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: $[n/n]_{\hat\mu}=P_n/Q_n$, $n=1,2,\dots$ .
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.