Localization and Perturbation of Zeros of Entire Functions (Lecture notes in pure and applied mathematics ; v. 258)

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. Localization and Perturbation of Zeros of Entire Functions 4 CONTENTS 6 Biography 10 Preface 11 Chapter 1: Finite Matrices 1 1.1 Inequalities for eigenvalues and singular numbers 1 1.2 Inequalities for convex functions 3 1.3 Traces of powers of matrices 4 1.4 A relation between determinants and resolvents 6 1.5 Estimates for norms of resolvents in terms of the distance to spectrum 8 1.6 Bounds for roots of some scalar equations 10 1.7 Perturbations of matrices 13 1.8 Preservation of multiplicities of eigenvalues 16 1.9 An identity for imaginary parts of eigenvalues 17 1.10 Additional estimates for resolvents 18 1.11 Gerschgorin’s circle theorem 22 1.12 Cassini ovals and related results 23 1.13 The Brauer and Perron theorems 27 1.14 Comments to Chapter 1 28 Chapter 2: Eigenvalues of Compact Operators 1 2.1 Banach and Hilbert spaces 1 2.2 Linear operators 3 2.3 Classification of spectra 5 2.4 Compact operators in a Hilbert space 7 2.5 Compact matrices 9 2.6 Resolvents of Hilbert-Schmidt operators 12 2.7 Operators with Hilbert-Schmidt powers 13 2.8 Resolvents of Schatten - von Neumann operators 15 2.9 Auxiliary results 15 2.10 Equalities for eigenvalues 19 2.11 Proofs of Theorems 2.6.1 and 2.8.1 20 2.12 Spectral variations 23 2.13 Preservation of multiplicities of eigenvalues 25 2.14 Entire Banach-valued functions and regularized determinants 26 2.15 Comments to Chapter 2 29 Chapter 3: Some Basic Results of the Theory of Analytic Functions 1 3.1 The Rouché and Hurwitz theorems 1 3.2 The Caratheodory inequalities 3 3.3 Jensen’s theorem 5 3.4 Lower bounds for moduli of holomorphic functions 8 3.5 Order and type of an entire function 11 3.6 Taylor coefficients of an entire function 13 3.7 The theorem of Weierstrass 15 3.8 Density of zeros 18 3.9 An estimate for canonical products in terms of counting functions 20 3.10 The convergence exponent of zeros 21 3.11 Hadamard’s theorem 23 3.12 The Borel transform 25 3.13 Comments to Chapter 3 27 Chapter 4: Polynomials 1 4.1 Some classical theorems 1 4.1.1 The Cauchy theorem 1 4.1.2 The Viéte and Varing formulas 3 4.1.3 The Enestróm-Kakeya theorem 4 4.1.4 The Gauss-Lucas theorem 4 4.1.5 Self-inversive polynomials 5 4.2 Equalities for real and imaginary parts of zeros 7 4.3 Partial sums of zeros and the counting function 10 4.4 Sums of powers of zeros 12 4.5 The Ostrowski type inequalities 13 4.6 Proof of Theorem 4.5.1 14 4.7 Higher powers of real parts of zeros 15 4.8 The Gerschgorin type sets for polynomials 16 4.9 Perturbations of polynomials 16 4.10 Proof of Theorem 4.9.1 19 4.11 Preservation of multiplicities 20 4.12 Distances between zeros and critical points 21 4.13 Partial sums of imaginary parts of zeros 22 4.14 Functions holomorphic on a circle 24 4.15 Comments to Chapter 4 26 Chapter 5: Bounds for Zeros of Entire Functions 1 5.1 Partial sums of zeros 2 5.2 Proof of Theorem 5.1.1 5 5.3 Functions represented in the root-factorial form 7 5.4 Functions represented in the Mittag-Leffler form 9 5.5 An additional bound for the series of absolute values of zeros 12 5.6 Proofs of Theorems 5.5.1 and 5.5.3 15 5.7 Partial sums of imaginary parts of zeros 17 5.8 Representation of ezr in the root-factorial form 20 5.9 The generalized Cauchy theorem for entire functions 21 5.10 The Gerschgorin type domains for entire functions 22 5.11 The series of powers of zeros and traces of matrices 24 5.12 Zero-free sets 25 5.13 Taylor coefficients of some infinite-order entire functions 27 5.14 Comments to Chapter 5 30 Chapter 6: Perturbations of Finite-Order Entire Functions 1 6.1 Variations of zeros 1 6.2 Proof of Theorem 6.1.2 5 6.3 Approximations by partial sums 8 6.4 Preservation of multiplicities 9 6.5 Distances between roots and critical points 10 6.6 Tails of Taylor series 12 6.7 Comments to Chapter 6 14 Chapter 7: Functions of Order Less than Two 1 7.1 Relations between real and imaginary parts of zeros 1 7.2 Proof of Theorem 7.1.1 4 7.3 Perturbations of functions of order less than two 6 7.4 Proof of Theorem 7.3.1 8 7.5 Approximations by polynomials 10 7.6 Preservation of multiplicities in the case ρ(f) < 2 12 7.7 Comments to Chapter 7 15 Chapter 8: Exponential Type Functions 1 8.1 Application of the Borel transform 1 8.2 The counting function 3 8.3 The case... 4 8.4 Variations of roots 7 8.5 Functions close to cos z and ez 9 8.6 Estimates for functions on the positive half-line 11 8.7 Difference equations 12 8.8 Comments to Chapter 8 14 Chapter 9: Quasipolynomials 1 9.1 Sums