Accuracy of Lattice Translates of Several Multidimensional Refinable Functions

CABRELLI, CARLOS; HEIL, CHRISTOPHER; MOLTER, URSULA

JOURNAL OF APPROXIMATION THEORY

ACADEMIC PRESS INC ELSEVIER SCIENCE

Lugar: Amsterdam; Año: 1998 vol. 95 p. 5 - 52

0021-9045

Complex-valued functions $f_1,\dots,f_r$ on ${\bold R}^d$ are {\it refinable} if they are linear combinations of finitely many of the rescaled 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 $k \in \Gamma$.