One of the most important problems in the theory of entire
functions is the distribution of the zeros of entire functions.
Localization and Perturbation of Zeros of Entire Functions is the
first book to provide a systematic exposition of the bounds for the
zeros of entire functions and variations of zeros under
perturbations. It also offers a new approach to the investigation
of entire functions based on recent estimates for the resolvents of
compact operators.
After presenting results about finite matrices and the spectral
theory of compact operators in a Hilbert space, the book covers the
basic concepts and classical theorems of the theory of entire
functions. It discusses various inequalities for the zeros of
polynomials, inequalities for the counting function of the zeros,
and the variations of the zeros of finite-order entire functions
under perturbations. The text then develops the perturbation
results in the case of entire functions whose order is less than
two, presents results on exponential-type entire functions, and
obtains explicit bounds for the zeros of quasipolynomials. The
author also offers additional results on the zeros of entire
functions and explores polynomials with matrix coefficients, before
concluding with entire matrix-valued functions.
This work is one of the first to systematically take the
operator approach to the theory of analytic functions.
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