Gelfond–Bézier curves

We show that the generalized Bernstein bases in Müntz spaces defined by Hirschman and Widder (1949) and extended by Gelfond (1950) can be obtained as pointwise limits of the Chebyshev–Bernstein bases in Müntz spaces with respect to an interval [a,1][a,1] as the positive real number a converges to zero. Such a realization allows for concepts of curve design such as de Casteljau algorithm, blossom, dimension elevation to be transferred from the general theory of Chebyshev blossoms in Müntz spaces to these generalized Bernstein bases that we termed here as Gelfond–Bernstein bases. The advantage of working with Gelfond–Bernstein bases lies in the simplicity of the obtained concepts and algorithms as compared to their Chebyshev–Bernstein bases counterparts.

► Show that Gelfond–Bernstein bases in Müntz spaces are limit of Chebyshev–Bernstein bases. ► Give a Schur function expression of Gelfond–Bernstein bases. ► Define the blossom of Gelfond–Bézier curves. ► Derive the de Casteljau algorithm in Müntz spaces. ► Show the conditions for the convergence of the dimension elevation algorithm to the Gelfond–Bézier curve.