Univariate multiquadric approximation: Quasi-interpolation to scattered data

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.