= Total nonnegativity of B-spline collocation matrices
For increasing sites and an ordered nonnegative normalized <B-spline> basis, the <matrix> $A_{ij}=N_j(x_i)$ is a <totally nonnegative matrix>. One way to see the minors' signs is <knot insertion>. Each inserted knot changes coefficients by $b_j=\alpha_j a_j+(1-\alpha_j)a_{j-1}$ with $0\leq\alpha_j\leq1$, using the standard constant endpoint pieces. This is a nonnegative rectangular bidiagonal map, whose minors are nonnegative. Products preserve that property by the <Cauchy–Binet formula>. Insert each site to full knot multiplicity; a suitable refined coefficient is then the value $s(x_i)$. Thus collocation is an ordered row submatrix of the refinement map, and has nonnegative minors. The strict support conditions $t_i<x_i<t_{i+k}$ give invertibility by the <Schoenberg–Whitney theorem>; its inverse has the <checkerboard inverse of a totally nonnegative matrix> sign pattern.
Back to article page
