Exploiting parity in converting to and from Bernstein polynomials and orthogonal polynomials

Abstract

The coefficients of a polynomial in the Bernstein basis can be converted to the coefficients of a Legendre or Chebyshev series by a simple matrix–vector multiply at a cost of O(2[ N + 1] 2) operations where N is the degree of the polynomial. In this note, we show that by exploiting parity with respect to the center of the interval x ∈ [0, 1], is possible to halve the cost. In d dimensions with a tensor product basis, the savings are a factor of two independent of d.