Optimal bases for a class of mixed spaces and their associated spline spaces

We consider spaces of the form spanh1; t; : : : ; tn􀀀4; u1.t/; u2.t/; u3.t/; u4.t/i; where the functions ui (i D 1; : : : ; 4) are algebraic polynomials, or trigonometric or hyperbolic functions. We find intervals T0; U where we can guarantee that the spaces possess normalized totally positive bases (and so, shape preserving representations). We construct their normalized B-bases (with optimal shape preserving properties). We also present a unified approach to deal with the associated spline spaces. A recursive procedure to obtain bases with optimal shape preserving and stability properties is presented.