New approach to study splines by blossoming method and application to the construction of a bivariate C1C1 quartic quasi-interpolant

This work is a contribution in the approximation theory for studying and analyzing piecewise polynomial functions (splines), which uses the blossoming approach. Some existing results in the literature are reformulated, such as the smoothness conditions between polynomials of a spline, by using the affinity property of the blossom. Some definitions of sub-splines are proposed which can be very useful in the study and the construction of splines such as macro-elements or quasi-interpolants. As an application of the proposed results, a C1C1 quartic spline quasi-interpolant with optimal approximation order is defined without using any mask for smoothness or B-spline basis. Numerical results are presented and compared with other methods given in the literature.