Constructing totally positive piecewise Chebyshevian B-spline bases

We consider piecewise Chebyshevian splines, in the sense of splines with pieces taken from any different five-dimensional Extended Chebyshev spaces, and with connection matrices at the knots. In this large context we establish necessary and sufficient conditions for the existence of totally positive refinable B-spline bases. These conditions are applied in many important special cases, e.g. symmetric cardinal geometrically continuous quartic B-spline, parametrically continuous mixed L-splines. The great variety of illustrations provided proves the richness of this class of splines for design. This richness can be exploited in various other fields as well.