of absolute values of zeros 1 9.2 Variations of roots 3 9.3 Trigonometric polynomials 6 9.4 Estimates for quasipolynomials on the positive halfline 8 9.5 Differential equations 8 9.6 Positive Green functions of functional differential equations 11 9.6.1 The first-order equations 11 9.6.2 The second-order equations 14 9.6.3 Higher-order equations 16 9.7 Stability conditions and lower bounds for some quasipolynomials 16 9.8 Comments to Chapter 9 18 Chapter 10: Transforms of Entire Functions and Canonical Products 1 10.1 Comparison functions 1 10.2 Transforms of entire functions 4 10.2.1 The ψ-transform 4 10.2.2 The Mittag-Leffler transform 5 10.2.3 The root-factorial transform 6 10.2.4 A relation between the Mittag-Leffler and root-factorial transforms 7 10.3 Relations between canonical products and Sp 9 10.4 Lower bounds for canonical products in terms of Sp 11 10.5 Proof of Theorem 10.4.1 12 10.6 Canonical products and determinants 14 10.7 Perturbations of canonical products 16 10.8 Comments to Chapter 10 19 Chapter 11: Polynomials with Matrix Coefficients 1 11.1 Partial sums of moduli of characteristic values 1 11.2 An identity for sums of characteristic values 4 11.3 Imaginary parts of characteristic values of polynomial pencils 7 11.4 Perturbations of polynomial pencils 9 11.5 Multiplicative representations of rational pencils 12 11.6 The Cauchy type theorem for polynomial pencils 17 11.7 The Gerschgorin type sets for polynomial pencils 18 11.8 Estimates for rational matrix functions 19 11.9 Coupled systems of polynomial equations 23 11.10 Vector difference equations 25 11.11 Comments to Chapter 11 27 Chapter 12: Entire Matrix-Valued Functions 1 12.1 Preliminaries 1 12.2 Partial sums of moduli of characteristic values 3 12.3 Proof of Theorem 12.2.1 6 12.4 Imaginary parts of characteristic values of entire pencils 9 12.5 Variations of characteristic values of entire pencils 11 12.6 Proof of Theorem 12.5.1 15 12.7 An identity for powers of characteristic values 17 12.8 Multiplicative representations of meromorphic matrix functions 18 12.9 Estimates for meromorphic matrix functions 19 12.10 Zero-free domains 23 12.11 Matrix-valued functions of a matrix argument 24 12.12 Green’s functions of differential equations 28 12.13 Comments to Chapter 12 30 Bibliography 1 List of main symbols 1 11.2 An identity for sums of characteristic values......Page 4 12.3 Proof of Theorem 12.2.1......Page 6 7.5 Approximations by polynomials......Page 10 12.5 Variations of characteristic values of entire pencils......Page 11 List of main symbols......Page 1 12.2 Partial sums of moduli of characteristic values......Page 3 9.5 Differential equations......Page 8 4.5 The Ostrowski type inequalities......Page 13 10.7 Perturbations of canonical products......Page 16 12.7 An identity for powers of characteristic values......Page 17 12.8 Multiplicative representations of meromorphic matrix functions......Page 18 5.10 The Gerschgorin type domains for entire functions......Page 22 12.10 Zero-free domains......Page 23 11.11 Comments to Chapter 11......Page 27 12.12 Green’s functions of differential equations......Page 28 10.2.2 The Mittag-Leffler transform......Page 5 11.3 Imaginary parts of characteristic values of polynomial pencils......Page 7 12.4 Imaginary parts of characteristic values of entire pencils......Page 9 11.5 Multiplicative representations of rational pencils......Page 12 12.6 Proof of Theorem 12.5.1......Page 15 12.9 Estimates for meromorphic matrix functions......Page 19 5.8 Representation of ezr in the root-factorial form......Page 20 11.10 Vector difference equations......Page 25 4.15 Comments to Chapter 4......Page 26 2.15 Comments to Chapter 2......Page 29 5.9 The generalized Cauchy theorem for entire functions......Page 21 10.6 Canonical products and determinants......Page 14 12.11 Matrix-valued functions of a matrix argument......Page 24 5.1 Partial sums of zeros......Page 2 12.13 Comments to Chapter 12......Page 30 "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."--Publisher's description One of the most important problems in the theory of entire functions is the distribution of the zeros of entire functions. This is the first book to provide a systematic exposition of the bounds for the zeros of entire functions and variations of zeros under perturbations. The book also offers a new approach for investigating entire functions based on recent estimates for the resolvents of compact operators. Along with describing applications to differential, functional differential, and difference equations, the author estimates the distance between the zeros of an entire function and its cri