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A Note on the Alexander Theorem on the Complex Plane

Sylwester Zając (2012)

Bulletin of the Polish Academy of Sciences. Mathematics

We investigate the Banach manifold consisting of complex r functions on the unit disc having boundary values in a given one-dimensional submanifold of the plane. We show that ∂/∂λ̅ restricted to that submanifold is a Fredholm mapping. Moreover, for any such function we obtain a relation between its homotopy class and the Fredholm index.

Admissible functions for the Dirichlet space

Javad Mashreghi, Mahmood Shabankhah (2010)

Studia Mathematica

Zero sets and uniqueness sets of the classical Dirichlet space are not completely characterized yet. We define the concept of admissible functions for the Dirichlet space and then apply them to obtain a new class of zero sets for . Then we discuss the relation between the zero sets of and those of .

Blaschke product generated covering surfaces

Ilie Barza, Dorin Ghisa (2009)

Mathematica Bohemica

It is known that, under very general conditions, Blaschke products generate branched covering surfaces of the Riemann sphere. We are presenting here a method of finding fundamental domains of such coverings and we are studying the corresponding groups of covering transformations.

Higher-order linear differential equations with solutions having a prescribed sequence of zeros and lying in the Dirichlet space

Li-Peng Xiao (2015)

Annales Polonici Mathematici

The aim of this paper is to consider the following three problems:i (1) for a given uniformly q-separated sequence satisfying certain conditions, find a coefficient function A(z) analytic in the unit disc such that f”’ + A(z)f = 0 possesses a solution having zeros precisely at the points of this sequence; (2) find necessary and sufficient conditions for the differential equation f ( k ) + A k - 1 f ( k - 1 ) + + A f ' + A f = 0 (*) in the unit disc to be Blaschke-oscillatory; (3) find sufficient conditions on the analytic coefficients of the...

Interpolation of bounded sequences

Francesc Tugores (2010)

Czechoslovak Mathematical Journal

This paper deals with an interpolation problem in the open unit disc 𝔻 of the complex plane. We characterize the sequences in a Stolz angle of 𝔻 , verifying that the bounded sequences are interpolated on them by a certain class of not bounded holomorphic functions on 𝔻 , but very close to the bounded ones. We prove that these interpolating sequences are also uniformly separated, as in the case of the interpolation by bounded holomorphic functions.

On Some Correspondence between Holomorphic Functions in the Unit Disc and Holomorphic Functions in the Left Halfplane

Ewa Ligocka (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

We study a correspondence L between some classes of functions holomorphic in the unit disc and functions holomorphic in the left halfplane. This correspondence is such that for every f and w ∈ ℍ, exp(L(f)(w)) = f(expw). In particular, we prove that the famous class S of univalent functions on the unit disc is homeomorphic via L to the class S(ℍ) of all univalent functions g on ℍ for which g(w+2πi) = g(w) + 2πi and l i m R e z - ( g ( w ) - w ) = 0 .

On some new sharp embedding theorems in minimal and pseudoconvex domains

Romi F. Shamoyan, Olivera R. Mihić (2016)

Czechoslovak Mathematical Journal

We present new sharp embedding theorems for mixed-norm analytic spaces in pseudoconvex domains with smooth boundary. New related sharp results in minimal bounded homogeneous domains in higher dimension are also provided. Last domains we consider are domains which are direct generalizations of the well-studied so-called bounded symmetric domains in n . Our results were known before only in the very particular case of domains of such type in the unit ball. As in the unit ball case, all our proofs are...

On the composition of Frostman Blaschke products

John R. Akeroyd, Pamela Gorkin (2013)

Studia Mathematica

We construct an infinite uniform Frostman Blaschke product B such that B ∘ B is also a uniform Frostman Blaschke product. We also show that the set of uniform Frostman Blaschke products is open in the set of inner functions with the uniform norm.

On the extendability of quadratic polynomial mappings of the plane

Ewa Ligocka (2009)

Annales Polonici Mathematici

We shall prove, using the result from our previous paper [Ann. Polon. Math. 88 (2006)], that for a quadratic polynomial mapping Q of ℝ² only the geometric shape of the critical set of Q determines whether the complexification of Q can be extended to an endomorphism of ℂℙ². At the end of the paper we describe some interesting classes of quadratic polynomial mappings of ℝ² and give some examples.

Restricted interpolation by meromorphic inner functions

Alexei Poltoratski, Rishika Rupam (2016)

Concrete Operators

Meromorphic Inner Functions (MIFs) on the upper half plane play an important role in applications to spectral problems for differential operators. In this paper, we survey some recent results concerning function theoretic properties of MIFs and show their connections with spectral problems for the Schrödinger operator.

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