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3x+1 inverse orbit generating functions almost always have natural boundaries

Jason P. Bell, Jeffrey C. Lagarias (2015)

Acta Arithmetica

The 3x+k function T k ( n ) sends n to (3n+k)/2, resp. n/2, according as n is odd, resp. even, where k ≡ ±1 (mod 6). The map T k ( · ) sends integers to integers; for m ≥1 let n → m mean that m is in the forward orbit of n under iteration of T k ( · ) . We consider the generating functions f k , m ( z ) = ∑ n > 0 , n → m z n , which are holomorphic in the unit disk. We give sufficient conditions on (k,m) for the functions f k , m ( z ) to have the unit circle |z|=1 as a natural boundary to analytic continuation. For the 3x+1 function these conditions hold for all m...

An analogue of Montel’s theorem for some classes of rational functions

R. K. Kovacheva, Julian Lawrynowicz (2002)

Czechoslovak Mathematical Journal

For sequences of rational functions, analytic in some domain, a theorem of Montel’s type is proved. As an application, sequences of rational functions of the best L p -approximation with an unbounded number of finite poles are considered.

Analytic continuation of Dirichlet series.

J. Milne Anderson, Dimitry Khavinson, Harold S. Shapiro (1995)

Revista Matemática Iberoamericana

The questions considered in this paper arose from the study [KS] of I. Fredholm's (insufficient) proof that the gap series Σ0∞ an ζn2 (where 0 < |a| < 1) is nowhere continuable across {|ζ| = 1}. The interest of Fredholm's method ([F],[ML]) is not so much its efficacy in proving gap theorems (indeed, much more general results can be got by other means, cf. the Fabry gap theorem in [Di]) as in the connection it made between certain special gap series and partial differential equations...

Boundary behavior and Cesàro means of universal Taylor series.

Frédéric Bayart (2006)

Revista Matemática Complutense

We study boundary properties of universal Taylor series. We prove that if f is a universal Taylor series on the open unit disk, then there exists a residual subset G of the unit circle such that f is unbounded on all radii with endpoints in G. We also study the effect of summability methods on universal Taylor series. In particular, we show that a Taylor series is universal if and only if its Cesàro means are universal.

Convolution operators on spaces of holomorphic functions

Tobias Lorson, Jürgen Müller (2015)

Studia Mathematica

A class of convolution operators on spaces of holomorphic functions related to the Hadamard multiplication theorem for power series and generalizing infinite order Euler differential operators is introduced and investigated. Emphasis is placed on questions concerning injectivity, denseness of range and surjectivity of the operators.

Determination of the pluripolar hull of graphs of certain holomorphic functions

Armen Edigarian, Jan Wiegerinck (2004)

Annales de l’institut Fourier

Let A be a closed polar subset of a domain D in ℂ . We give a complete description of the pluripolar hull Γ D × ℂ * of the graph Γ of a holomorphic function defined on D ∖ A . To achieve this, we prove for pluriharmonic measure certain semi-continuity properties and a localization principle.

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