# Hector Pijeira CabreraUniversity Carlos III de Madrid | UC3M · Department of Mathematics

Hector Pijeira Cabrera

PhD

## About

38

Publications

3,625

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277

Citations

Citations since 2016

Introduction

Additional affiliations

June 2006 - present

September 1993 - February 2006

## Publications

Publications (38)

We study the relation between certain non-degenerate lower Hessenberg infinite matrix $\mathcal{G}$ and the existence of sequences of orthogonal polynomials with respect to Sobolev inner products. In other words, we extend the well-known Favard theorem for the different cases of Sobolev orthogonality. The structure of $\mathcal{G}$ and the associat...

The first part of this paper complements previous results on characterization of polynomials of least deviation from zero in Sobolev p -norm ( $$1<p<\infty $$ 1 < p < ∞ ) for the case $$p=1$$ p = 1 . Some relevant examples are indicated. The second part deals with the location of zeros of polynomials of least deviation in discrete Sobolev p -norm....

The first part of this paper complements previous results on characterization of polynomials of least deviation from zero in Sobolev $p$-norm ($1<p<\infty$) for the case $p=1$. Some relevant examples are indicated. The second part deals with the location of zeros of polynomials of least deviation in discrete Sobolev $p$-norm. The asymptotic distrib...

In this paper, we study the sequence of orthogonal polynomials {Sn}n=0∞ with respect to the Sobolev-type inner product 〈f,g〉=∫−11f(x)g(x)dμ(x)+∑j=1Nηjf(dj)(cj)g(dj)(cj)where μ is a finite positive Borel measure whose support suppμ⊂[−1,1] contains an infinite set of points, ηj>0, N,dj∈Z+ and {c1,…,cN}⊂R∖[−1,1]. Under some restriction of order in the...

We study the zero location and the asymptotic behavior of iterated integrals of polynomials. Borwein–Chen–Dilcher’s polynomials play an important role in this issue. For these polynomials we find their strong asymptotics and give the limit measure of their zero distribution. We apply these results to describe the zero asymptotic distribution of ite...

Let \(p\ge 1, \ell \in \mathbb {N}, \alpha ,\beta >-1\) and \(\varpi =(\omega _0,\omega _1, \ldots , \omega _{\ell -1})\in \mathbb {R}^{\ell }\). Given a suitable function f, we define the discrete–continuous Jacobi–Sobolev norm of f as: $$\begin{aligned} \Vert f \Vert _{{\scriptscriptstyle \mathsf {s}},{\scriptscriptstyle p}}:= \left( \sum _{k=0}^...

We study the zero location and the asymptotic behavior of iterated integrals of polynomials. Borwein-Chen-Dilcher's polynomials play an important role in this issue. For these polynomials we find their strong asymptotics and give the limit measure of their zero distribution. We apply these results to describe the zero asymptotic distribution of ite...

Let $p\geq 1$, $\ell\in \mathbb{N}$, $\alpha,\beta>-1$ and $\varpi=(\omega_0,\omega_1, \dots, \omega_{\ell-1})\in \mathbb{R}^\ell$. Given a suitable function $f$, we define the discrete-continuous Jacobi-Sobolev norm of $f$ as: $$\Vert f\Vert_{s,p} := \left(\sum_{k=0}^{\ell-1} \left|f^{(k)}(\omega_{k})\right|^{p} + \int_{-1}^{1} \left|f^{(\ell)}(x)...

Let Pn(α,β) be the n-th monic Jacobi polynomial with α,β>-1. Given m numbers ω1,…,ωm∈C\[-1,1], let Ωm=(ω1,…,ωm) and Pn,m,Ωm(α,β) be the m-th iterated integral of (n+m)!n!Pn(α,β) normalized by the conditions dkPn,m,Ωm(α,β)dzk(ωm-k)=0,fork=0,1,…,m-1.The main purpose of the paper is to study the algebraic and asymptotic properties of the sequence of m...

