Unicity of best mean approximation by second order splines with variable knots
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- by D. L. Barrow, C. K. Chui, P. W. Smith and J. D. Ward PDF
- Math. Comp. 32 (1978), 1131-1143 Request permission
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.References
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Additional Information
- © Copyright 1978 American Mathematical Society
- Journal: Math. Comp. 32 (1978), 1131-1143
- MSC: Primary 41A15
- DOI: https://doi.org/10.1090/S0025-5718-1978-0481754-1
- MathSciNet review: 0481754