On approximation of functions by harmonic polynomials

For certain finite continua $\mathfrak M\subset\mathbf R^2$ with simply connected complements $\Omega=C\mathfrak M$, the direct problem of using harmonic polynomials to approximate realvalued functions continuous on $\mathfrak M$, harmonic on its interior, and having a specified majorant for their moduli of continuity is solved. As in the case of approximation of functions continuous on $\mathfrak M$ and analytic in $\mathring{\mathfrak M}$ by analytic polynomials, the estimates obtained depend on the distance from the boundary points of $\mathfrak M$ to the level curves of the function mapping $\Omega$ conformally onto the exterior of the unit disk with the standard normalization at $\infty$.
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