Mixed hyperbolic/trigonometric spaces for design

We investigate the class of five-dimensional null spaces of linear differential operators with constant coefficients and odd characteristic polynomials. One of the advantages of this class is that it permits to mix trigonometric and hyperbolic functions within the same space, and we will more specially focus on this interesting blending. Whenever necessary we determine the critical lengths for design. This yields the largest possible intervals on which existence of Bernstein bases is guaranteed, such bases being then automatically the optimal normalised totally positive bases. This also enables us to show the interest of this class of spaces for geometric design.