Introduction to Nonharmonic Fourier Series (eBook)

Front Cover 1
An Introduction to Nonharmonic Fourier Series 4
Copyright Page 5
CONTENTS 8
Preface 10
Chapter 1. Bases in Banach Spaces 12
1 Schauder Bases 12
2 Schauder’s Basis for C[a,b] 14
3 Orthonormal Bases in Hilbert Space 17
4 The Reproducing Kernel 26
5 Complete Sequences 30
6 The Coefficient Functionals 33
7 Duality 38
8 Reisz Bases 41
9 The Stability of Bases in Banach Spaces 48
10 The Stability of Orthonormal Bases in Hilbert Space 52
Chapter 2. Entire Functions of Exponential Type 64
Part One. The Classical Factorization Theorems 65
1 Weierstrass’s Factorization Theorem 65
2 Jensen’s Formula 70
3 Functions of Finite Order 73
4 Estimates for Canonical Products 80
5 Hadamard’s Factorization Theorem 85
Part Two. Restrictions along a Line 91
1 The “Phragmén-Lindelöf” Method 91
2 Carleman’s Formula 98
3 Integrability on a Line 104
4 The Paley–Wiener Theorem 111
5 The Paley–Wiener Space 116
Chapter 3. The Completeness of Sets of Complex Exponentials 122
1 The Trigonometric System 123
2 Exponentials Close to the Trigonometric System 129
3 A Counterexample 133
4 Some Intrinsic Properties of Sets of Complex Exponentials 137
5 Stability 142
6 Density and the Completeness Radius 148
Chapter 4. Interpolation and Bases in Hilbert Space 156
1 Moment Sequences in Hilbert Space 157
2 Bessel Sequences and Riesz–Fischer Sequences 164
3 Applications to Systems of Complex Exponentials 173
4 The Moment Space and Its Relation to Equivalent Sequences 178
5 Interpolation in the Paley–Wiener Space: Functions of Sine Type 181
6 Interpolation in the Paley–Wiener Space: Stability 190
7 The Theory of Frames 195
8 The Stability of Nonharmonic Fourier Series 201
9 Pointwise Convergence 208
Notes and Comments 214
References 236
List of Special Symbols 246
Author Index 248
Subject Index 252