Optimal bases for a class of mixed spaces and their associated spline spaces
We consider spaces of the form span ? 1 , t ,  , t n ? 4 , u 1 ( t ) , u 2 ( t ) , u 3 ( t ) , u 4 ( t ) ? , where the functions ui u i (i=1,,4 i = 1 ,  , 4 ) are algebraic polynomials, or trigonometric or hyperbolic functions. We find intervals [0,?] [ 0 , ? ] where we can guarantee that the spaces possess normalized totally positive bases (and so, shape preserving representations). We construct their normalized B 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.
total positivity shape preserving representations splines trigonometric and hyperbolic functions