Lacunary Interpolation by Splines
A. Meir, Abhinav Sharma · SIAM Journal on Numerical Analysis · 1973
It is shown that for arbitrary lacunary data $\{ y_ u \} _{ u = 0}^n ,\{ y''_ u \} _{ u = 0}^n $ there exist unique (up to boundary conditions) quintic splines $S_n (x) \in C^3 [0,1]$ with joints at ${ u / n}$ such that $S_n ({ u / n}) = y_ u ,S''_n ({ u / n}) = y''_ u $. Moreover, if the given data $y_ u ,y''_ u $ are the values, respectively second derivatives, of a function f satisfying certain smoothness conditions, then \[ \left\| {S_n^{(r)} (x) - f^{(r)} (x)} \right\| \to 0 \] as $n \to \infty $, for every $r,0 \leqq r \leqq 3$.