Simultaneous approximation and interpolation of functions on continua in the complex plane

We construct polynomial approximations of Dzjadyk type (in terms of the k-th modulus of continuity, $k \ge 1$) for analytic functions defined on a continuum E in the complex plane, which simultaneously interpolate at given points of E. Furthermore, the error in this approximation is decaying as $e^{-cn^α}$ strictly inside E, where c and $α$ are positive constants independent of the degree n of the approximating polynomial.

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