Sequences Of Holomorphic Functions Research Articles

Integral representation of superoscillations via complex Borel measures and their convergence

In the last decade there has been a growing interest in superoscillations in various fields of mathematics, physics and engineering. However, while in applications as optics the local oscillatory behaviour is the important property, some convergence to a plane wave is the standard characterizing feature of a superoscillating function in mathematics and quantum mechanics. Also there exists a certain discrepancy between the representation of superoscillations either as generalized Fourier series, as certain integrals or via special functions. The aim of this work is to close these gaps and give a general definition of superoscillations, covering the well-known examples in the existing literature. Superoscillations will be defined as sequences of holomorphic functions, which admit integral representations with respect to complex Borel measures and converge to a plane wave in the space A 1 ( C ) \mathcal {A}_1(\mathbb {C}) of entire functions of exponential type.

Wandering Montel theorems for Hilbert space valued holomorphic functions

We prove that if { u k } \{ u^k \} is a sequence of holomorphic functions that takes values in an infinite dimensional Hilbert space H \mathcal {H} , there are unitaries { U k } \{ U^k \} on H \mathcal {H} so that U k u k U^k u^k has a subsequence that converges locally uniformly. We also prove a non-commutative version of this result.

Normality concerning the sequence of functions

Let {fn} be a sequence of functions meromorphic in a domain D, let {hn} be a sequence of holomorphic functions in D, such that \({h_n}\left( z \right)\mathop \Rightarrow \limits^\chi h\left( z \right)\) , where h(z) ≢ 0 is holomorphic in D, and let k be a positive integer. If for each n ∈ ℕ+, fn(z) ≠ 0 and fn(k) − hn(z) has at most k distinct zeros (ignoring multiplicity) in D, then {fn} is normal in D.

Boundary properties of sequences of holomorphic functions

Boundary properties of sequences of holomorphic functions

The multiple residue and the weight filtration on the logarithmic de Rham complex

We study the multiple residue of logarithmic differential forms with poles along a reducible divisor and compute the kernel and the image of the multiple residue map. As an application we describe the weight filtration on the logarithmic de Rham complex for divisors whose irreducible components are given locally by a regular sequence of holomorphic functions. In particular, this allows us to compute the mixed Hodge structure on the cohomology of the complement of divisors of certain types without the use of theorems on resolution of singularities and the standard reduction to the case of normal crossings.

Non-Normal Sequences of Holomorphic Functions and Universality

Inspired by a lemma of Zalcman, we show that certain universal sequences of holomorphic functions can always be chosen to be non-normal. This leads to a new vision of universality in connection with non-normality.

Sequences of Exponents in Constructing Strongly Annular Products

This paper gives conditions on the behavior of a sequence of holomorphic functions {fk(z)} and a strictly increasing sequence of positive integers {mk} that assures the infinite product \({\Pi f_k(z^{m_k})}\) is strongly annular. A constructive proof is given that shows if the sequence {fk(z)} exhibits certain boundary behavior along with a uniform boundedness condition then a number p > 1 exists such that if {mk} satisfies mk+1/mk ≥ p then the above product is strongly annular.

Uniform Convergence of Sequence of Holomorphic Functions in the Unite Disk

In this paper, suppose that ƒ m (z)(m = 1,2,…) are of infinite order. Applying the notion of infinite type [4], we obtain the sufficient conditions for uniform convergence of sequence of holomorphic functions in the unite disk.