157 research outputs found
What is a multiple orthogonal polynomial?
This is an extended version of our note in the Notices of the American
Mathematical Society 63 (2016), no. 9, in which we explain what multiple
orthogonal polynomials are and where they appear in various applications.Comment: 5 pages, 2 figure
Strong asymptotics for Sobolev orthogonal polynomials
14 pages, no figures.-- MSC1991 codes: 42C05, 33C25.Zbl 0937.42011In this paper we obtain the strong asymptotics for the sequence of orthogonal polynomials with respect to the inner product \langle f,g,\rangle=\sum_{k=0}\sp n \int_{\Delta_k} f\sp (k)}(x)g\sp (k)}(x) d\mu_k(x), where \{\mu_k\}_k=0\sp m, with m ∈ Z+ are measures supported on [−1,1] which satisfy Szegö's condition.Research by first author (A.M.F.) was partially supported by a research grant from Dirección General de Enseñanza Superior (DGES) of Spain, project code PB95-1205, a research grant from the European Economic Community, INTAS-93-219-ext, and by Junta de Andalucía, Grupo de Investigación FQM 0229.Publicad
Riemann-Hilbert analysis for Jacobi polynomials orthogonal on a single contour
Classical Jacobi polynomials , with , have a number of well-known properties, in particular the location
of their zeros in the open interval . This property is no longer valid
for other values of the parameters; in general, zeros are complex. In this
paper we study the strong asymptotics of Jacobi polynomials where the real
parameters depend on in such a way that with . We
restrict our attention to the case where the limits are not both positive
and take values outside of the triangle bounded by the straight lines A=0, B=0
and . As a corollary, we show that in the limit the zeros distribute
along certain curves that constitute trajectories of a quadratic differential.
The non-hermitian orthogonality relations for Jacobi polynomials with varying
parameters lie in the core of our approach; in the cases we consider, these
relations hold on a single contour of the complex plane. The asymptotic
analysis is performed using the Deift-Zhou steepest descent method based on the
Riemann-Hilbert reformulation of Jacobi polynomials.Comment: 37 pages, 10 figure
Asymptotics of polynomial solutions of a class of generalized Lamé differential equations
In this paper we study the asymptotic behavior of sequences of Heine-Stieltjes and Van Vleck polynomials for a class of generalized Lamé differential equations connected with certain equilibrium problems on the unit circle.The research of A.M.F. and P.M.G. was partially supported by the Ministry of Science and Technology (MCYT) of Spain through the grant BFM2001-3878-C02-02, and by Junta de Andalucía through Grupo de Investigación FQM 0229. A.M.F. acknowledges also the support of the European Research Network on Constructive Complex Approximation (NeCCA), INTAS 03-51-6637, and of NATO Collaborative Linkage Grant “Orthogonal Polynomials: Theory, Applications and Generalizations,” ref.
PST.CLG.979738. The research of R.O. was partially supported by grants from Spanish MCYT (Research Project BFM2001-3411) and Gobierno Autónomo de Canarias (Research Project PI2002/136)
The semiclassical--Sobolev orthogonal polynomials: a general approach
We say that the polynomial sequence is a semiclassical
Sobolev polynomial sequence when it is orthogonal with respect to the inner
product where is a semiclassical linear functional,
is the differential, the difference or the --difference
operator, and is a positive constant. In this paper we get algebraic
and differential/difference properties for such polynomials as well as
algebraic relations between them and the polynomial sequence orthogonal with
respect to the semiclassical functional . The main goal of this article
is to give a general approach to the study of the polynomials orthogonal with
respect to the above nonstandard inner product regardless of the type of
operator considered. Finally, we illustrate our results by
applying them to some known families of Sobolev orthogonal polynomials as well
as to some new ones introduced in this paper for the first time.Comment: 23 pages, special issue dedicated to Professor Guillermo Lopez
lagomasino on the occasion of his 60th birthday, accepted in Journal of
Approximation Theor
Shannon entropy of symmetric Pollaczek polynomials
We discuss the asymptotic behavior (as ) of the entropic integrals and when is the symmetric Pollaczek weight on with main parameter , and is the corresponding orthonormal polynomial of degree . It is well known that does not belong to the Szeg\H{o} class, which implies in particular that . For this sequence we find the first two terms of the asymptotic expansion. Furthermore, we show that , proving that this ``universal behavior'' extends beyond the Szeg\H{o} class. The asymptotics of has also a curious interpretation in terms of the mutual energy of two relevant sequences of measures associated with 's
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