INVESTIGADORES
MOLTER ursula Maria
artículos
Título:
Accuracy of Lattice Translates of Several Multidimensional Refinable Functions
Autor/es:
CABRELLI, CARLOS A.; HEIL, CHRISTOPHER; MOLTER, URSULA M.
Revista:
JOURNAL OF APPROXIMATION THEORY
Editorial:
Academic Press
Referencias:
Lugar: Boca Raton; Año: 1998 vol. 95 p. 5 - 52
ISSN:
0021-9045
Resumen:
Complex-valued functions $f_1,\dots,f_r$ on ${\bold R}^d$ are{\it refinable} if they are linear combinations of finitely many of therescaled and translated functions $f_i(Ax-k)$, where the translates $k$are taken along a lattice $\Gamma \subset {\bold R}^d$ and $A$ is a{\it dilation matrix} that expansively maps $\Gamma$ into itself.Refinable functions satisfy a {\it refinement equation}$f(x) = \sum_{k \in \Lambda} c_k \, f(Ax-k)$, where$\Lambda$ is a finite subset of $\Gamma$, the $c_k$ are $r \times r$matrices, and $f(x) = (f_1(x),\dots,f_r(x))^{\text{T}}$.The {\it accuracy} of $f$ is the highest degree $p$such that all multivariate polynomials $q$  with degree$(q) < p$are exactly reproduced from linear combinations of translates of$f_1,\dots,f_r$ along the lattice $\Gamma$.In this paper, we determine the accuracy $p$ from the matrices $c_k$.Moreover, we determine explicitly the coefficients $y_{\alpha,i}(k)$ such that$x^\alpha = \sum_{i=1}^r \sum_{k \in \Gamma} y_{\alpha,i}(k) \, f_i(x+k)$.These coefficients are multivariate polynomials $y_{\alpha,i}(x)$of degree $|\alpha|$ evaluated at lattice points $k \in \Gamma$.