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

We consider spaces of the form span〈1,t,…,tn−4,u1(t),u2(t),u3(t),u4(t)〉, where the functions uiui (i=1,…,4i=1,…,4) are algebraic polynomials, or trigonometric or hyperbolic functions. We find intervals [0,α][0,α] where we can guarantee that the spaces possess normalized totally positive bases (and so, shape preserving representations). We construct their normalized BB-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.