Approximation by GP box-splines on a four-direction mesh

On a four-directional uniform mesh of the plane, we define and study C0C0 Gori–Pitolli (abbreviated as GP) quadratics and C1C1 GP quartics generated by integer translates of GP box-splines generalizing respectively the classical C1C1 quadratic box-spline and one of the C2C2 quartic box-splines. More specifically, we study the approximation of a bivariate function ff by some quasi-interpolants and interpolants with values in these generalized spline spaces.