Rational Askey–Wilson Bernstein bases and a multirational Askey–Wilson blossom

Abstract

We introduce and study the properties of new negative degree rational Bernstein bases associated with the Askey–Wilson operator and we use these bases to define new types of rational Bernstein-Bézier curves. We also introduce a new type of blossom, the multirational Askey–Wilson blossom. We prove that four axioms uniquely characterize this blossom and we provide an explicit formula for this multirational blossom involving a right inverse of the Askey–Wilson operator. A formula for the coefficients of a function expanded in a rational Askey–Wilson Bernstein basis in terms of certain values of the Askey–Wilson operator is derived. We also establish a dual functional property that expresses the coefficients of these new types of rational Bernstein–Bézier curves in terms of values of their multirational Askey–Wilson blossom.