Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 
 

 

Asymptotic properties of best $ L_{2}[0,\,1]$ approximation by splines with variable knots


Authors: D. L. Barrow and P. W. Smith
Journal: Quart. Appl. Math. 36 (1978), 293-304
MSC: Primary 41A15
DOI: https://doi.org/10.1090/qam/508773
MathSciNet review: 508773
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Abstract: Let $ S_N^k$ be the set of $ k$th-order splines on $ \left[ {0, 1} \right]$ having at most $ N - 1$ interior knots, counting multiplicities. We prove the following sharp asymptotic behavior of the error for the best $ {L_2}\left[ {0, 1} \right]$ approximation of a sufficiently smooth function $ f$ by the set $ S_N^k$:

$\displaystyle \mathop {\lim }\limits_{N \to \infty } {N^k}dist\left( {f, S_N^k}... ...ight)}}\left( \tau \right)} \right\vert} }^\sigma }d\tau } \right)^{1/\sigma }}$

, where $ \sigma = {\left( {k + 1/2} \right)^{ - 1}}$ and $ {B_{2k}}$ is the $ 2k$th Bernoulli number. Similar results have previously been obtained for piecewise polynomial (i.e., with no continuity constraints) approximation, but with different constant before the integral term. The approach we use is first to study the asymptotic behavior of dist($ f$, $ S_N^k\left( t \right)$), where $ S_N^k\left( t \right)$ is the linear space of $ k$th-order splines having simple knots determined from the fixed function $ t$ by the rule $ {t_i} \\ = t\left( {i/N} \right),i = 0,...,N$.

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DOI: https://doi.org/10.1090/qam/508773
Article copyright: © Copyright 1978 American Mathematical Society

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