Contrained interpolants with minimal \(W^{k,p}\)-norm
The problem of minimizing \(\| f^{(k)}\|_ p\) \((1<p<\infty)\) under the constraints \(f^{(k)}(t)\geq 0\) and \(f^{(k)}(t_ i)=y_ i\) for given points \((t_ i,y_ i)\), \(i=1,2,...,N+k\), with positive kth divided difference is considered. It is shown that f solves this problem if and only if \[ f^{(k)}(t)=(\sum^{N}_{i=1}\alpha_ iM_{i,k}(t))_+^{1/(p-1)} \] where \(\alpha_ 1,...,\alpha_ N\) are real numbers and the \(M_{i,k}\) are basic spline functions.
scientific article; zbMATH DE number 3981740