Ana Peña

On linearly related orthogonal polynomials and their functionals

Let {Pn} be a sequence of polynomials orthogonal with respect a linear functional u and {Qn} a sequence of polynomials defined by Pn(x) + snPn−1(x) = Qn(x) + tnQn−1(x). We find necessary and sufficient conditions in order to {Qn} be a sequence of polynomials orthogonal with respect to a linear functional v. Furthermore we prove that the relation between these linear functionals is (x −˜ a)u = λ(x − a)v. Even more, if u and v are linked in this way we get that {Pn} and {Qn} satisfy a formula as above.

An extension of Milman's reverse Brunn-Minkowski inequality

compact convex sets, followed by an analytical proof by Minkowski [Min]. The inequality (1)for compact sets, not necessarily convex, was first proved by Lusternik [Lu]. A very simple proof ofit can be found in [Pi 1], Ch. 1.It is easy to see that one cannot expect the reverse inequality to hold at all, even if it is perturbedby a fixed constant and we restrict ourselves to balls (i.e. convex symmetric compact sets with theorigin as an interior point). Take for instance A

Full length article: A new approach to the asymptotics of Sobolev type orthogonal polynomials

This paper deals with Mehler-Heine type asymptotic formulas for the so-called discrete Sobolev orthogonal polynomials whose continuous part is given by Laguerre and generalized Hermite measures. We use a new approach which allows to solve the problem when the discrete part contains an arbitrary (finite) number of mass points.

Inequalities for the Gamma function and estimates for the volume of sections of B~p^n

Let B n p = {(x i ) E R n ; Σ n 1 |x i | p < 1} and let E be a k-dimensional subspace of R n . We prove that |E ∩ B n p | 1/k k ≥ |B n p | 1/n n , for 1 ≤ k ≤ (n - 1)/2 and k = n - 1 whenever 1 < p < 2. We also consider 0 < p < 1 and other related cases. We obtain sharp inequalities involving Gamma function in order to get these results.

Discrete Laguerre–Sobolev expansions: A Cohen type inequality☆

Abstract C. Markett proved a Cohen type inequality for the classical Laguerre expansions in the appropriate weighted L p spaces. In this paper, we get a Cohen type inequality for the Fourier expansions in terms of discrete Laguerre–Sobolev orthonormal polynomials with an arbitrary (finite) number of mass points. So, we extend the result due to B.Xh. Fejzullahu and F. Marcellan.

THE THEOREMS OF CARATHEODORY AND GLUSKIN FOR 0 < p < 1

In this note we prove the p-convex analogue of both Caratheodorys convexity theorem and Gluskins theorem concerning the diameter of Minkow- ski compactum. Throughout this note X will denote a real vector space and p will be a real number, 0 0, with Xp + pp = 1. Given A c X, the p-convex hull of A is defined as the intersection of all p-convex sets that contain A. This set is denoted by p-conv(A). A (real) p-normed space (X, \\ • ||) is a (real) vector space equipped with a quasi-norm such that ||x + v||p en for some absolute constant c. Our purpose is to study this problem in the p-convex setting. In (Pe), Peck gave an upper bound of the diameter of JAP , namely, diam(^f ) < n2/p~x. We will show that this bound is optimal (Theorem 2). When proving it, in order to compute some volumetric estimates, it will be necessary to have the corresponding version for p < 1 of Caratheodorys convexity theorem (Theorem 1). The results of this note are the following:

Asymptotics for a generalization of Hermite polynomials

We consider a generalization of the classical Hermite polynomials by the addition of terms involving derivatives in the inner product. This type of generalization has been studied in the literature from the point of view of the algebraic properties. Thus, our aim is to study the asymptotics of this sequence of nonstandard orthogonal polynomials. In fact, we obtain Mehler{Heine type formulas for these polynomials and, as a consequence, we prove that there exists an acceleration of the convergence of the smallest positive zeros of these generalized Hermite polynomials towards the origin.

Connection formulas for general discrete Sobolev polynomials

In this paper the discrete Sobolev inner product { p , q } = ? p ( x ) q ( x ) d µ + ? i = 0 r M i p ( i ) ( c ) q ( i ) ( c ) is considered, where µ is a finite positive Borel measure supported on an infinite subset of the real line, c ? R and ?Mi ? 0,?i = 0, 1, ?, r.Connection formulas for the orthonormal polynomials associated with {., .} are obtained. As a consequence, for a wide class of measures µ, we give the Mehler-Heine asymptotics in the case of the point c is a hard edge of the support of µ. In particular, the case of a symmetric measure µ is analyzed. Finally, some examples are presented.

On linearly related orthogonal polynomials in several variables

Let {ℙn}n≥0

Orthogonal polynomials generated by a linear structure relation: Inverse problem

\{\mathbb{P}_{n}\}_{n\ge 0}