulas

AbstractAs is well known, the n-point Szegö quadrature formula integrates correctly any Laurent polynomial in the subspace span{1/zn-1,…,1/z,1,z,…,zn-1}. In this paper we enlarge this subspace. We prove that a set of 2n linearly independent Laurent polynomials are integrated correctly. The obtained result is used for the construction of Szegö quadrature formulas. Illustrative examples are given

Multipoint Schur algorithm and orthogonal rational functions: convergence properties, I

Classical Schur analysis is intimately connected to the theory of orthogonal polynomials on the circle [Simon, 2005]. We investigate here the connection between multipoint Schur analysis and orthogonal rational functions. Specifically, we study the convergence of the Wall rational functions via the development of a rational analogue to the Szeg\H o theory, in the case where the interpolation points may accumulate on the unit circle. This leads us to generalize results from [Khrushchev,2001], [Bultheel et al., 1999], and yields asymptotics of a novel type.Comment: a preliminary version, 39 pages; some changes in the Introduction, Section 5 (Szeg\H o type asymptotics) is extende

The impact of Stieltjes' work on continued fractions and orthogonal polynomials

Stieltjes' work on continued fractions and the orthogonal polynomials related to continued fraction expansions is summarized and an attempt is made to describe the influence of Stieltjes' ideas and work in research done after his death, with an emphasis on the theory of orthogonal polynomials

Orthogonal Rational Functions and Interpolatory Product Rules on the Unit Circle. - III: Convergence of General Sequences

: Let R be the space of rational functions with poles among fff k ; 1=¯ff k g 1 k=0 with ff 0 = 0 and jff 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. We continue our investigation of the convergence as n ! 1 of quadrature rules which are exact in R n . In Part II we have discussed the convergence for a particular nesting of the subspaces R n . In this part we prove similar convergence results for a more general sequence of subspaces. By similar arguments as in part II, we can also here derive related results about the convergence of multipoint rational approximants to the RieszHerglotz transform associated with a complex measure. Keywords: orthogonal rational functions, multipoint Pad'e approximants, numerical quadrature. AMS Classification: 41A55, 33C45 1 Introduction We use T= fz 2 C : jzj = 1g, D = fz 2 C : jzj ! 1g, E = fz 2 C : jzj ? 1g to denote the unit circle, its interior and its exterior in the complex plane C . We s..

A Rational Moment Problem on the Unit Circle

Let fff k g 1 k=1 be a sequence of not necessarily distinct points on the complex unit circle. We consider the moment problem where it is to find a positive measure on [\Gammaß; ß] such that for ! 0 = 1 and ! n (z) = (z \Gamma ff 1 ) \Delta \Delta \Delta (z \Gamma ff n ), n = 1; 2; : : : we have Z ß \Gammaß d¯(`) = 1; Z ß \Gammaß d¯(`) ! n (e i` ) = ¯ n ; n = 1; 2; : : : for a given sequence of moments f¯ n g 1 n=0 . This paper gives results which to some extend generalise the limit point - limit circle situation of classical moment problems. Keywords: general moment problem, orthogonal rational functions, nested disks. AMS Classification: 30E05, 42C10, 42A99, 47A57 1 Introduction Let T = fz 2 C : jzj = 1g be the complex unit circle and let fff k g 1 k=1 be a sequence of not necessarily distinct points on Tn f1g. Introduction of the "forbidden" point 1 is not a severe restriction because there is only a countable number of ff k 's so that there always exists such a poi..

An indeterminate rational moment problem and Carathéodory functions

AbstractLet {αn}n=1∞ be a sequence of points in the open unit disk in the complex plane and letB0=1andBn(z)=∏k=0nαk¯|αk|αk-z1-αk¯z,n=1,2,…,(αk¯/|αk|=-1 when αk=0). We put L=span{Bn:n=0,1,2,…} and we consider the following “moment” problem:Given a positive-definite Hermitian inner product 〈·,·〉 in L, find all positive Borel measures ν on [-π,π) such that〈f,g〉=∫-ππf(eiθ)g(eiθ)¯dν(θ)forf,g∈L.We assume that this moment problem is indeterminate. Under some additional condition on the αn we will describe a one-to-one correspondence between the collection of all solutions to this moment problem and the collection of all Carathéodory functions augmented by the constant ∞