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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

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


Author: M. J. Marsden
Journal: Math. Comp. 38 (1982), 113-126
MSC: Primary 41A15; Secondary 65D07
MathSciNet review: 637290
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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\Vert P \right\Vert$ 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\Vert P \right\Vert$ with k are also discussed.


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DOI: http://dx.doi.org/10.1090/S0025-5718-1982-0637290-3
PII: S 0025-5718(1982)0637290-3
Article copyright: © Copyright 1982 American Mathematical Society