On best partial bases
The partial basis problem is to determine a subsequence of given length n of a given basis \(h_ 1,...,h_ N\) in a normed linear space X for which best approximation to a given f in X is closest. Conditions guaranteeing that the initial segment \(h_ 1,...,h_ n\) (or terminal segment \(h_{N-n+1},...,h_ N)\) is the unique best partial basis are obtained. The results are closely connected with the theory of Chebyshev systems and extended complete Chebyshev systems. One of the nicest results is the following: Theorem. Let \(0<a<b<\infty\); n, N integers, \(1\leq n<N\). Let f be a real function, positive and continuous on [a,b] and assume that for \(k=1,2,...,n\), \((-1)^ kf^{(k)}(x)>0\) throughout (a,b). Let \(1\leq p\leq \infty\). Then \(1,x,...,x^{n-1}\) is the unique best partial basis of \(1,x,...,x^{N-1}\) of length n to approximate f in the \(L^ p(a,b)\) norm.
scientific article; zbMATH DE number 3979697