Spline interpolation at knot averages on a two-sided geometric mesh

Abstract:

For splines of degree $k \geqslant 1$ with knots $- {t_i} = {t_{2m + 1 - i}} = 1 + q + {q^2} + \cdots + {q^{m - i}}$, $i = 1, \ldots ,m$, where $0 < q < 1$, it is shown that spline interpolation to continuous functions at nodes ${\tau _i} = \Sigma _1^k{w_j}{t_{i + j}}$, $i = 1, \ldots ,n = 2m - k - 1$, has operator norm $\left \| P \right \|$ which is bounded independently of q and m as q tends to zero if and only if ${(1 - {w_1})^k} < \frac {1}{2}$, ${(1 - {w_k})^k} < \frac {1}{2}$, and ${w_j} > 0$, $j = 1, \ldots ,k$. The choice of nodes ${\tau _i} = \Sigma _0^{k + 1}{w_j}{t_{i + j}}$ and the growth rate of $\left \| P \right \|$ with k are also discussed.