A Bernstein-like operator for a mixed algebraic-trigonometric space

Bernstein-like operators are useful to measure the approximation properties of a vector space of functions. Especially important for measuring the degree of approximation and describing these approximations are their spectral properties. A property usually required for these approximation spaces is that there exists a normalized totally positive basis because it allows us to provide shape preserving representations. The normalized B-basis of a space has optimal shape preserving properties. We present the normalized B-basis of the space T̄1/2 generated by 1, t, cos t, sin t, cos(t/2), sin(t/2), computed using the techniques in the paper [2]. The spectral properties of the Bernstein-like operator associated to this space are discussed and indicate how close is the curve to its control polygon. The third greatest eigenvalue gives a rough idea of the approximation power of the space. For the classical Bernstein operator, the third eigenvalue is λ2 = 0.8 and for the space T̄1/2 we find λ2 ≈ 0.7842. In contrast to the classical Bernstein operator, whose corresponding Greville abscissae are strictly increasing, we find coincident abscissae in this case, leading to an example of a Bernstein-like operator which is not a bijection from the space to itself.