Univariate multiquadric approximation: Quasi-interpolation to scattered data |
| |
Authors: | R. K. Beatson M. J. D. Powell |
| |
Affiliation: | 1. Department of Mathematics, University of Canterbury, Christchurch, New Zealand 2. Department of Applied Mathematics and Theoretical Physics, University of Cambridge, CB3 9EW, Cambridge, England
|
| |
Abstract: | The univariate multiquadric function with centerx j ∈R has the form {? j (x)=[(x?x j )2+c 2]1/2, x∈R} wherec is a positive constant. We consider three approximations, namely, ? A f, ?? f, and ? C f, to a function {f(x),x 0≤x≤x N } from the space that is spanned by the multiquadrics {? j :j=0, 1, ...,N} and by linear polynomials, the centers {x j :j=0, 1,...,N} being given distinct points of the interval [x 0,x N ]. The coefficients of ? A f and ?? f depend just on the function values {f(x j ):j=0, 1,...,N}. while ? A f, ? C f also depends on the extreme derivativesf′(x 0) andf′(x N ). These approximations are defined by quasi-interpolation formulas that are shown to give good accuracy even if the distribution of the centers in [x 0,x N ] is very irregular. Whenf is smooth andc=O(h), whereh is the maximum distance between adjacent centers, we find that the error of each quasi-interpolant isO(h 2|logh|) away from the ends of the rangex 0≤x≤x N. Near the ends of the range, however, the accuracy of ? A f and ?? f is onlyO(h), because the polynomial terms of these approximations are zero and a constant, respectively. Thus, some of the known accuracy properties of quasiinterpolation when there is an infinite regular grid of centers {x j =jh:j ∈F} given by Buhmann (1988), are preserved in the case of a finite rangex 0≤x≤x N , and there is no need for the centers {x j :j=0, 1, ...,N} to be equally spaced. |
| |
Keywords: | |
本文献已被 SpringerLink 等数据库收录! |
|