Accuracy of Lattice Translates of Several Multidimensional Refinable Functions

✍ Scribed by Carlos Cabrelli; Christopher Heil; Ursula Molter

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 554 KB
• Volume: 95
• Category: Article
• ISSN: 0021-9045
• DOI: 10.1006/jath.1997.3211

✦ Synopsis

Complex-valued functions f 1 , ..., f r on R d are 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 1/R d and A is a dilation matrix that expansively maps 1 into itself. Refinable functions satisfy a refinement equation f

, where 4 is a finite subset of 1, the c k are r_r matrices, and f (x)=( f 1 (x), ..., f r (x)) T . The 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 , ..., f r along the lattice 1. In this paper, we determine the accuracy p from the matrices c k . Moreover, we determine explicitly the coefficients y :, i (k) such that x : = r i=1 k # 1 y :, i (k) f i (x+k). These coefficients are multivariate polynomials y :, i (x) of degree |:| evaluated at lattice points k # 1.