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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

Another extremal property of perfect splines


Authors: T. N. T. Goodman and S. L. Lee
Journal: Proc. Amer. Math. Soc. 70 (1978), 129-135
MSC: Primary 41A15
MathSciNet review: 0481760
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Abstract: Let $ {\mathbf{t}} = \{ {t_i}\} ,i = 1,2, \ldots ,n + k$, be a given sequence in [a, b], $ {f_0} \in W_\infty ^k[a,b],A > {\left\Vert {f_0^{(k)}} \right\Vert _\infty }$, and $ F = \{ f \in W_\infty ^k[a,b]:f\vert{\mathbf{t}} = {f_0}\vert{\mathbf{t}}$ and $ {\left\Vert {{f^{(k)}}} \right\Vert _\infty } \leqslant A\} $. We show that F contains precisely two perfect splines g, h of degree k with $ \vert{g^{(k)}}\vert = \vert{h^{(k)}}\vert = A$ and n interior nodes, and for all $ f \in F$, $ \min (g(x),h(x)) \leqslant f(x) \leqslant \max (g(x),h(x));\forall x \in [a,b]$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1978-0481760-9
PII: S 0002-9939(1978)0481760-9
Keywords: Perfect splines
Article copyright: © Copyright 1978 American Mathematical Society