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

Abstract

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 u i ( i = 1 , … , 4 ) are algebraic polynomials, or trigonometric or hyperbolic functions. We find intervals [ 0 , α ] 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.