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Steklov Mathematical Institute Seminar
October 25, 2007 16:00, Moscow, Steklov Mathematical Institute of RAS, Conference Hall (8 Gubkina)
 


On some fundamental properties and applications of Chebyshëv continued fractions

S. P. Suetin

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
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S. P. Suetin
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Abstract: In his study in 1855 of the problem of the optimal approximate recovery of a function $F$ from its values given at finitely many nodes on real axis $\mathbb R$, P. Chebyshëv gave a precise answer in terms of the parameters of a continued (“Chebyshëv”) fraction. That fraction can be constructed directly from the Loran coefficients of the expansion at the point $z=\infty$ of the function
$$ \hat\mu(z):=\int_S\frac{d\mu(x)}{z-x} , $$
where $\mu$ is a positive Borel measure with support $\operatorname{supp}\mu=:S\Subset\mathbb R$.
It is in this way that P. Chebyshëv discovered general orthogonal polynomials corresponding to an arbitrary positive Borel measure $\mu$. Such polynomials arised in a natural way as the denominators $Q_n$ of the $n$th convergents $P_n/Q_n$ to the Chebyshëv continued fraction.
For the function $\hat\mu$ the Chebyshëv continued fraction produces precisely the sequence of its diagonal Padé approximants: $[n/n]_{\hat\mu}=P_n/Q_n$, $n=1,2,…$ .
In the talk we shall consider some fundamental properties and numerical applications of Chebyshëv continued fractions for the functions of more general type then $\hat\mu$ is.

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