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Approximation by harmonic polynomials

In this work we studied the approximation of the fundamental solution of the Laplace operator in two-dimensional space u = log(|x|) by harmonic polynomials. We analyzed the best approximation in the semi-ring with fixed outer radius and inner radius ? tending to zero. We observed exponential convergence in the degree of polynomials used for approximation. However, with inner radius tending to zero the rate of convergence gets worse. The convergence still remains exponential but with some exponent for the degree of polynomials.