In this paper, we study the sequence of orthogonal polynomials $\{S_n\}_{n=0}^{\infty}$ with respect to the Sobolev-type inner product $$\langle f,g \rangle= \int_{-1}^{1} f(x) g(x) \,d\mu(x) +\sum_{j=1}^{N} \eta_{j} \,f^{(d_j)}(c_{j}) g^{(d_j)}(c_{j}), $$ where $\mu$ is in the Nevai class $\mathbf{M}(0,1)$, $\eta_j >0$, $N,d_j \in \mathbb{Z}_{+}$...

We consider extremal polynomials with respect to a Sobolev-type [Formula: see text]-norm, with [Formula: see text] and measures supported on compact subsets of the real line. For a wide class of such extremal polynomials with respect to mutually singular measures (i.e. supported on disjoint subsets of the real line), it is proved that their critica...

We consider extremal polynomials with respect to a Sobolev-type $p$-norm, with $1<p<\infty$ and measures supported on compact subsets of the real line. For a wide class of such extremal polynomials with respect to mutually singular measures (i.e. supported on disjoint subsets of the real line), it is proved that their critical points are simple and...

Discrete orthogonal polynomials are useful tools in digital image processing to extract visual object contours in different application contexts. This paper proposes an alternative method that extends beyond classic first-order differential operators, by using the properties of Krawtchouk orthogonal polynomials to achieve a first order differential...

Let $\mu$ be a finite positive Borel measure supported on R, $\LL[f]
=xf''+(\alpha+1-x)f'$ with $\alpha>-1$, or $\LL[f] =\frac{1}{2}f''-xf'$, and
$m$ a natural number. We study algebraic, analytic and asymptotic properties of
the sequence of monic polynomials $\{Q_n\}_{n>m}$ orthogonal with respect to
the operator $\LL$. We also provide a fluid dyn...

In this paper we consider the sequences of polynomials orthogonal with respect to the Laguerre measure modified by m Dirac mass points located in the negative real semiaxis. We first focus our attention in the representation of these polynomials in terms of the standard Laguerre polynomials. Next we find the explicit formula for their outer relativ...

It is very well known that a sequence of polynomials orthogonal with respect to a Sobolev discrete inner product where μ is a finite Borel measure and I is an interval of the real line, satisfies a five-term recurrence relation. In this contribution we study other three families of polynomials which are linearly independent solutions of such a five...

Let p, be a finite positive Borel measure on [-1, 1], m a fixed natural number and L-(alpha,L-beta)[f]= (1-x(2))f ''+ (beta - alpha - (alpha + beta + 2)x)f', with alpha, beta > -1. We study algebraic and analytic properties of the sequence of monic polynomials (Q)(n>m) that satisfy the orthogonality relations integral L-(alpha,L-beta)[Q(n)](x)x(k)d...

Sobolev formal orthogonality on harmonic algebraic curves (Im(h(z))=0, h(z)∈ℂ[z]) and equipotential curves (|h(z)|=1, h(z)∈ℂ[z]) is defined and studied. In each case, such a study is done through the characterization of bilinear forms whose associated functional annihilates at the multiples of (h(z)-h(z) ¯) 2n+1 for harmonic algebraic curves, or th...

In this paper, we study some algebraic, differential and asymptotic properties of the orthogonal polynomials with respect to discrete–continuous Sobolev-type inner product with respect to Gegenbauer measures. These polynomials are also primitives of Gegenbauer polynomials. A hydrodynamic model for source points location of a flow of an incompressib...

We obtain the (contracted) weak zero asymptotics for orthogonal polynomials with respect to Sobolev inner products with exponential weights in the real semiaxis, of the form , with γ>0, which include as particular cases the counterparts of the so-called Freud (i.e., when φ has a polynomial growth at infinity) and Erdös (when φ grows faster than any...

We study the asymptotic behavior of the zeros of a sequence of polynomials whose weighted norms, with respect to a sequence of weight functions, have the same nth root asymptotic behavior as the weighted norms of certain extremal polynomials. This result is applied to obtain the (contracted) weak zero distribution for orthogonal polynomials with re...

We obtain the strong asymptotics for the sequence of monic polynomials minimizing the norm{norm of matrix} q {norm of matrix}S = (underover(∑, k = 0, N) {norm of matrix} q(k) {norm of matrix}k2)1 / 2, where {norm of matrix} ṡ {norm of matrix}k, k = 0, ..., N - 1, are L2 norms with respect to measures supported on the same rectifiable Jordan closed...

