On the explicit representation of orthonormal Bernstein polynomials

In this work we present an explicit representation of the orthonormal Bernstein polynomials and demonstrate that they can be generated from a linear combination of non-orthonormal Bernstein polynomials. In addition, we report a set of $n$ Sturm-Liouville eigenvalue equations, where each of the $n$ eigenvalue equations have the orthonormal Bernstein polynomials of degree $n$ as their solution set. We also show that each of the $n$ Sturm-Liouville operators are naturally self-adjoint. While the orthonormal Bernstein polynomials can be used in a variety of different applications, we demonstrate the utility of these polynomials here by using them in a generalized Fourier series to approximate curves and surfaces. Using the orthonormal Bernstein polynomial basis, we show that highly accurate approximations to curves and surfaces can be obtained by using small sized basis sets. Finally, we demonstrate how the orthonormal Bernstein polynomials can be used to find the set of control points of Bezier curves or Bezier surfaces that best approximate a function.