Convergence of Univariate Quasi-Interpolation Using Multiquadrics |
| |
Authors: | BUHMANN M D |
| |
Institution: |
Department of Applied Mathematics and Theoretical Physics Silver Street, Cambridge CB3 9EW, England
|
| |
Abstract: | Quasi-interpolants to a function f: R R on an infinite regularmesh of spacing h can be defined by
where :R R is a function with fast decay for large argument. In the approach employing the radial-basis-function : R R, thefunction is a finite linear combination of basis functions (|jh|) (j Z). Choosing Hardy's multiquadrics (r)=(r2+c2)?,we show that sufficiently fast-decaying exist that render quasi-interpolationexact for linear polynomials f. Then, approximating f C2(R),we obtain uniform convergence of s to f as (h, c) 0, and convergenceof s' to f' as (h, c2/h) 0. However, when c stays bounded awayfrom 0 as h 0, there are f C(R) for which s does not convergeto f as h 0. We also show that, for all which vanish at infinity but arenot integrable over R, there are no finite linear combinations of the given basis functions allowing the construction of admissiblequasi-interpolants. This includes the case of the inverse multiquadncs (r)=(r2+c2)?. |
| |
Keywords: | |
本文献已被 Oxford 等数据库收录! |
|