arXiv subjects

J. L. Varona

Publications and source records attributed to J. L. Varona.

8 recordsLinked to original sources

arXiv2025

A class of weighted Delannoy numbers

The weighted Delannoy numbers are defined by the recurrence relation $f_{m,n}=αf_{m-1,n}+ βf_{m,n-1}+ γf_{m-1,n-1}$ if $m n>0 $, with $f_{m,n}=α^m β^n$ if $n m=0$. In this work, we study a generalization of these numbers considering the same recurrence relation but with $f_{m,n}=A^m B^n$ if $n m=0$. More particularly, we focus on the diagonal sequence $f_{n,n}$. With some ingenuity, we are able to make use of well-established methods by Pemantle and Wilson, and by Melczer in order to determine its asymptotic behavior in the case $A,B,α,β,γ\geq 0$. In addition, we also study its P-recursivity with the help of symbolic computation tools.

math.CO↗

arXiv2018

Nonlocal discrete diffusion equations and the fractional discrete Laplacian, regularity and applications

The analysis of nonlocal discrete equations driven by fractional powers of the discrete Laplacian on a mesh of size $h>0$ \[ (-Δ_h)^su=f, \] for $u,f:\mathbb{Z}_h\to\mathbb{R}$, $0<s<1$, is performed. The pointwise nonlocal formula for $(-Δ_h)^su$ and the nonlocal discrete mean value property for discrete $s$-harmonic functions are obtained. We observe that a characterization of $(-Δ_h)^s$ as the Dirichlet-to-Neumann operator for a semidiscrete degenerate elliptic local extension problem is valid. Regularity properties and Schauder estimates in discrete Hölder spaces as well as existence and uniqueness of solutions to the nonlocal Dirichlet problem are shown. For the latter, the fractional discrete Sobolev embedding and the fractional discrete Poincaré inequality are proved, which are of independent interest. We introduce the negative power (fundamental solution) \[ u=(-Δ_h)^{-s}f, \] which can be seen as the Neumann-to-Dirichlet map for the semidiscrete extension problem. We then prove the discrete Hardy--Littlewood--Sobolev inequality for $(-Δ_h)^{-s}$. As applications, the convergence of our fractional discrete Laplacian to the (continuous) fractional Laplacian as $h\to0$ in Hölder spaces is analyzed. Indeed, uniform estimates for the error of the approximation in terms of $h$ under minimal regularity assumptions are obtained. We finally prove that solutions to the Poisson problem for the fractional Laplacian \[ (-Δ)^sU=F, \] in $\mathbb{R}$, can be approximated by solutions to the Dirichlet problem for our fractional discrete Laplacian, with explicit uniform error estimates in terms of~$h$.

math.AP↗

arXiv2015

Fractional discrete Laplacian versus discretized fractional Laplacian

We define and study some properties of the fractional powers of the discrete Laplacian $$(-Δ_h)^s,\quad\hbox{on}~\mathbb{Z}_h = h\mathbb{Z},$$ for $h>0$ and $0<s<1$. A comparison between our fractional discrete Laplacian and the \textit{discretized} continuous fractional Laplacian as $h\to0$ is carried out. We get estimates in $\ell^\infty$ for the error of the approximation in terms of $h$ under minimal regularity assumptions. Moreover, we provide a pointwise formula with an explicit kernel and deduce Hölder estimates for $(-Δ_h)^s$. A study of the negative powers (or discrete fractional integral) $(-Δ_h)^{-s}$ is also sketched. Our analysis is mainly performed in dimension one. Nevertheless, we show certain asymptotic estimates for the kernel in dimension two that can be extended to higher dimensions. Some examples are plotted to illustrate the comparison in both one and two dimensions.

math.AP↗

arXiv2014

Harmonic Analysis associated with a discrete Laplacian

It is well-known that the fundamental solution of $$ u_t(n,t)= u(n+1,t)-2u(n,t)+u(n-1,t), \quad n\in\mathbb{Z}, $$ with $u(n,0) =δ_{nm}$ for every fixed $m \in\mathbb{Z}$, is given by $u(n,t) = e^{-2t}I_{n-m}(2t)$, where $I_k(t)$ is the Bessel function of imaginary argument. In other words, the heat semigroup of the discrete Laplacian is described by the formal series $$ W_tf(n) = \sum_{m\in\mathbb{Z}} e^{-2t} I_{n-m}(2t) f(m). $$ By using semigroup theory, this formula allows us to analyze some operators associated with the discrete Laplacian. In particular, we obtain the maximum principle for the discrete fractional Laplacian, weighted $\ell^p(\mathbb{Z})$-boundedness of conjugate harmonic functions, Riesz transforms and square functions of Littlewood-Paley. Interestingly, it is shown that the Riesz transforms coincide essentially with the so called discrete Hilbert transform defined by D. Hilbert at the beginning of the XX century. We also see that these Riesz transforms are limits of the conjugate harmonic functions. The results rely on a careful use of several properties of Bessel functions.

math.CA↗

arXiv2013

A simple computation of $ζ(2k)$ by using Bernoulli polynomials and a telescoping series

We present a new proof of Euler's formulas for $ζ(2k)$, where $k = 1,2,3,...$, which uses only the defining properties of the Bernoulli polynomials, obtaining the value of $ζ(2k)$ by summing a telescoping series. Only basic techniques from Calculus are needed to carry out the computation. The method also applies to $ζ(2k+1)$ and the harmonic numbers, yielding integral formulas for these.

math.NT↗

arXiv2012

Bilinear biorthogonal expansions and the Dunkl kernel on the real line

We study an extension of the classical Paley-Wiener space structure, which is based on bilinear expansions of integral kernels into biorthogonal sequences of functions. The structure includes both sampling expansions and Fourier-Neumann type series as special cases, and it also provides a bilinear expansion for the Dunkl kernel (in the rank 1 case) which is a Dunkl analogue of Gegenbauer's expansion of the plane wave and the corresponding sampling expansions. In fact, we show how to derive sampling and Fourier-Neumann type expansions from the results related to the bilinear expansion for the Dunkl kernel.

math.FA↗

arXiv2010

Mean convergence of Fourier-Dunkl series

In the context of the Dunkl transform a complete orthogonal system arises in a very natural way. This paper studies the weighted norm convergence of the Fourier series expansion associated to this system. We establish conditions on the weights, in terms of the $A_p$ classes of Muckenhoupt, which ensure the convergence. Necessary conditions are also proved, which for a wide class of weights coincide with the sufficient conditions.

math.FA↗

arXiv2005

Heat and Poisson semigroups for Fourier-Neumann expansions

Given $α> -1$, consider the second order differential operator in $(0,\infty)$, $$L_αf \equiv (x^2 \frac{d^2}{dx^2} + (2α+3)x \frac{d}{dx} + x^2 + (α+1)^2)(f), $$ which appears in the theory of Bessel functions. The purpose of this paper is to develop the corresponding harmonic analysis taking $L_α$ as the analogue to the classical Laplacian. Namely we study the boundedness properties of the heat and Poisson semigroups. These boundedness properties allow us to obtain some convergence results that can be used to solve the Cauchy problem for the corresponding heat and Poisson equations.

math.FA↗