arXiv subjects

Amadeo Irigoyen

Publications and source records attributed to Amadeo Irigoyen.

7 recordsLinked to original sources

arXiv2019

Multidimensional intertwining Leja sequences and applications in bidimensional Lagrange interpolation

We first give a method to get multidimensional Leja sequences by considering intertwining sequences from one-dimensional ones. An application is the existence of explicit Leja sequences for the closed unit polydisc. Next, we deal with some applications in bidimensional Lagrange interpolation with intertwining Leja sequences. These results also require an explicit formula for the associated fundamental Lagrange polynomials with uniform estimates.

math.CV↗

arXiv2015

Geometric conditions for the reconstruction of a holomorphic function by an interpolation formula

We give here some precisions and improvements about the validity of the explicit reconstruction of any holomorphic function on a ball of $\mathbb{C}^2$ from its restrictions on a family of complex lines. Such validity depends on the mutual distribution of the lines. This condition can be geometrically described and is equivalent to a stronger stability of the reconstruction formula in terms of permutations and subfamilies of these lines. The motivation of this problem comes from possible applications in mathematical economics and medical imaging.

math.CV↗

arXiv2015

A uniform bound for the Lagrange polynomials of Leja points for the unit disk and applications in multivariate Lagrange interpolation

We study uniform estimates for the family of fundamental Lagrange polynomials associated with any Leja sequence for the complex unit disk. The main result claims that all these polynomials are uniformly bounded on the disk, i.e. independently on the range $N$ of the associated $N$-Leja section. We also give an improved estimate for special values of $N$. As a first application, we get an analogous estimate for any compact subset whose boundary is an Alper-smooth Jordan curve. We then deal with some applications in multivariate Lagrange interpolation. We first give some estimates of the Lebesgue constant for the intertwining sequence of Leja sequences, a result that also requires an explicit formula for the associated fundamental Lagrange polynomials with uniform estimates. We finish by dealing with a method to get explicit bidimensional Leja sequences and then give a positive example for the bidisk.

math.CV↗

arXiv2010

A criterion for the explicit reconstruction of a holomorphic function from its restrictions on lines

We deal with a problem of the explicit reconstruction of any holomorphic function $f$ on $\mathbb{C}^2$ from its restricions on a union of complex lines. The validity of such a reconstruction essentially depends on the mutual repartition of these lines, condition that can be analytically described. The motivation of this problem comes also from possible applications in mathematical economics and medical imaging.

math.CV↗

arXiv2008

An approximation formula for holomorphic functions by interpolation on the ball

We deal with a problem of the reconstruction of any holomorphic function $f$ on the unit ball of $\mathbb{C}^2$ from its restricions on a union of complex lines. We give an explicit formula of Lagrange interpolation's type that is constructed from the knowledge of $f$ and its derivatives on these lines. We prove that this formula approximates any function when the number of lines increases. The motivation of this problem comes also from possible applications in mathematical economics and medical imaging.

math.CV↗

arXiv2006

Application of approximation theory by nonlinear manifolds in Sturm-Liouville inverse problems

We give here some negative results in Sturm-Liouville inverse theory, meaning that we cannot approach any of the potentials with $m+1$ integrable derivatives on $\mathbb{R}^+$ by an $ω$-parametric analytic family better than order of $(ω\lnω)^{-(m+1)}$. Next, we prove an estimation of the eigenvalues and characteristic values of a Sturm-Liouville operator and some properties of the solution of a certain integral equation. This allows us to deduce from [Henkin-Novikova] some positive results about the best reconstruction formula by giving an almost optimal formula of order of $ω^{-m}$.

math-ph↗

arXiv2006

Resultat negatif en theorie d'approximation de compacts fonctionnels par des varietes analytiques et application a un probleme inverse

In the theory of approximation there are some problems on approximation of compacts in functional spaces by nonlinear families : first we deal with the polynomial case, and then we consider the analytic case. We demonstrate a negative result in which we claim that an analytic familie of functions with $N$ parameters can not approach the compact $Λ_l(I^s)$ closer than of order $(N\log N)^{\frac{l}{s}}$, when $N$ increases. As applied to an inverse problem in Sturm-Liouville theory, this assertion provides an answer to a question about the best possible reconstruction of the negative potential $Q$ with $m+1$ integrable derivatives, from its eigenvalues and characteristic values of the equation $-y''+ω^2Qy=λy$, when $ω$ increases : we show that it is impossible to get an analytic approximating formula with precision better than of order $(ω\logω)^{-(m+1)}$. Moreover there is from Henkin-Novikova formulas which are almost optimal.

math.FA↗