A new approach to deal with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e4605" altimg="si163.svg"><mml:msup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math> cubic splines and its application to super-convergent quasi-interpolation

In this paper, we construct a novel normalized B-spline-like representation for C2-continuous cubic spline space defined on an initial partition refined by inserting two new points inside each sub-interval. The basis functions are compactly supported non-negative that geometrically constructed and form convex of unity. With the help control polynomial theory introduced herein, Marsden identity is derived, from which several families super-convergent quasi-interpolation operators defined.