8,824 research outputs found

    Bounded Fatou and Julia components of meromorphic functions

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    We completely characterise the bounded sets that arise as components of the Fatou and Julia sets of meromorphic functions. On the one hand, we prove that a bounded domain is a Fatou component of some meromorphic function if and only if it is regular. On the other hand, we prove that a planar continuum is a Julia component of some meromorphic function if and only if it has empty interior. We do so by constructing meromorphic functions with wandering continua using approximation theory.Comment: 15 pages, 4 figures. V2: We have revised the introduction, and introduced two new sections: Section 2 discusses and compare topological properties of Fatou components, while Section 3 establishes that certain bounded regular domains cannot arise as eventually periodic Fatou components of meromorphic function

    An Lp Analog to AAK Theory for p⩾2

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    AbstractWe develop an Lp analog to AAK theory on the unit circle that interpolates continuously between the case p=∞, which classically solves for best uniform meromorphic approximation, and the case p=2, which is equivalent to H2-best rational approximation. We apply the results to the uniqueness problem in rational approximation and to the asymptotic behaviour of poles of best meromorphic approximants to functions with two branch points. As pointed out by a referee, part of the theory extends to every p∈[1, ∞] when the definition of the Hankel operator is suitably generalized; this we discuss in connection with the recent manuscript by V. A. Prokhorov, submitted for publication

    Super-optimal approximation by meromorphic functions.

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    Let G be a matrix function of type m × n and suppose that G is expressible as the sum of an H∞ function and a continuous function on the unit circle. Suppose also that the (k – 1)th singular value of the Hankel operator with symbol G is greater than the kth singular value. Then there is a unique superoptimal approximant to G in : that is, there is a unique matrix function Q having at most k poles in the open unit disc which minimizes s∞(G – Q) or, in other words, which minimizes the sequence with respect to the lexicographic ordering, where and Sj(·) denotes the jth singular value of a matrix. This result is due to the present authors [PY1] in the case k = 0 (when the hypothesis on the Hankel singular values is vacuous) and to S. Treil[T2] in general. In this paper we give a proof of uniqueness by a diagonalization argument, a high level algorithm for the computation of the superoptimal approximant and a recursive parametrization of the set of all optimal solutions of a matrix Nehari—Takagi problem

    RG solutions for \alpha_s at large N_c in d=3+1 QCD

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    Solutions of RG equations for β(α)\beta(\alpha) and α(Q)\alpha(Q) are found in the class of meromorphic functions satisfying asymptotic conditions at large QQ (resp. small α)\alpha), and analyticity properties in the Q2Q^2 plane. The resulting αR(Q)\alpha_R(Q) is finite in the Euclidean Q2Q^2 region and agrees well at Q≥1Q\geq 1 GeV with the MSˉαs(Q)\bar{MS} \alpha_s(Q).Comment: 11 pages, no figures, dedicated to the 70th birthday of Professor Francesco Calogero, subm. to the Journal of Nonlinear Mathematical Physic
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