82 research outputs found
The Beurling--Malliavin Multiplier Theorem and its analogs for the de Branges spaces
Let be a non-negative function on . We are looking for a
non-zero from a given space of entire functions satisfying The
classical Beurling--Malliavin Multiplier Theorem corresponds to and the
classical Paley--Wiener space as . We survey recent results for the case
when is a de Branges space \he. Numerous answers mainly depend on the
behaviour of the phase function of the generating function .Comment: Survey, 25 page
A Hardy's Uncertainty Principle Lemma in Weak Commutation Relations of Heisenberg-Lie Algebra
In this article we consider linear operators satisfying a generalized
commutation relation of a type of the Heisenberg-Lie algebra. It is proven that
a generalized inequality of the Hardy's uncertainty principle lemma follows.
Its applications to time operators and abstract Dirac operators are also
investigated
Orthonormal sequences in and time frequency localization
We study uncertainty principles for orthonormal bases and sequences in
. As in the classical Heisenberg inequality we focus on the product
of the dispersions of a function and its Fourier transform. In particular we
prove that there is no orthonormal basis for for which the time and
frequency means as well as the product of dispersions are uniformly bounded.
The problem is related to recent results of J. Benedetto, A. Powell, and Ph.
Jaming.
Our main tool is a time frequency localization inequality for orthonormal
sequences in . It has various other applications.Comment: 18 page
Complex analysis I: entire and meromorphic functions polyanalytic functions and their generalizations
The first part of the volume contains a comprehensive description of the theory of entire and meromorphic functions of one complex variable and its applications. It includes the fundamental notions, methods and results on the growth of entire functions and the distribution of their zeros, the Rolf Nevanlinna theory of distribution of values of meromorphic functions including the inverse problem, the theory of completely regular growth, the concept of limit sets for entire and subharmonic functions. The authors describe the interpolation by entire functions, to entire and meromorphic solutions of ordinary differential equations, to the Riemann boundary problem with an infinite index and to the arithmetic of the convolution semigroup of probability distributions. Polyanalytic functions form one of the most natural generalizations of analytic functions and are described in Part II. They emerged for the first time in plane elasticity theory where they found important applications (due to Kolossof, Mushelishvili etc.). This book contains a detailed review of recent investigations concerning the function-theoretical pecularities of polyanalytic functions (boundary behavour, value distributions, degeneration, uniqueness etc.). Polyanalytic functions have many points of contact with such fields of analysis as polyharmonic functions, Nevanlinna Theory, meromorphic curves, cluster set theory, functions of several complex variables etc
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