Bounding the support of a measure from its marginal moments

Given all moments of the marginals of a measure on Rn, one provides (a) explicit bounds on its support and (b), a numerical scheme to compute the smallest box that contains the support. It consists of solving a hierarchy of generalized eigenvalue problems associated with some Hankel matrices.Comment: To appear in Proc. Amer. Math. So

Computing the Hessenberg matrix associated with a self-similar measure

We introduce in this paper a method to calculate the Hessenberg matrix of a sum of measures from the Hessenberg matrices of the component measures. Our method extends the spectral techniques used by G. Mantica to calculate the Jacobi matrix associated with a sum of measures from the Jacobi matrices of each of the measures. We apply this method to approximate the Hessenberg matrix associated with a self-similar measure and compare it with the result obtained by a former method for self-similar measures which uses a fixed point theorem for moment matrices. Results are given for a series of classical examples of self-similar measures. Finally, we also apply the method introduced in this paper to some examples of sums of (not self-similar) measures obtaining the exact value of the sections of the Hessenberg matrix

On computational aspects of discrete Sobolev inner products on the unit circle

In this paper, we show how to compute in O(n² )steps the Fourier coefficients associated with the Gelfand-Levitan approach for discrete Sobolev orthogonal polynomials on the unit circle when the support of the discrete component involving derivatives is located outside the closed unit disk. As a consequence, we deduce the outer relative asymptotics of these polynomials in terms of those associated with the original orthogonality measure. Moreover, we show how to recover the discrete part of our Sobolev inner product.The research of K. Castillo was supported by CNPq Program/Young Talent Attraction, Ministério da Ciência, Tecnologia e Inovação of Brazil, Project 370291/2013 1. The research of K. Castillo and F. Marcellán was supported by Dirección General de Investigación, Ministerio de Economía y Competitividad of Spain, Grant MTM2012 36732 C03 01. F. Marcellán also acknowledges the financial support of CAPES Program/Special Visiting Researcher by Ministério da Educação of Brasil, Project 107/201

Ratio and Plancherel-Rotach asymptotics for Meixner-Sobolev orthogonal polynomials

13 pages, no figures.-- MSC codes: 42C05; 33C25; 39A10.MR#: MR1741786 (2000k:42032)Zbl#: Zbl 0984.42016We study the analytic properties of the monic Meixner-Sobolev polynomials $\{Q_n\}$ orthogonal with respect to the inner product involving differences $$(p,q)_S=\sum infty_{i=0}[p(i)q(i)+\lambda\Delta p(i)\Delta q(i)] {\mu (\gamma)_i\over i!},$$ $\gamma>0,\ 0<\mu<1$, where $\lambda\geq0,\ \Delta$ is the forward difference operator $(\Delta f(x)=f(x+1)-f(x))$ and $(\gamma)_n$ denotes the Pochhammer symbol. Relative asymptotics for Meixner-Sobolev polynomials with respect to Meixner polynomials is obtained. This relative asymptotics is also given for the scaled polynomials. Moreover, a zero distribution for the scaled Meixner-Sobolev polynomials and Plancherel-Rotach asymptotics for $\{Q_n\}$ are deduced.The work of E.G. has been partially supported by Dirección General de Enseñanza Superior (DGES) of Spain under Grant PB-96-0952. The work of F.M. is partially supported by PB96-0120-C03-01 and INTAS-93-0219 Ext. The work of J.J.M.-B. is partially supported by Junta de Andalucía, G.I. FQM0229.Publicad