It’s well-known that there is a so-called high-level error bound for multiquadric and inverse multiquadric interpolations, which was put forward by Madych and Nelson in 1992. It’s of the form |f(x)− s(x)| ≤ λ 1 d ‖f‖h where 0 < λ < 1 is a constant, d is the fill distance which roughly speaking measures the spacing of the data points, s(x) is the interpolating function of f(x), and h denotes the...

In this note a local thinning of the data locations is proposed, in order to construct a least squares multiquadric approximant stable and close to the multiquadric interpolant

This note concerns density properties of the general multiquadric, (x + c2)k−1/2, where k is a fixed natural number. We establish that scattered translates of the general multiquadric are dense in C([a, b]), where a and b are finite. As a corollary, we show that scattered translated of the general multiquadric are dense in the function spaces L([a, b]), for 1 ≤ p <∞.

Numerical simulation of the high order derivatives based on the sampling data is an important and basic problem in numerical approximation, especially for solving the differential equations numerically. The classical method is the divided difference method. However, it has been shown strongly unstable in practice. Actually, it can only be used to simulate the lower order derivatives in applicat...

A generalised multiquadric radial basis function is a function of the form s(x) = ∑N i=1 diφ(|x − ti|), where φ(r) = ( r2 + τ2 )k/2 , x ∈ Rn, and k ∈ Z is odd. The direct evaluation of an N centre generalised multiquadric radial basis function at m points requires O(mN) flops, which is prohibitive when m and N are large. Similar considerations apparently rule out fitting an interpolating N cent...

In this paper, we identify univariate prewavelets on spaces spanned by translates of multiquadric functions and other radial basis functions with nonequally spaced centers (or "knots"). Although the multiquadric function and its relations are our prime examples, the theory is sufficiently broad to admit prewavelets from other radial basis function spaces as well.

This paper presents numerical solution of elliptic partial differential equations Poisson’s equation using a combination of logarithmic and multiquadric radial basis function networks. This method uses a special combination between logarithmic and multiquadric radial basis functions with a parameter r. Further, the condition number which arises in the process is discussed, and a comparison is m...

Among other things, we prove that multiquadric surface interpolation is always solvable, thereby settling a conjecture of R. Franke.