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Mathematical Proceedings of the Cambridge Philosophical Society, 2004
Let \(\mathcal B\) be the set of all functions holomorphic in the unit disk \(\mathbb D\) and bounded by one. Each sequence \(\{z_n\}\) of points in \(\mathbb D\) gives rise to a family of Möbius transforms \(w_n(z)=(z+z_n)/(1+\bar{z}_n z)\). A Blaschke product \(B\) is said to be universal for \(\{z_n\}\) if the set of compositions \(\{B\circ w_n ...
Gorkin, Pamela, Mortini, Raymond
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Let \(\mathcal B\) be the set of all functions holomorphic in the unit disk \(\mathbb D\) and bounded by one. Each sequence \(\{z_n\}\) of points in \(\mathbb D\) gives rise to a family of Möbius transforms \(w_n(z)=(z+z_n)/(1+\bar{z}_n z)\). A Blaschke product \(B\) is said to be universal for \(\{z_n\}\) if the set of compositions \(\{B\circ w_n ...
Gorkin, Pamela, Mortini, Raymond
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Zero Tracts of Blaschke Products
Canadian Journal of Mathematics, 1966Let ﹛an﹜ be a sequence of complex numbers such thatandThen {an} is called a Blaschke sequence. For each Blaschke sequence {an} a Blaschke product is defined asThus a Blaschke product B(z, ﹛an﹜) is a function regular in the open unit disk D = {z: |z| < 1﹜ and having a zero at each point of the sequence ﹛an﹜.
Linden, C. N., Somadasa, H.
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On Interpolating Blaschke Products and Blaschke-Oscillatory Equations
Constructive Approximation, 2010This research is partially a continuation of the author's paper [``Growth estimates for logarithmic derivatives of Blaschke products and of functions in the Nevanlinna class'', Kodai Math. J. 30, No. 2, 263--279 (2007; Zbl 1134.30025)]. If \(\{z_n\}\) is an infinite sequence of nonzero points in the unit disc \(D = \{z: |z|
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Finite Products of Interpolating Blaschke Products
Journal of the London Mathematical Society, 1994The main result of this paper is a characterization of the Blaschke products \(B\) which are such that \(\tau_ \alpha (B)\) is a finite product of interpolating Blaschke products for all \(\alpha \in D\), the unit disc. That is Theorem. Let \(B\) be a finite product of interpolating Blaschke products. Let \(\{z_ n\}\) be the sequence of zeros of \(B\),
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Blaschke Products and Circumscribed Conics
Computational Methods and Function Theory, 2017By a canonical Blaschke product of degree \(d\), the author means a function of the form \[ B(z)=z\prod_{k=1}^{d-1}\frac{z-a_k}{1-\bar{a_k}z}\,, \qquad |a_k|
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Minimal Interpolation by Blaschke Products
Journal of the London Mathematical Society, 1985A sequence \(z_ n\) in the unit disc D is called an interpolating sequence if for every bounded sequence \(w_ n\) the problem (*) \(f(z_ n)=w_ n\), for all n is solvable with a function \(f\in H^{\infty}(D)\). Theorem 1: Assume (1) \(z_ n\) is an interpolating sequence, (2) any minimal solution (i.e. with least uniform norm) of problem (*) has norm 1, (
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Polynomials Versus Finite Blaschke Products
2013The aim of this chapter is to compare polynomials of one complex variable and finite Blaschke products and demonstrate that they share many similar properties. In fact, we collect many known results as well as some very recent results for finite Blaschke products here to establish a dictionary between polynomials and finite Blaschke products.
Tsang, CY, Ng, PTW
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On the Growth of Blaschke Products
Canadian Journal of Mathematics, 1969It is well known that the distribution of the zeros of an analytic function affects its rate of growth. The literature is too extensive to indicate here. We only point out (1, p. 27; 2; 3; 5), where the angular distribution of the zeros plays a role, as it will in this paper. In private communication, A.
MacLane, G. R., Rubel, L. A.
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