An Introduction to Nonharmonic Fourier Series

Contents

• Preface to the Revised First Edition
• Preface to the First Edition

• Schauder Bases 1
• Schauder's Basis for C[a,b] 3
• Orthonormal Bases in Hilbert Space 5
• The Reproducing Kernel 13
• Complete Sequences 16
• The Coefficient Functionals 19
• Duality 22
• Riesz Bases 25
• The Stability of Bases in Banach Spaces 31
• The Stability of Orthonormal Bases in Hilbert Space 35

• Weierstrass's Factorization Theorem 46
• Jensen's Formula 50
• Functions of Finite Order 53
• Estimates for Canonical Products 59
• Hadamard's Factorization Theorem 63
• The "Phragmen-Lindelof" Method 68
• Carleman's Formula 74
• Integrability on a Line 79
• The Paley-Wiener Theorem 85
• The Paley-Wiener Space 90

• The Trigonometric System 94
• Exponentials Close to the Trigonometric System 99
• A Counterexample 102
• Some Intrinsic Properties of Sets of Complex Exponentials 106
• Stability 110
• Density and the Completeness Radius 115

• Moment Sequences in Hilbert Space 122
• Bessel Sequences and Riesz-Fischer Sequences 128
• Applications to Systems of Complex Exponentials 135
• The Moment Space and Its Relation to Equivalent Sequences 139
• Interpolation in the Paley-Wiener Space: Functions of Sine Type 142
• Interpolation in the Paley-Wiener Space: Stability 149
• The Theory of Frames 154
• The Stability of Nonharmonic fourier Series 160
• Pointwise Convergence 165
• Notes and Comments 171
• References 199
• List of Special Symbols 221
• Author Index 223
• Subject Index 229