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 (( - )ⁱ). 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 (( - )i + jA- 1(i,j)) or its inverse is again totally positive.
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Carl de Boor (1982) studied this question.
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