Extremal and optimal properties of B-bases collocation matrices

Totally positive matrices are related with the shape preserving representations of a space of functions. The normalized B-basis of the space has optimal shape preserving properties. Bernstein polynomials, B-splines and rational Bernstein bases are examples of normalized B-bases. It is proven that the minimal eigenvalue and singular value of a collocation matrix of a normalized B-basis is bounded below by the minimal eigenvalue and singular value of the corresponding collocation matrix of any normalized totally positive basis of the same space. The optimal conditioning for the $$\infty $$ -norm of a collocation matrix of a normalized B-basis among all the normalized totally positive bases of a space of functions is also shown. Numerical examples confirm the theoretical results and answer related questions.