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Electronic Resource
Springer
Constructive approximation 8 (1992), S. 275-288
ISSN: 1432-0940
Keywords: 41A25 ; 41A30 ; 65D10 ; Multiquadric approximation ; Order of convergence ; Quasi-interpolation ; Radial basis functions
Source: Springer Online Journal Archives 1860-2000
Topics: Mathematics
Notes: Abstract The univariate multiquadric function with centerx j ∈R has the form {ϕ j (x)=[(x−x j )2+c 2]1/2, x∈R} wherec is a positive constant. We consider three approximations, namely, ℒ A f, ℒℬ f, and ℒ C f, to a function {f(x),x 0≤x≤x N } from the space that is spanned by the multiquadrics {ϕ 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 ℒ A f and ℒℬ f depend just on the function values {f(x j ):j=0, 1,...,N}. while ℒ A f, ℒ 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 ℒ A f and ℒℬ 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.
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