Optimal interpolation in a general setting - a beginning
Let the subspace Y of C[a,b] of dimension \(n+1\) (n\(\geq 0)\) be spanned by a complete extended Chebyshev system \(u_ 0,...,u_ n\). It is possible to find for given nodes \(a=t_ 0<...<t_ n=b\), a basis \(\{y_ 0,...,y_ n\}\) of Y such that \(y_ i(t_ j)=\delta_{ij}\) (Kronecker delta), and to obtain an interpolating projection P: C[a,b]\(\to Y\) defined by \(Pf=\sum^{n}_{i=0}f(t_ i)y_ i\). Let \(L:=\sum^{n}_{i=1}| y_ i|\). Then \(\| P\| =\| L\|\) and L is maximized on \([t_{i-1},t_ i]\) at a unique point \(T_ i\) with \(t_{i-1}<t_ i<T_ i\). The author has conjectured [J. Approximation Theory 41, 279-290 (1984; Zbl 0566.41003)] that, under the hypothesis laid down above on Y and P, the norm of P is minimal if \(L(T_ 1)=...=L(T_ n)=C_ Y\) for some unique value \(C_ Y\) (generalization of the Bernstein conjecture regarding Lagrange interpolation), and furthermore if \(\| P\| >C_ Y\), then at least one of the local maximum values of L is less than \(C_ Y\) (generalization of the Erdős conjecture on Lagrangian interpolation). There are several strong arguments for the plausibility of the Kilgore conjectures, including the upholding of the two original conjectures in their original contexts as theorems. Kilgore [loc. cit.] pointed out that there exist two impediments to the successful generalization of these two conjectures, although there are at this time several choices of the range space Y for which the above generalized conjectures are indeed valid, as well as examples of interpolation with spaces of functions on a complex domain for which the conditions of Bernstein and Erdős have been shown to characterize minimal norm interpolation. The first of these impediments is the problem of reduction of the matrix \((\partial \lambda_ i/\partial t_ j:1\leq i\leq n,1\leq j\leq n-1)\), where \(\lambda_ i=L(T_ i)=\max \{L:t_{i-1}\leq t\leq t_ i\}\), to a matrix of evaluation. In this paper the author solves this problem in complete generality. It is hoped that in the framework of the new technique it will be possible to solve particular problems in which algebraic complexity has caused previous methods of solution to be totally inapplicable, or to fail.
scientific article; zbMATH DE number 4105504