521 research outputs found

    Prescribing inner parts of derivatives of inner functions

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    Let ℐ be the set of inner functions whose derivative lies in the Nevanlinna class. We show that up to a post-composition with a Möbius transformation, an inner function F ∈ ℐ is uniquely determined by the inner part of its derivative. We also characterize inner functions which can be represented as Inn Fâ€Č for some F ∈ ℐ in terms of the associated singular measure, namely, it must live on a countable union of Beurling–Carleson sets. This answers a question raised by K. Dyakonov

    Prescribing inner parts of derivatives of inner functions

    Get PDF
    Let ℐ be the set of inner functions whose derivative lies in the Nevanlinna class. We show that up to a post-composition with a Möbius transformation, an inner function F ∈ ℐ is uniquely determined by the inner part of its derivative. We also characterize inner functions which can be represented as Inn Fâ€Č for some F ∈ ℐ in terms of the associated singular measure, namely, it must live on a countable union of Beurling–Carleson sets. This answers a question raised by K. Dyakonov

    Calculating non-equidistant discretizations generated by Blaschke products

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    The argument functions of Blaschke products provide a very elegant way of handling non-uniformity of discretizations. In this paper we analyse the efficiency of numerical methods as the bisection method and Newton's method in the case of calculating non-equidistant discretizations generated by Blaschke products. By taking advantage of the strictly increasing property of argument functions we may calculate the discrete points in an enhanced order — to be introduced here. The efficiency of the discrete points' sequential calculation in this order is significantly increased compared to the naive implementation. In our research we are primarily motivated by ECG curves which usually have alternating regions of high or low variability, and therefore different degree of discretization is needed at different regions of the signals

    Schwarz reflections and the Tricorn

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    We continue our study of the family S\mathcal{S} of Schwarz reflection maps with respect to a cardioid and a circle which was started in [LLMM1]. We prove that there is a natural combinatorial bijection between the geometrically finite maps of this family and those of the basilica limb of the Tricorn, which is the connectedness locus of quadratic anti-holomorphic polynomials. We also show that every geometrically finite map in S\mathcal{S} arises as a conformal mating of a unique geometrically finite quadratic anti-holomorphic polynomial and a reflection map arising from the ideal triangle group. We then follow up with a combinatorial mating description for the "periodically repelling" maps in S\mathcal{S}. Finally, we show that the locally connected topological model of the connectedness locus of S\mathcal{S} is naturally homeomorphic to such a model of the basilica limb of the Tricorn
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