The Inverse of a Totally Positive Biinfinite Band Matrix

摘要

Spline approximation is often most effective when the breakpoint (knot) sequence can be chosen suitably non-uniform. At the same time, the standard approximation schemes (such as least-squares approximation, or interpolation at suitable interpolation points by splines) are so far only known to be usable and bounded as long as the breakpoint sequence is almost uniform. The problem of showing existence and uniqueness of bounded spline approximants to bounded data boils down to showing invertibility of a certain infinite matrix A. The distinguished features of this matrix are its bandedness and its total positivity, i.e., all minors of A are nonnegative. In this paper we show that if there is exactly one bounded sequence mapped by a biinfinite totally positive banded matrix A to the particular sequence (...,-1,1,-1,1,-1,...), then every bounded sequence is contained in the range of A. In spline terms, this result says, for example, that any bounded data sequence can be interpolated, and in exactly one way, with a bounded spline (with a given knot sequence, at a given interpolation point sequence) provided that the periodic data (+1,-1) can be interpolated, and in exactly one way, by a bounded spline from that class. Further, our arguments show that such an interpolating spline can be constructed as the limit of splines which satisfy finitely many of the given interpolation conditions.