arXiv subjects

Abel Díaz-González

Publications and source records attributed to Abel Díaz-González.

5 recordsLinked to original sources

arXiv2023

Electrostatic models for zeros of Laguerre-Sobolev polynomials

Let {$\{S_n\}_{n\geqslant 0}$} be the sequence of orthogonal polynomials with respect to the Laguerre-Sobolev inner product $$ \langle f,g\rangle_S =\!\int_{0}^{+\infty}\! f(x) g(x)x^αe^{-x}dx+\sum_{j=1}^{N}\sum_{k=0}^{d_j}λ_{j,k} f^{(k)}(c_j)g^{(k)}(c_j), $$ where $λ_{j,k}\geqslant 0$, $α>-1$ and $c_i \in (-\infty, 0)$ for $i=1,2,\dots,N$. We provide a formula that relates the Laguerre-Sobolev polynomials $S_n$ to the standard Laguerre orthogonal polynomials. We find the ladder operators for the polynomial sequence $\{S_n\}_{n\geqslant 0}$ and a second-order differential equation with polynomial coefficients for $\{S_n\}_{n\geqslant 0}$. We establish a sufficient condition for an electrostatic model of the zeros of orthogonal Laguerre-Sobolev polynomials. Some examples are given where this condition is either satisfied or not.

math.CA↗

arXiv2023

Sequentially-ordered Sobolev inner product and Laguerre-Sobolev polynomials

We study the sequence of polynomials $\{S_n\}_{n\geq 0}$ that are orthogonal with respect to the general discrete Sobolev-type inner product $$ \langle f,g \rangle_{\mathsf{s}}=\!\int\! f(x) g(x)dμ(x)+\sum_{j=1}^{N}\sum_{k=0}^{d_j}λ_{j,k} f^{(k)}(c_j)g^{(k)}(c_j), $$ where $μ$ is a finite Borel measure whose support $\suppμ$ is an infinite set of the real line, $λ_{j,k}\geq 0$, and the mass points $c_i$, $i=1,\dots,N$ are real values outside the interior of the convex hull of $\suppμ$ ($c_i\in\RR\setminus\inter{\ch{\suppμ}}$). Under some restriction of order in the discrete part of $\langle \cdot, \cdot \rangle_{\mathsf{s}}$, we prove that $S_n$ has at least $n-d^*$ zeros on $\inter{\ch{\suppμ}}$, being $d^*$ the number of terms in the discrete part of $\langle \cdot, \cdot \rangle_{\mathsf{s}}$. Finally, we obtain the outer relative asymptotic for $\{S_n\}$ in the case that the measure $μ$ is the classical Laguerre measure, and for each mass point, only one order derivative appears in the discrete part of $\langle \cdot, \cdot \rangle_{\mathsf{s}}$.

math.CA↗

arXiv2021

Polynomials of least deviation from zero in 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 distribution of zeros is established on general conditions. Under some order restriction in the discrete part, we prove that, the $n$-th polynomial of least deviation has at least $n-\mathbf{d}^*$ zeros on the convex hull of the support of the measure, where $\mathbf{d}^*$ denotes the number of terms in the discrete part.

math.CV↗

arXiv2020

Discrete-Continuous Jacobi-Sobolev Spaces and Fourier Series

Let $p\geq 1$, $\ell\in \NN$, $α,β>-1$ and $\varpi=(ω_0,ω_1, \dots, ω_{\ell-1})\in \RR^{\ell}$. Given a suitable function $f$, we define the discrete-continuous Jacobi-Sobolev norm of $f$ as: $$ \normSp{f}:= \left(\sum_{k=0}^{\ell-1} \left|f^{(k)}(ω_{k})\right|^{p} + \int_{-1}^{1} \left|f^{(\ell)}(x)\right|^{p} d\Jm(x)\right)^{\frac{1}{p}}, $$ where $ d\Jm(x)=(1-x)^α (1+x)^βdx$. Obviously, $\normSp[2]{\cdot}= \sqrt{\IpS{\cdot}{\cdot}}$, where $\IpS{\cdot}{\cdot}$ is the inner product. $$ \IpS{f}{g}:= \sum_{k=0}^{\ell-1} f^{(k)}(ω_{k}) \, g^{(k)}(ω_{k}) + \int_{-1}^{1} f^{(\ell)}(x) \,g^{(\ell)}(x) d\Jm(x). $$ In this paper, we summarize the main advances on the convergence of the Fourier-Sobolev series, in norms of type $L^p$, cases continuous and discrete. We study the completeness of the Sobolev space of functions associated with the norm $\normSp{\cdot}$ and the denseness of the polynomials. Furthermore, we obtain the conditions for the convergence in $\normSp{\cdot}$ norm of the partial sum of the Fourier-Sobolev series of orthogonal polynomials with respect to $\IpS{\cdot}{\cdot}$ .

math.CA↗

arXiv2019

Rational Approximation and Sobolev-type Orthogonality

In this paper, we study the sequence of orthogonal polynomials $\{S_n\}_{n=0}^{\infty}$ with respect to the Sobolev-type inner product $$\langle f,g \rangle= \int_{-1}^{1} f(x) g(x) \,dμ(x) +\sum_{j=1}^{N} η_{j} \,f^{(d_j)}(c_{j}) g^{(d_j)}(c_{j}), $$ where $μ$ is in the Nevai class $\mathbf{M}(0,1)$, $η_j >0$, $N,d_j \in \mathbb{Z}_{+}$ and $\{c_1,\dots,c_N\}\subset \mathbb{R} \setminus [-1,1]$. Under some restriction of order in the discrete part of $\langle\cdot,\cdot \rangle$, we prove that for sufficiently large $n$ the zeros of $S_n$ are real, simple, $n-N$ of them lie on $(-1,1)$ and each of the mass points $c_j$ ``attracts'' one of the remaining $N$ zeros. The sequences of associated polynomials $\{S_n^{[k]}\}_{n=0}^{\infty}$ are defined for each $k\in \mathbb{Z}_{+}$. We prove an analogous of Markov's Theorem on rational approximation to a function of certain class of holomorphic functions and we give an estimate of the ``speed'' of convergence.

math.CA↗