Which spline spaces for design?

International audienceWe recently determined the largest class of spaces of sufficient regularity which are suitable for design. How can weconnect different such spaces, possibly with the help of connection matrices, to produce the largest class of splinesusable for design? We present the answer to this question, along with some of the major difficulties encountered toestablish it.We would like to stress that the results we announce are far from being a straightforward generalisationof previous work on piecewise Chebyshevian splines