The Inverse of a Totally Positive Bi-infinite Band Matrix1

It is shown that a bounded bi-infinite banded totally positive matrix A is boundedly invertible iff there is one and only one bounded sequence mapped by A to the sequence ((-)')■ The argument shows that such a matrix has a main diagonal, i.e., the inverse of A is the bounded pointwise limit of inverses of finite sections of A principal with respect to a particular diagonal; hence ((—)'+^4~ (j, 7» or its inverse is again totally positive. 0. Introduction. This paper is a further step in a continuing effort to understand certain linear spline approximation schemes. Convergence of such processes is intimately tied to their stability, i.e., to their boundedness, as maps on C, say. Use of the B-spline basis shows this question to be equivalent to bounding the inverse of certain totally positive band matrices. The calculation of bounds on the inverse of a given matrix is in general a difficult task. It is hoped that the present investigation into the consequences of bandedness and total positivity for the structure of the inverse may ultimately prove helpful in obtaining such bounds. The results in this paper were obtained in the study of a conjecture due to C. A. Micchelli [7]. In connection with his work on the specific approximation scheme of interpolation at a (strictly increasing) point sequence t by elements of Sm t, i.e., by splines of some order m with some knot sequence T = (/,), Micchelli became convinced that every bounded function has one and only one bounded spline interpolant iff the particular function which satisfies /(t,) = (-)', all /', has a bounded spline interpolant in Sm,. If (N¡) = (N¡ mt) denotes the corresponding B-spline basis for Sm,, then Micchelli's conjecture can be phrased thus: The matrix A : = (Nj(Tj)) is boundedly invertible iff the linear system (1) Ax = ((-)'") has a bounded solution. Micchelli points out that, for a finite A, this conjecture is indeed true and can be established using the total positivity of A. Whether A = (N(t¡)) is finite or not, it is not difficult to see that A fails to be invertible unless A is ///-banded, i.e., unless t and t so harmonize that at most m + 1 consecutive bands of A are not identically zero. It is then a small step to the conjecture Received by the editors November 21, 1980. 1980 Mathematics Subject Classification. Primary 47B37; Secondary 15A09, 15A48.