TABLE OF CONTENTS We have discussed a constructive approach to the introduction of B-splines using a recursive definition. We shall introduce several approaches below that stem from very different views.
This view of splines was first noted by Curry and Schoenberg [25] and later expanded by deBoor [28].
Select n > 0. Define P : 
? R by P( t 1, , t n) = t 1, the projection operator on the first coordinate. Given { x 0, , x n}, in R, not all the same value, choose points P i in 
such that P[ P i] = x i and { P 0, , P n} form the vertices of an arbitrary n-simplex, ?. It is easily shown that this can be done by a constructive example. Note, however that the simplex obtained is only one of the infinite number possible. For the purposes of example, suppose x 0 ? x 1, and choose P 0 = ( x 0, 0, , 0), P 1 = ( x 1, 0, , 0). Let P i = ( x i, 0, , ? i , j, 0, , 0), i = 2, , n, where ? i , j = 1 if i = j and ? i
