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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

$ H\sp{r,}\,\sp{\infty }(R)$- and $ W\sp{r,\infty }(R)$-splines


Author: Philip W. Smith
Journal: Trans. Amer. Math. Soc. 192 (1974), 275-284
MSC: Primary 41A65
DOI: https://doi.org/10.1090/S0002-9947-1974-0367538-6
MathSciNet review: 0367538
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Abstract: Let E be a subset of R the real line and $ f:E \to R$. Necessary and sufficient conditions are derived for $ \inf (\left\Vert{D^r}x\right\Vert _{{L^\infty }}:x{\vert _E} = f)$ to have a solution. When restricted to quasi-uniform partitions E, necessary and sufficient conditions are derived for the solution to be in $ {L^\infty }$. For finite partitions E it is shown that a solution to the $ {L^\infty }$ infimum problem can be obtained by solving $ \inf (\left\Vert{D^r}x\right\Vert _{{L^p}}:x{\vert _E} = f)$ and letting p go to infinity. In this way it was discovered that solutions to the $ {L^\infty }$ problem could be chosen to be piecewise polynomial (of degree r or less). The solutions to the $ {L^p}$ problem are called $ {H^{r,p}}$-splines and were studied extensively by Golomb in [3].


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DOI: https://doi.org/10.1090/S0002-9947-1974-0367538-6
Keywords: Spline
Article copyright: © Copyright 1974 American Mathematical Society