ON THE OPTIMAL STABILITY OF THE BERNSTEIN BASIS

We show that the Bernstein polynomial basis on a given interval is ''o ptimally stable'' in the sense that no other nonnegative basis yields systematically smaller condition numbers for the values or roots of ar bitrary polynomials on that interval. This result follows from a parti al ordering of the set of all nonnegative bases that is induced by non negative basis transformations. We further show, by means of some low- degree examples, that the Bernstein form is not uniquely optimal in th is respect. However, it is the only optimally stable basis whose eleme nts have no roots on the interior of the chosen interval. These ideas are illustrated by comparing the stability properties of the power, Be rnstein, and generalized Ball bases.