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

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.