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Localization and Perturbation of Zeros of Entire Functions [Kõva köide]

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(Ben Gurion University of the Negev, Israel)
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Localization and Perturbation of Zeros of Entire FunctionsSuurem pilt
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Püsilink: https://www.kriso.ee/db/9781439800324.html
Märksõnad:
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.
Preface
Finite Matrices
Inequalities for eigenvalues and singular numbers
1(2)
Inequalities for convex functions
3(1)
Traces of powers of matrices
4(2)
A relation between determinants and resolvents
6(2)
Estimates for norms of resolvents in terms of the distance to spectrum
8(2)
Bounds for roots of some scalar equations
10(3)
Perturbations of matrices
13(3)
Preservation of multiplicities of eigenvalues
16(1)
An identity for imaginary parts of eigenvalues
17(1)
Additional estimates for resolvents
18(4)
Gerschgorin's circle theorem
22(1)
Cassini ovals and related results
23(4)
The Brauer and Perron theorems
27(1)
Comments to
Chapter 1
28(1)
Eigenvalues of Compact Operators
Banach and Hilbert spaces
29(2)
Linear operators
31(2)
Classification of spectra
33(2)
Compact operators in a Hilbert space
35(2)
Compact matrices
37(3)
Resolvents of Hilbert-Schmidt operators
40(1)
Operators with Hilbert-Schmidt powers
41(2)
Resolvents of Schatten -von Neumann operators
43(1)
Auxiliary results
43(4)
Equalities for eigenvalues
47(1)
Proofs of Theorems 2.6.1 and 2.8.1
48(3)
Spectral variations
51(2)
Preservation of multiplicities of eigenvalues
53(1)
Entire Banach-valued functions and regularized determinants
54(3)
Comments to
Chapter 2
57(2)
Some Basic Results of the Theory of Analytic Functions
The Rouche and Hurwitz theorems
59(2)
The Caratheodory inequalities
61(2)
Jensen's theorem
63(3)
Lower bounds for moduli of holomorphic functions
66(3)
Order and type of an entire function
69(2)
Taylor coefficients of an entire function
71(2)
The theorem of Weierstrass
73(3)
Density of zeros
76(2)
An estimate for canonical products in terms of counting functions
78(1)
The convergence exponent of zeros
79(2)
Hadamard's theorem
81(2)
The Borel transform
83(2)
Comments to
Chapter 3
85(2)
Polynomials
Some classical theorems
87(6)
Equalities for real and imaginary parts of zeros
93(3)
Partial sums of zeros and the counting function
96(2)
Sums of powers of zeros
98(1)
The Ostrowski type inequalities
99(1)
Proof of Theorem 4.5.1
100(1)
Higher powers of real parts of zeros
101(1)
The Gerschgorin type sets for polynomials
102(1)
Perturbations of polynomials
102(3)
Proof of Theorem 4.9.1
105(1)
Preservation of multiplicities
106(1)
Distances between zeros and critical points
107(1)
Partial sums of imaginary parts of zeros
108(2)
Functions holomorphic on a circle
110(2)
Comments to
Chapter 4
112(2)
Bounds for Zeros of Entire Functions
Partial sums of zeros
114(3)
Proof of Theorem 5.1.1
117(2)
Functions represented in the root-factorial form
119(2)
Functions represented in the Mittag-Leffler form
121(3)
An additional bound for the series of absolute values of zeros
124(3)
Proofs of Theorems 5.5.1 and 5.5.3
127(2)
Partial sums of imaginary parts of zeros
129(3)
Representation of ezr in the root-factorial form
132(1)
The generalized Cauchy theorem for entire functions
133(1)
The Gerschgorin type domains for entire functions
134(2)
The series of powers of zeros and traces of matrices
136(1)
Zero-free sets
137(2)
Taylor coefficients of some infinite-order entire functions
139(3)
Comments to
Chapter 5
142(1)
Perturbations of Finite-Order Entire Functions
Variations of zeros
143(4)
Proof of Theorem 6.1.2
147(3)
Approximations by partial sums
150(1)
Preservation of multiplicities
151(1)
Distances between roots and critical points
152(2)
Tails of Taylor series
154(2)
Comments to
Chapter 6
156(1)
Functions of Order Less than Two
Relations between real and imaginary parts of zeros
157(3)
Proof of Theorem 7.1.1
160(2)
Perturbations of functions of order less than two
162(2)
Proof of Theorem 7.3.1
164(2)
Approximations by polynomials
166(2)
Preservation of multiplicities of in the case p(f) < 2
168(3)
Comments to
Chapter 7
171(2)
Exponential Type Functions
Application of the Borel transform
173(2)
The counting function
175(1)
The case α(f) < ∞
176(3)
Variations of roots
179(2)
Functions close to cos z and ez
181(2)
Estimates for functions on the positive half-line
183(1)
Difference equations
184(2)
Comments to
Chapter 8
186(1)
Quasipolynomials
Sums of absolute values of zeros
187(2)
Variations of roots
189(3)
Trigonometric polynomials
192(2)
Estimates for quasipolynomials on the positive half-line
194(1)
Differential equations
194(3)
Positive Green functions of functional differential equations
197(5)
Stability conditions and lower bounds for quasipolynomials
202(2)
Comments to
Chapter 9
204(3)
Transforms of Finite Order Entire Functions and Canonical Products
Comparison functions
207(3)
Transforms of entire functions
210(5)
Relations between canonical products and Sp
215(2)
Lower bounds for canonical products in terms of Sp
217(1)
Proof of Theorem 10.4.1
218(2)
Canonical products and determinants
220(2)
Perturbations of canonical products
222(3)
Comments to
Chapter 10
225(2)
Polynomials With Matrix Coefficients
Partial sums of moduli of characteristic values
227(3)
An identity for sums of characteristic values
230(3)
Imaginary parts of characteristic values of polynomial pencils
233(2)
Perturbations of polynomial pencils
235(3)
Multiplicative representations of rational pencils
238(5)
The Cauchy type theorem for polynomial pencils
243(1)
The Gerschgorin type sets for polynomial pencils
244(1)
Estimates for rational matrix functions
245(4)
Coupled systems of polynomial equations
249(2)
Vector difference equations
251(2)
Comments to
Chapter 11
253(2)
Entire Matrix-Valued Functions
Preliminaries
255(2)
Partial sums of moduli of characteristic values
257(3)
Proof of Theorem 12.2.1
260(3)
Imaginary parts of characteristic values of entire pencils
263(2)
Variations of characteristic values of entire pencils
265(4)
Proof of Theorem 12.5.1
269(2)
An identity for powers of characteristic values
271(1)
Multiplicative representations of meromorphic matrix functions
272(1)
Estimates for meromorphic matrix functions
273(4)
Zero free domains
277(1)
Matrix-valued functions of a matrix argument
278(4)
Green's functions of differential equations
282(2)
Comments to
Chapter 12
284(3)
Bibliography 287(10)
List of Main Symbols 297(2)
Index 299
Michael Gil is a professor in the Department of Mathematics at Ben Gurion University of the Negev in Israel.