CONVERGENCE OF UNIVARIATE QUASI-INTERPOLATION USING MULTIQUADRICS

Quasi-interpolants to a function f: R→R on an infinite regular mesh of spacing h can be defined by 'Equation Presented' where ψ:R→R is a function with fast decay for large argument. In the approach employing the radial-basis-function φ: R→R, the function ψ is a finite linear combination of basis functions φ(|•-jh|) (jϵZ). Choosing Hardy's multiquadrics φ(r) = (r2+c2)1/2, we show that sufficiently fast-decaying ψ exist that render quasi-interpolation exact for linear polynomials f. Then, approximating f ϵ C2(R), we obtain uniform convergence of s to f as (h, c)→0, and convergence of s′ to f′ as (h, c2/h)→0. However, when c stays bounded away from 0 as h→0, there are f ϵ C(R) for which s does not converge to f as h→0. We also show that, for all φ which vanish at infinity but are not integrable over R, there are no finite linear combinations ψ of the given basis functions allowing the construction of admissible quasi-interpolants. This includes the case of the inverse multiquadncs φ(r) = (r2+c2)-1/2 © 1988 Oxford University Press.