Univariate multiquadric approximation: Quasi-interpolation to scattered data View Full Text


Ontology type: schema:ScholarlyArticle     


Article Info

DATE

1992-09

AUTHORS

R. K. Beatson, M. J. D. Powell

ABSTRACT

The univariate multiquadric function with centerxj∈R has the form {ϕj(x)=[(x−xj)2+c2]1/2, x∈R} wherec is a positive constant. We consider three approximations, namely, ℒAf, ℒℬf, and ℒCf, to a function {f(x),x0≤x≤xN} from the space that is spanned by the multiquadrics {ϕj:j=0, 1, ...,N} and by linear polynomials, the centers {xj:j=0, 1,...,N} being given distinct points of the interval [x0,xN]. The coefficients of ℒAf and ℒℬf depend just on the function values {f(xj):j=0, 1,...,N}. while ℒAf, ℒCf also depends on the extreme derivativesf′(x0) andf′(xN). These approximations are defined by quasi-interpolation formulas that are shown to give good accuracy even if the distribution of the centers in [x0,xN] 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(h2|logh|) away from the ends of the rangex0≤x≤xN. Near the ends of the range, however, the accuracy of ℒAf and ℒℬf 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 {xj=jh:j ∈F} given by Buhmann (1988), are preserved in the case of a finite rangex0≤x≤xN, and there is no need for the centers {xj:j=0, 1, ...,N} to be equally spaced. More... »

PAGES

275-288

References to SciGraph publications

Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/bf01279020

DOI

http://dx.doi.org/10.1007/bf01279020

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1005480549


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