Zeros and Coefficients
Zeros and coefficients Alexandre Eremenko∗ 10.28.2015 Abstract Asymptotic distribution of zeros of sequences of analytic functions is studied. Let fn be a sequence of analytic functions in a region D in the complex plane. The case of entire functions, D = C will be the most important. Sup- pose that there exist positive numbers Vn → ∞, such that the subharmonic functions 1 log |fn| areuniformlyboundedfromabove (1) Vn on every compact subset of D, and for some z0 ∈ D the sequence 1 log |fn(z0)| isboundedfrombelow. (2) Vn Under these conditions, one can select a subsequence such that for this sub- sequence 1 log |fn| → u, (3) Vn where u is a subharmonic function in D, u =6 −∞. This convergence can be understood in various senses, for example, quasi-everywhere, which means at every point except a set of points of zero logarithmic capacity, and also in ′ p D (Schwartz distributions), or in Lloc, 1 ≤ p< ∞. The corresponding Riesz measures converge weakly. The limit measure, −1 which is (2π) ∆u, describes the limit distribution of zeros of fn. ∗Supported by NSF grant DMS-1361836. 1 For all these facts we refer to [9] or [10]. This setting is useful in many cases when one is interested in asymptotic distribution of zeros of sequences of analytic functions. Let us mention several situations to which our results apply. 1. Let fn be monic polynomials of degree n, and Vn = n. We denote by µn the so-called empirical measures, which are the Riesz measures of (1/n)log |fn|. The measure µn is discrete, with finitely many atoms, with an atom of mass m at every zero of fn of multiplicity m.
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