ABSTRACT
The semi-norm has a fundamental property that provides excellent corrob oration for the claim that it is very suitable for the Krylov subspace method. The property is taken from Schaback (1993), and will not be employed by our algorithm, because it applies to a linear space of functions S that is larger than the one that has been defined already. Indeed, just for the remainder of this section, we let s be in S if it has the form
(^^ .) — ^ y A? (11-~ ¿ 7112) ^ ^7j Pj (—)? xEl Z , (2*7) j = i j - 1
the parameters being subject to the usual constraints
n = 0 , i = 1, 2, . . . , ra. (2.8)
Here (j) and pj, j = 1,2, . . . , ra, are as before, but h can be much larger than n. The only restriction on the set : ¿ =1 , 2 , . . . , n} is that it has to include all the original data points x{) i = 1, 2, . . . , n. It follows that the previous S is an n-dimensional linear subspace of the new that the polynomial unisolvency
condition is preserved, and that at least one element of the new S interpolates the data. Further, s E S is an interpolant if and only if it can be written in the form s = s*+t, where s* is still the required interpolant, and where t £ S satisfies % i ) = 0, i = l , 2, . . . ,n.
