As is now well known for some basic functions φ, hierarchical and fast multipole-like methods can greatly reduce the storage and operation counts for fitting and evaluating radial basis functions. In particular, for spline functions of the form

The Beppo-Levi native spaces which arise when using polyharmonic splines to interpolate in many space dimensions are embedded in Hölder-Zygmund spaces. Convergence rates for radial basis function interpolation are inferred in some special cases.

We introduce a class of matrix-valued radial basis functions (RBFs) of compact support that can be customized, e.g. chosen to be divergence-free. We then derive and discuss error estimates for interpolants and derivatives based on these matrixvalued RBFs.

The behavior of Radial Basis Function (RBF) Networks greatly depends on how the center points of the basis functions are selected. In this work we investigate the use of instance reduction techniques, originally developed to reduce the storage requirements of instance based learners, for this purpose. Five Instance-Based Reduction Techniques were used to determine the set of center points, and ...

We present a combinatorial characterization of sets of integers (rO,rl, ... ,TII-d, with OSri~"_2, " such that a'O,({1, ... ,a'·-1 fonn a basis of GF(q") over GF(q) for some ae GF'(q"). We use this characterization to prove the following generalization of the nonnal basis theorem fpr finite -fields of characteristic two: Let Ao.Al; ... ,A.,.-1 be integers in the range~<q, with at most one A.; e...

We prove a conjecture of K. Monks 4] on the relation between the admissible basis and the Milnor basis of the mod 2 Steenrod algebra A 2 , and generalise the result to the mod p Steenrod algebra A p where p is prime. This establishes a necessary and suucient condition for the Milnor basis element P(r 1 ; r 2 ; : : : ; r k) and the admissible basis element P t 1 P t 2 : : : P t k to coincide. Th...

Let q be a prime power, m ≥ 2 an integer and A = ( a b c d ) ∈ GL2(Fq), where A 6= ( 1 1 0 1 ) if q = 2 and m is odd. We prove an extension of the primitive normal basis theorem and its strong version. Namely, we show that, except for an explicit small list of genuine exceptions, for every q, m and A, there exists some primitive x ∈ Fqm such that both x and (ax+b)/(cx+d) produce a normal basis ...

This contribution gives a partial survey over the native spaces associated to (not necessarily radial) basis functions. Starting from reproducing kernel Hilbert spaces and invariance properties, the general construction of native spaces is carried out for both the unconditionally and the conditionally positive definite case. The definitions of the latter are based on finitely supported function...

A new scheme of data-driven segmentation is proposed, which is based on detection of object boundary, and volumetric pattern reconstruction as implicit function by using the detected object boundary and the radial basis functions (RBF). By using clinical X-ray CT data, applications in visualization of the pancreatic duct by MINIP of curved thin-slab and in liver segmentation are shown.

In the field of radial basis functions mathematicians have been endeavouring to find infinitely differentiable and compactly supported radial functions. This kind of functions is extremely important. One of the reasons is that its error bound will converge very fast. However there is hitherto no such function which can be expressed in a simple form. This is a famous question. The purpose of thi...