Advanced Topics in System and Signal Theory (eBook)

Preface 6
Contents 8
Part I Mathematical Preliminaries 10
Function Spaces and Operators 11
Banach and Hilbert spaces 11
Operators on Banach spaces 15
Spaces of Smooth Functions 19
Fourier Analysis and Analytic Functions 23
Trigonometric Series 23
Hardy Spaces on the Unit Disk 37
Vector-valued Hardy Spaces 50
Operator-valued Analytic Functions 54
Banach Algebras 59
The Invertible Elements 62
Complex Homomorphisms and Ideals 68
Involutions 73
Signal Models and Linear Systems 75
Signal Models 75
Linear Systems -- Properties and Representation 78
Part II Fundamental Operators 87
Poisson Integral and Hilbert Transformation 88
Definitions 88
The Poisson Integral 90
The Conjugate Poisson Integral 96
Causal Projections 106
Complemented Subspaces and Projections 107
Projections from Lp to Hp 110
Projections in Spaces of Smooth Functions 113
Inner-Outer Factorization on Subspaces of H 118
Part III Causality Aspects in Signal and System Theory 126
Disk Algebra Bases 127
On the Existence of Disk Algebra Bases 128
Robust Approximation in Disk Algebra Bases 130
Bases in Spaces of Smooth Functions 135
Uniformly Stable Basis Representations 140
Causal Approximations 143
Non-linear and Causal Approximations 145
Non-causal, Linear Approximations 146
Behavior of Causal Approximations 147
Causal Approximations for Smooth Functions 156
On Algorithms for Calculating the Hilbert Transform 158
Spectral Factorization 168
Regularity of Stochastic Sequences 168
Definition and Basic Properties 173
Factorization on Algebras of Continuous Functions 176
Error Bounds for Polynomial Data 196
Approximation of Spectral Densities 213
Spectral Factorization of Approximated Spectra 222
List of Symbols 238
References 241