We introduce a new class of polynomials {Pn}, that we call polar Legendre polynomials, they appear as solutions of an inverse Gauss problem of equilibrium position of a field of forces with n + 1 unit masses. We study algebraic, differential and asymptotic properties of this class of polynomials, that are simultaneously orthogonal with respect to a...

For a wide class of Sobolev type norms with respect to measures with unbounded support on the real line, the contracted zero distribution and the logarithmic asymptotic of the corresponding re-scaled Sobolev orthogonal polynomials is given.

We study the zero location and asymptotic zero distribution of sequences of polynomials which satisfy an extremal condition with respect to a norm given on the space of all polynomials.

7 pages, no figures.-- MSC2000 codes: 42C05, 33C25. We study the zero location and the asymptotic behavior of the primitives of the standard orthogonal polynomials with respect to a finite positive Borel measure concentrate on [−1,1]. Research of second (H.P.) and third (W.U.) authors partially supported by a research grant # PI 03-14-4744-2000 fro...

Decimocuarta Escuela Venezolana de Matemáticas Incluye bibliografía

Sobolev orthogonal polynomials with respect to measures supported on compact subsets of the complex plane are considered. For a wide class of such Sobolev orthogonal polynomials, it is proved that their zeros are contained in a compact subset of the complex plane and their asymptotic-zero distribution is studied. We also find the nth-root asymptoti...

We consider the Sobolev orthogonal polynomials associated to the Jacobi’s measure on [-1,1]. It is proved that for the class of monic Jacobi-Sobolev orthogonal polynomials, the smallest closed interval that contains its real zeros is [-1+2C,1+2C] with C a constant explicitly determined. The asymptotic distribution of those zeroes is studied and als...

In this paper we obtain the strong asymptotics for the sequence of orthogonal polynomials with respect to the inner product
f,g
s = åk - 0m òDk f( k ) ( x )g( k ) ( x )dmk ( x )\left\langle {f,g} \right\rangle s = \sum\limits_{k - 0}^m {\int\limits_{\Delta _k } {f^{\left( k \right)} \left( x \right)g^{\left( k \right)} \left( x \right)d\mu \kappa...

The concepts of definite and determinate Sobolev moment problem are introduced. The study of these questions is reduced to the definiteness or determinacy, respectively, of a system of classical moment problems by means of a canonical decomposition of the moment matrix associated with a Sobolev inner product in terms of Hankel matrices.

For a wide class of Sobolev orthogonal polynomials, it is proved that their zeros are contained in a compact subset of the complex plane and the asymptotic zero distribution is obtained. With this information, the nth root asymptotic behavior outside the compact set containing all the zeros is given.

17 pages, no figures.-- MSC2000 code: 44A60. MR#: MR1715024 (2000m:44013) Zbl#: Zbl 0980.44008 The concepts of definite and determinate Sobolev moment problem are introduced. The study of these questions is reduced to the definiteness or determinacy, respectively, of a system of classical moment problems by means of a canonical decomposition of the...

El objetivo de estudio se centra en las propiedades asintómicas y sus consecuencias, Para ello se estudia el problema de momentos para productos de Sokolev. Tambien se abordan las propiedades asintóticas de los polinomios ortogonales de Sobolev, considerando clases amplias de medidas. El estudio del comportamiento de los polinomios ortogonales, cua...

We study the asymptotic behaviour of the monic orthogonal polynomials with respect to the Gegenbauer-Sobolev inner product (f,g)s = 〈f,g〉 + λ〈f′,g′〉 where with and λ>0. The asymptotics of the zeros and norms of these polynomials are also established.

23 pages, no figures. [ES] En este trabajo se da una condición suficiente para la convergencia de la diagonal principal de la tabla de aproximantes multipuntuales de Padé asociada a funciones meromorfas del tipo μ'+r, donde r es una fracción racional con coeficientes complejos y polos en C\[a,b], r(∞)=0 y μ'= μ*(1/z), siendo μ una medida positiva d...