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zsplins.aux
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zsplins.aux
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\relax
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\providecommand\babel@aux[2]{}
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\@nameuse{bbl@beforestart}
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\catcode `"\active
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\catcode `<\active
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\catcode `>\active
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\@nameuse{es@quoting}
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\@writefile{toc}{\contentsline {section}{\numberline {1}Los Z-splines}{1}{}\protected@file@percent }
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\newlabel{eq:aiz1}{{1}{4}}
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\gdef \@abspage@last{4}
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82
zsplins.fdb_latexmk
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zsplins.fdb_latexmk
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zsplins.fls
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zsplins.fls
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\documentclass{article}
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% --- PAQUETES ESENCIALES ---
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\usepackage[utf8]{inputenc} % Para reconocer tildes y ñ directamente
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\usepackage[T1]{fontenc}
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\usepackage{geometry} % (Opcional) Para mejorar los márgenes
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% ---------------------------
|
||||
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||||
\title{Proyecto 1: Métodos numéricos del Z-spline cúbico}
|
||||
\author{Reyes Garcia Joel}
|
||||
\date{\today}
|
||||
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\begin{document}
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\maketitle
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\begin{abstract}
|
||||
En este trabajo se presentan los métodos numéricos asociados a...
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||||
\end{abstract}
|
||||
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||||
\section{Los Z-splines}
|
||||
|
||||
\subsection{Introducción}
|
||||
|
||||
Los Z-splines constituyen una familia de splines cardinales de soporte
|
||||
compacto que preservan momentos discretos y poseen alto orden de
|
||||
regularidad. La idea central es construir una base spline
|
||||
cuya definición incorpora explícitamente operadores de diferencias
|
||||
finitas derivados de desarrollos de Taylor. Su creador,
|
||||
Julián T. Becerra-Sagredo, desarrolló esta teoría en dos trabajos clave
|
||||
(2003 y 2020), en los cuales se presentan tanto la construcción como
|
||||
sus propiedades analíticas y aplicaciones.
|
||||
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||||
\subsection{Definición cardinal y construcción básica}
|
||||
|
||||
Sea $\{x_j\} = \{j\}$ una malla equiespaciada.
|
||||
Los Z-splines se definen mediante un núcleo cardinal $Z_m(x)$ tal que la
|
||||
interpolación de un conjunto de datos $\{y_j\}$ está dada por
|
||||
|
||||
\[
|
||||
f_m(x) = \sum_{j\in\mathbb{Z}} y_j \, Z_m(x - j).
|
||||
\]
|
||||
|
||||
Cada función base $Z_m$ es un polinomio por tramos de grado $2m-1$ y con
|
||||
regularidad $C^{m-1}$. El parámetro $m$ determina el orden del spline.
|
||||
|
||||
La construcción impone condiciones de interpolación entre nodos y
|
||||
conservación de momentos discretos. Para cada intervalo
|
||||
$[j,j+1]$, el polinomio está determinado por un conjunto de condiciones
|
||||
de tipo Hermite–Birkhoff derivadas de operadores de diferencias finitas.
|
||||
|
||||
\subsection{Conservación de momentos}
|
||||
|
||||
Una de las propiedades fundamentales es que las Z-splines preservan los
|
||||
primeros $2m-1$ momentos discretos. Sea
|
||||
|
||||
\[
|
||||
M_k = \sum_j y_j j^k, \qquad k=0,\dots,2m-2,
|
||||
\]
|
||||
|
||||
entonces la interpolante satisface
|
||||
|
||||
\[
|
||||
\int_{-\infty}^{\infty} x^k f_m(x) \, dx = M_k,
|
||||
\]
|
||||
|
||||
lo que asegura que la interpolación no altera estos invariantes
|
||||
(discretos o físicos).
|
||||
|
||||
\subsection{Reproducción polinómica}
|
||||
|
||||
El artículo demuestra que las Z-splines reproducen exactamente
|
||||
polinomios hasta grado $2m-2$:
|
||||
|
||||
\[
|
||||
p(x) = x^n, \qquad n \le 2m-2 \quad \Rightarrow \quad
|
||||
f_m(x) = p(x).
|
||||
\]
|
||||
|
||||
Esto implica un orden de aproximación
|
||||
|
||||
\[
|
||||
\| f - f_m \| = \mathcal{O}(h^{\,2m-1}),
|
||||
\]
|
||||
|
||||
cuando $f$ es suficientemente regular.
|
||||
|
||||
\subsection{Soporte compacto y regularidad}
|
||||
|
||||
Cada $Z_m$ tiene soporte compacto mínimo:
|
||||
|
||||
\[
|
||||
Z_m(x) = 0 \qquad \text{si } |x| > m,
|
||||
\]
|
||||
|
||||
y regularidad
|
||||
|
||||
\[
|
||||
Z_m \in C^{m-1}.
|
||||
\]
|
||||
|
||||
Esto permite su uso eficiente en contextos numéricos, especialmente en
|
||||
métodos de remapeo o mallas adaptativas.
|
||||
|
||||
\subsection{Relación con diferencias finitas}
|
||||
|
||||
Una contribución teórica clave es que los coeficientes de los polinomios
|
||||
que forman $Z_m$ coinciden con los coeficientes de operadores de
|
||||
diferencias finitas de orden elevado. En particular, se demuestra que
|
||||
para derivadas,
|
||||
|
||||
\[
|
||||
f_m^{(k)}(x_j) = \sum_{\ell=-m}^m c^{(k)}_{\ell} \, y_{j+\ell},
|
||||
\]
|
||||
|
||||
donde los $c^{(k)}_\ell$ son coeficientes de diferencias finitas
|
||||
centradas obtenidos por expansión de Taylor.
