Approximation in a  \(\mathcal {C}^1\) Cubic Super-Spline Space

In this paper, a suitable normalized B-spline-like basis is constructed for a \(\mathcal {C}^1\) cubic super-spline space defined on an initial partition refined by adding a point in each sub-interval. These compactly supported non-negative functions are geometrically constructed and form a convex partition of unity. As an application, explicit formulae for the coefficients in the B-spline-like basis representation of the Hermite interpolant of given data are derived. Moreover, by blossoming Marsden’s identity, we investigate local quasi-interpolation schemes reproducing cubic polynomials.