Weighted holomorphic polynomial approximation
Résumé
For G an open set in C and W a non-vanishing holomorphic function in G, in the late 1990 's, Pritsker and Varga [13] characterized pairs (G, W ) having the property that any f holomorphic in G can be locally uniformly approximated in G by weighted holomorphic polynomials {W (z) n p n (z)}, deg(p n ) ≤ n. We further develop their theory in first proving a quantitative Bernstein-Walsh type theorem for certain pairs (G, W ). Then we consider the special case where W (z) = 1/(1 + z) and G is a loop of the lemniscate {z ∈ C : |z(z +1)| = 1/4}. We show the normalized measures associated to the zeros of the n-th order Taylor polynomial about 0 of the function (1 + z) -n converge to the weighted equilibrium measure of G with weight |W | as n → ∞. This mimics the motivational case of Pritsker and Varga [12] where G is the inside of the Szegő curve and W (z) = e -z . Lastly, we initiate a study of weighted holomorphic polynomial approximation in C n , n > 1.
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