|
||||
|
||||
\subsection{Límite hacia la función sinc}
|
||||
|
||||
A medida que $m \to \infty$, las Z-splines convergen hacia la función
|
||||
ideal de reconstrucción cardinal:
|
||||
|
||||
\[
|
||||
Z_m(x) \longrightarrow \operatorname{sinc}(x)
|
||||
= \frac{\sin(\pi x)}{\pi x}.
|
||||
\]
|
||||
|
||||
Esto establece un vínculo entre splines compactos y teoría de muestreo.
|
||||
|
||||
\subsection{Z-splines en mallas no uniformes}
|
||||
|
||||
Los artículos extienden la teoría a mallas arbitrarias
|
||||
$\{x_j\}$ no equidistantes. En ese caso, el núcleo cardinal ya no es
|
||||
trasladable, y se construyen polinomios locales mediante matrices de
|
||||
diferencias finitas adaptadas a la geometría de los nodos.
|
||||
|
||||
En regiones de frontera se emplean splines ``one-sided'', lo cual permite
|
||||
resolver la falta de simetría.
|
||||
|
||||
\subsection{Aplicaciones numéricas}
|
||||
|
||||
Las Z-splines son especialmente útiles en contextos donde es esencial la
|
||||
conservación de cantidades físicas discretas:
|
||||
|
||||
\begin{itemize}
|
||||
\item métodos de remapeo ALE y semi-Lagrangianos,
|
||||
\item interpolaciones conservativas en dinámica de fluidos,
|
||||
\item regularización de funciones delta,
|
||||
\item remallado adaptativo,
|
||||
\item filtrado numérico (denoising) preservando momentos.
|
||||
\end{itemize}
|
||||
|
||||
En estas aplicaciones, la preservación de momentos juega un papel
|
||||
fundamental para mantener estabilidad física del sistema.
|
||||
|
||||
\subsection{Conclusión}
|
||||
|
||||
Los Z-splines constituyen una herramienta moderna de
|
||||
interpolación de gran interés tanto teórico como práctico, especialmente
|
||||
en contextos de simulación numérica donde la preservación de invariantes
|
||||
es esencial.
|
||||
|
||||
|
||||
\section{El Z-spline cúbico}
|
||||
|
||||
La función base Z-spline cúbica para datos a intervalos arbitrarios $a_{1}, a_{2}, a_{3}$ y $a_{4}$
|
||||
está dada por:
|
||||
|
||||
\begin{equation} \label{eq:aiz1}
|
||||
\widetilde{Z}_{1} (x) = \left\{
|
||||
\begin{array}{lcl}
|
||||
0 & & x < -a_{1}-a_{2} \\ [0.3cm]
|
||||
|
||||
\left( \frac{a_{2}+a_{1}}{a_{1}} \right) + \left(
|
||||
\frac{3a_{2}+a_{1}}{a_{2}a_{1}} \right) x & & \\ [0.1cm]
|
||||
\quad + \frac{3a_{2}+2a_{1}}{a_{2}a_{1}(a_{2}+a_{1})}
|
||||
x^2 + \frac{1}{a_{2}a_{1}(a_{2}+a_{1})}x^{3} & & -a_{1}-a_{2} \leq x \leq -a_{2}, \\ [0.3cm]
|
||||
|
||||
1- \left( \frac{1}{a_{3}}-\frac{1}{a_{2}} \right) x & & \\ [0.1cm]
|
||||
\quad - \frac{a_{3}+2(a_{2}+a_{1})}{a_{3}a_{2}(a_{2}+a_{1})} x^2
|
||||
- \frac{a_{3}+a_{2}+a_{1}}{a_{3}a_{2}^2(a_{2}+a_{1})} x^3 & & -a_{2} \leq x \leq 0, \\ [0.3cm]
|
||||
|
||||
1+ \left( \frac{1}{a_{2}}-\frac{1}{a_{3}} \right) x & & \\ [0.1cm]
|
||||
\quad - \frac{a_{2}+2(a_{3}+a_{4})}{a_{2}a_{3}(a_{3}+a_{4})} x^2
|
||||
+ \frac{a_{2}+a_{3}+a_{4}}{a_{2}a_{3}^2(a_{3}+a_{4})} x^3 & & 0 \leq x \leq a_{3}, \\ [0.3cm]
|
||||
|
||||
\left( \frac{a_{3}+a_{4}}{a_{4}} \right) - \left(
|
||||
\frac{3a_{3}+a_{4}}{a_{3}a_{4}} \right) x & & \\ [0.1cm]
|
||||
\quad + \frac{3a_{3}+2a_{4}}{a_{3}a_{4}(a_{3}+a_{4})} x^2
|
||||
- \frac{1}{a_{3}a_{4}(a_{3}+a_{4})}x^{3} & & a_{3} \leq x \leq a_{3}+a_{4}, \\ [0.3cm]
|
||||
|
||||
0 & & x > a_{3}+a_{4}.
|
||||
\end{array}
|
||||
\right.
|
||||
\end{equation}
|
||||
|
||||
|
||||
\subsection{Z-spline cúbico en los extremos de un intervalo}
|
||||
|
||||
\section{Fórmula de integración numérica}
|
||||
|
||||
\section{Fórmula de derivación numérica}
|
||||
\section{Convergencia}
|
||||
\subsection{Integración}
|
||||
\subsection{Derivación}
|
||||
\subsection{Interpolación equidistante en 2D}
|
||||
\subsection{Interpolación en 1D con puntos arbitrarios}
|
||||
|
||||
\end{document}
|
||||
Reference in New Issue
Block a user