Equilibrium distributions and degree of rational approximation of analytic functions

A theorem is proved on the degree of rational approximation of sequences of analytic functions given by Cauchy-type integrals of the form
$$ f_n(z)=\oint_F\Phi_n(t)f(t)(t-z)^{-1}\,dt,\qquad z\in E. $$
The theorem is formulated in terms connected with the equilibrium distribution of the charge on the plates of a capacitor $(E,F)$ under the assumption that an external field $\varphi=\lim_{n\to\infty}(2n)^{-1}\log|\Phi_n|^{-1}$ acts on the plate $F$, and this plate satisfies a certain symmetry condition in the field $\varphi$. The theorem is used to solve the problem of the degree of rational approximation of the function $e^{-x}$ on $[0,+\infty)$.
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