Thạc Sĩ Linear and Non-linear Operators, and The Distribution of Zeros of Entire Functions

TABLE OF CONTENTS
Acknowledgments . ii
Abstract . iii
Index of Notation . 1
1 Introduction 2
1.1 Historical remarks . 2
1.2 Synopsis . 3
2 Polynomials and transcendental entire functions . 5
2.1 Zeros of polynomials 5
2.1.1 Resultants and discriminants . 6
2.2 Orthogonal polynomials 9
2.2.1 Jacobi polynomials . 13
2.3 Composition theorem . 16
2.4 Transcendental entire functions 20
2.4.1 The Laguerre-Polya class . 23
2.5 Generalized Laguerre inequality 28
3 Linear operators acting on entire functions . 34
3.1 Multiplier sequences 34
3.1.1 Complex zero decreasing sequences 39
3.2 Hyperbolicity and stability preservers 41
3.2.1 Stability preservation . 42
3.3 Dierential operators 44
3.3.1 Proof of Proposition 118 49
3.3.2 Quadratic dierential operators 59
iv
4 Multiplier sequences with various polynomial bases . 72
4.1 General polynomial base 72
4.2 Orthogonal polynomial base 73
4.3 Jacobi polynomial base . 76
5 Non-linear operators acting on entire functions 82
5.1 Non-linear operators preserving stability . 82
5.2 Related results . 87
5.3 Applications . 93
Bibliography 95