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The Theory of Best Approximation of Functions

✍ Scribed by V.L. Goncharov


Publisher
Elsevier Science
Year
2000
Tongue
English
Weight
301 KB
Volume
106
Category
Article
ISSN
0021-9045

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Research in approximation theory in Russia dates back to P. L. Chebyshev's memoir ``The orie des me canismes connus sous le nom de paralle logrammes'' (Me m. Pre s. Acad. Imp. Sci. Pe tersb. Divers Savants, 1854, VII, 539 568). This memoir posed the problem of the best approximation of functions by

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The object of this paper is to prove the following theorem: Let \(Y\) be a closed subspace of the Banach space \(X,(S, \Sigma, \mu)\) a \(\sigma\)-finite measure space, \(L(S, Y)\) (respectively, \(L(S, X)\) ) the space of all strongly measurable functions from \(S\) to \(Y\) (respectively, \(X\) ),

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Let \(\xi\) be an irrational number with simple continued fraction expansion \(\xi=\left[a_{0} ; a_{1}, a_{2}, \ldots, a_{i}, \ldots\right]\). Let the \(i\) th convergent \(p_{i} / q_{i}=\left[a_{0} ; a_{1}, a_{2}, \ldots, a_{i}\right]\). Let \(\mu=\) \(\left|\left[0 ; a_{n+2}, a_{n+3}, \ldots\right