Polynomials beyond polynomials
Résumé
On any given non-trivial real intervals, there exist linear spaces of functions with the same upper bounds on the numbers of zeroes as polynomial spaces. These spaces, called Extended Chebyshev (EC) spaces are useful in Approximation Theory as well as Geometric Design. There exists a famous procedure to build EC-spaces by means of weight functions and associated generalised derivatives. When all the weight functions are the constant function 1, the generalised derivatives are simply the ordinary derivatives, and the associated EC-spaces are polynomial spaces. Let us recall two well-known facts concerning the procedure in question: 1- two different systems of weight functions may lead to the same EC-space; 2- on a given closed bounded interval it yields all EC-spaces. However, identifying all systems of weight functions which can be associated with a given EC-space on a closed bounded interval remained an open question for a long time. This problem was recently solved. In particular, we now know all possible ways to associate generalised derivatives with polynomial spaces on closed bounded intervals. What we can we learn from this is the precise subject of the present talk.