On the rate of approximation of closed Jordan curves by lemniscatesстатья
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Дата последнего поиска статьи во внешних источниках: 7 декабря 2016 г.
Аннотация:For any natural number n and any C > 0, we obtain an integral formula for calculating the lengths |L(Pn, C)| of the lemniscates L(Pn,C):={z:|Pn(z)|=C} of algebraic polynomials Pn(z):= zn + cn−1zn−1 + ... + c0 in the complex variable z with complex coefficients cj, j = 0, ..., n − 1, and establish the upper bound for the quantities λn:=sup{|L(Pn,1)|:Pn(z)}, which is currently best for 3 ≤ n ≤ 1014. We also study the properties of the derivative S′(C) of the area function S(C) of the set {z: |Pn(z)| ≤ C}.