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. 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