Re nement of Vectors of Bernstein Polynomials

For the case of Bernstein polynomials the re nement mask is calculated recursively and the re nement matrices are given explicitely Moreover the eigenvectors of the transposed re nement matrices are constructed whereas the eigenvectors of the re nement matrices themselves can be determined by a theorem of Micchelli and Prautzsch INTRODUCTION Let n N and let b t b t bn t T be a vector of uniformly re nable real functions on i e there are n n matrices A Ak k N k such that b n t m k Am b n t is satis ed for m k and t These equations are called re nement equations and the matrices Am re nement matrices cf Micchelli and Prautzsch It is well known that the re nement equations are closely connected with corresponding subdivision algorithms which provide important techniques for the fast generation of curves cf In and some applications of such equations in the theory of wavelets are discussed For polynomials bi t i n spanning the vector space of all polynomials of degree n the matrices Am in always exist and are uniquely determined Here we consider these matrices in the case of Bernstein polynomials bi t n i t t n i i n and study their spectral properties In particular we prove a recursion formula for the re nement mask of b