By Winter P. A.

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In this case, any polynomial in P:;' can uniquely be reconstructed from its function values sampled at the points in X. The point set X is then said to be P:;'-unisolvent. 13) for the points in X is rather weak for small m. Indeed, when m = 0, the condition is empty, for m = 1 it is trivial, and for m = 2 the points in X must not lie on a straight line. We summarize the above discussion as follows. Theorem 2. 13). 0 Note that radial basis function interpolation is meshfree. of radial basis functions is in contrast to many other methods for scattered data interpolation, such as splines over triangulations.

The computational costs required for building a heap of size n by using build-heap can be bounded above by the following simple calculation. This is the number of O(n) iterations in build-heap times the running time of heapify. But the running time of heapify can uniformly be bounded above by O(1og(n)) operations. Altogether, the running time of build-heap can thus be bounded above by O(n log(n)) . g. 3. 3 Heapsort Although we are not mainly interested in sorting data objects, it is useful to briefly discuss heapsort , Algorithm 7.

Ad . 7), we first consider its corres po nding homogeneous system xr A¢,x . c + Px . 10). 10), we imm ediately obt ain t he identity CT. A¢,x . c = O. 10), we require that the matrix A¢,x is, for any finite set X = {Xl , . 10). Thi s can be rest ated as cT . A¢ ,x . c > 0, for all X and c E JRN \ {O} with pl · c = O. 8), on th e order m of t he polynom ial space P '/;" . 12) is empty, in which case is, according to Definition 8, positi ve definite . This altogether leads us to t he following definiti on.