Convergence of Univariate Quasi-Interpolation Using Multiquadrics

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)^–?.     

On the dynamics of a discretized neutral equation

In this paper, we investigate the stability properties of numericalmethods for solving the differential equation of the neutraltype y'(t) = Ay(t) + By(qt) + Cy'(pt), y(0) = 1, where p, q (0, 1), A, B, C and A 0. Sufficient conditions for the asymptoticstability of a discretized model of this equation are givenwhen p = q = L^–1, where L 2 is an integer. These coincidewith conditions for asymptotic stability of the original equation,as reported in (Iserles, 1991), except that the step-lengthneed be restricted and, at the limit, asymptotic stability isretained only if |C| < 1. Moreover, for a particular choiceof A these conditions are also shown to be necessary.