Convergence of Univariate Quasi-Interpolation Using Multiquadrics |
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Authors: | BUHMANN M. D. |
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Affiliation: | Department of Applied Mathematics and Theoretical Physics Silver Street, Cambridge CB3 9EW, England |
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Abstract: | Quasi-interpolants to a function f: RR on an infinite regularmesh of spacing h can be defined by where :RR is a function with fast decay for large argument. In the approach employing the radial-basis-function : RR, thefunction is a finite linear combination of basis functions(|jh|) (jZ). 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 h0, there are f C(R) for which s does not convergeto f as h0. 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)?. |
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