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Unicity of best mean approximation by second order splines with variable knots

Authors: D. L. Barrow, C. K. Chui, P. W. Smith and J. D. Ward
Journal: Math. Comp. 32 (1978), 1131-1143
MSC: Primary 41A15
MathSciNet review: 0481754
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Abstract: Let $ S_N^2$ denote the nonlinear manifold of second order splines defined on [0, 1] having at most $ N$ interior knots, counting multiplicities. We consider the ques tion of unicity of best approximations to a function $ f$ by elements of $ S_N^2$. Approximation relative to the $ {L_2}[0,1]$ norm is treated first, with the results then extended to the best $ {L_1}$ and best one-sided $ {L_1}$ approximation problems. The conclusions in each case are essentially the same, and can be summarized as follows: a sufficiently smooth function $ f$ satisfying $ f'' > 0$ has a unique best approximant from $ S_N^2$ provided either $ \log f''$ is concave, or $ N$ is sufficiently large, $ N \geqslant {N_0}(f)$; for any $ N$, there is a smooth function $ f$, with $ f'' > 0$, having at least two best approximants. A principal tool in the analysis is the finite dimensional topological degree of a mapping.

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Article copyright: © Copyright 1978 American Mathematical Society