Inverses of Infinite Sign Regular Matrices' by C. De Boor, S. Friedland and A. Pinkus

Let A be an infinite sign regular (sr) matrix which can be viewed as a bounded linear operator from lx to itself. It is proved here that if the range of A contains the sequence (...,1,-1,1,-1,...), then A is onto. If A'' exists, then DA~'D is also sr, where D is the diagonal matrix with diagonal entries alternately 1 and -1. In case A is totally positive (tp), then DA~]D is also tp under additional assumptions on A. 0. Introduction. If the problem of spline interpolation is expressed in terms of B-splines, then the question of existence of a bounded spline interpolant to bounded data is seen to be equivalent to the question of whether a certain bounded band matrix has all bounded sequences in its range. In [5], C A. Micchelli conjectured that there exists a unique bounded spline interpolant (of a given order and a given knot sequence) to any data sequence (t,, y,),<=z in the plane, with (t() strictly increasing and (y) bounded, provided only that it is possible to interpolate the particular data sequence (t;,(-l)'),ez by sucn a sP'ineThere is apparently nothing special about the particular spline problem other than that it leads to a banded totally positive matrix. Therefore one of us quoted this conjecture in [3, p. 319, Problem 4] as "A bi-infinite banded totally positive matrix A is boundedly invertible if and only if the linear system Ax = ((-1)') has a bounded solution."2 Micchelli gave a simple argument for the case when A is a Toeplitz matrix. Cavaretta, Dahmen, Micchelli and Smith [2] recently proved the conjecture in case A is a block Toeplitz matrix. This is all the more remarkable since it is easy to see in hindsight that the conjecture is faulty even in the original context of spline interpolation. For example, interpolation by bounded broken lines with breakpoint sequence Z at the sequence t = Z \ {0} is possible to any bounded ordinate sequence y, but not uniquely so since the value of the interpolant at 0 is freely choosable. In matrix terms, this corresponds to the matrix obtained from the bi-infinite identity matrix by dropping one row. But, with the condition changed to " ... has a unique bounded solution", the conjecture was proved in [1]. Received by the editors November 21, 1980. 1980 Mathematics Subject Classification. Primary 47B37; Secondary 15A09, 15A48.