1st Edition
Localization and Perturbation of Zeros of Entire Functions
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.
Finite Matrices
Inequalities for eigenvalues and singular numbers
Inequalities for convex functions
Traces of powers of matrices
A relation between determinants and resolvents
Estimates for norms of resolvents in terms of the distance to spectrum
Bounds for roots of some scalar equations
Perturbations of matrices
Preservation of multiplicities of eigenvalues
An identity for imaginary parts of eigenvalues
Additional estimates for resolvents
Gerschgorin’s circle theorem
Cassini ovals and related results
The Brauer and Perron theorems
Eigenvalues of Compact Operators
Banach and Hilbert spaces
Linear operators
Classification of spectra
Compact operators in a Hilbert space
Compact matrices
Resolvents of Hilbert–Schmidt operators
Operators with Hilbert–Schmidt powers
Resolvents of Schatten–von Neumann operators
Auxiliary results
Equalities for eigenvalues
Proofs of Theorems 2.6.1 and 2.8.1
Spectral variations
Preservation of multiplicities of eigenvalues
Entire Banach-valued functions and regularized determinants
Some Basic Results of the Theory of Analytic Functions
The Rouché and Hurwitz theorems
The Caratheodory inequalities
Jensen’s theorem
Lower bounds for moduli of holomorphic functions
Order and type of an entire function
Taylor coefficients of an entire function
The theorem of Weierstrass
Density of zeros
An estimate for canonical products in terms of counting functions
The convergence exponent of zeros
Hadamard’s theorem
The Borel transform
Polynomials
Some classical theorems
Equalities for real and imaginary parts of zeros
Partial sums of zeros and the counting function
Sums of powers of zeros
The Ostrowski-type inequalities
Proof of Theorem 4.5.1
Higher powers of real parts of zeros
The Gerschgorin type sets for polynomials
Perturbations of polynomials
Proof of Theorem 4.9.1
Preservation of multiplicities
Distances between zeros and critical points
Partial sums of imaginary parts of zeros
Functions holomorphic on a circle
Bounds for Zeros of Entire Functions
Partial sums of zeros
Proof of Theorem 5.1.1
Functions represented in the root-factorial form
Functions represented in the Mittag–Leffler form
An additional bound for the series of absolute values of zeros
Proofs of Theorems 5.5.1 and 5.5.3
Partial sums of imaginary parts of zeros
Representation of ezr in the root-factorial form
The generalized Cauchy theorem for entire functions
The Gerschgorin-type domains for entire functions
The series of powers of zeros and traces of matrices
Zero-free sets
Taylor coefficients of some infinite-order entire functions
Perturbations of Finite-Order Entire Functions
Variations of zeros
Proof of Theorem 6.1.2
Approximations by partial sums
Preservation of multiplicities
Distances between roots and critical points
Tails of Taylor series
Functions of Order Less than Two
Relations between real and imaginary parts of zeros
Proof of Theorem 7.1.1
Perturbations of functions of order less than two
Proof of Theorem 7.3.1
Approximations by polynomials
Preservation of multiplicities of in the case p(f) < 2
Exponential-Type Functions
Application of the Borel transform
The counting function
The case a(f) < ∞
Variations of roots
Functions close to cos z and ez
Estimates for functions on the positive half-line
Difference equations
Quasipolynomials
Sums of absolute values of zeros
Variations of roots
Trigonometric polynomials
Estimates for quasipolynomials on the positive half-line
Differential equations
Positive Green functions of functional differential equations
Stability conditions and lower bounds for quasipolynomials
Transforms of Finite-Order Entire Functions and Canonical Products
Comparison functions
Transforms of entire functions
Relations between canonical products and Sp
Lower bounds for canonical products in terms of Sp
Proof of Theorem 10.4.1
Canonical products and determinants
Perturbations of canonical products
Polynomials with Matrix Coefficients
Partial sums of moduli of characteristic values
An identity for sums of characteristic values
Imaginary parts of characteristic values of polynomial pencils
Perturbations of polynomial pencils
Multiplicative representations of rational pencils
The Cauchy type theorem for polynomial pencils
The Gerschgorin type sets for polynomial pencils
Estimates for rational matrix functions
Coupled systems of polynomial equations
Vector difference equations
Entire Matrix-Valued Functions
Preliminaries
Partial sums of moduli of characteristic values
Proof of Theorem 12.2.1
Imaginary parts of characteristic values of entire pencils
Variations of characteristic values of entire pencils
Proof of Theorem 12.5.1
An identity for powers of characteristic values
Multiplicative representations of meromorphic matrix functions
Estimates for meromorphic matrix functions
Zero free domains
Matrix-valued functions of a matrix argument
Green’s functions of differential equations
Bibliography
Index
Biography
Michael Gil’ is a professor in the Department of Mathematics at Ben Gurion University of the Negev in Israel.
