Localization and Perturbation of Zeros of Entire Functions

ISBN: 9781439800324 出版年：2009 页码：318 Gil', Michael CRC Press

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 Rouche 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