A Generalized Asymptotic Upper Bound on Fast Polynomial Evaluation and Interpolation

It is shown in this paper that the evaluation and interpolation problems corresponding to a set of points, $\{ x_i \} _{i = 0}^{n - 1} $, with $(c_i - 1)$ higher derivatives at each $x_i $ such that $\sum _{i = 1}^{n - 1} c_i = N$, can be solved in $O([N\log N][(\log n) + 1])$ steps.l This upper bound matches perfectly with the known upper bounds of the two extreme cases, which are $O(N\log ^2 N)$ and $O(N\log N)$ steps when $n = N$ and $n = 1$, respectively.