Generalized (0,m1,m2, . . . ,ml) interpolation and extrapolation based on splines

In this article, we introduce some explicit formulas for the generalized case of (0,m1,m2, . . . ,ml) Birkhoff interpolation and extrapolation via splines, where the elements of {mi}^l_ i=1 are not limited to the successive and finite natural numbers. Error analysis is presented and an example is given to illuminate the details and efficiency of the procedure. As a special case the existing research works for the nominal and limited values of {mi}^l_ i=1 can be obtained.