On the Approximation Order of Principal Shift-Invariant Subspaces ofLp(Rd)

✍ Scribed by Michael J. Johnson

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 482 KB
• Volume: 91
• Category: Article
• ISSN: 0021-9045
• DOI: 10.1006/jath.1996.3061

✦ Synopsis

For 1 p , sufficient conditions on the generators [, h ] h>0 are given which ensure that the h-dilates of the shift-invariant space generated by , h provide L p -approximation of order k>0. Examples where , h is an exponential box spline or certain dilates of the Gaussian e &| } | 2 are considered; it is shown that our sufficient condition then provides an optimal lower bound on their approximation order. 1997 Academic Press 1. INTRODUCTION Let d # N :=[1, 2, ...], and let C :=(&1Â2 } } } 1Â2) d denote the open unit cube in R d . Following [19], define

. It was shown in [19] that if , # L p , then the semi-discrete convolution operator