Discrete Fourier Analysis

2. Auflage 2027
ISBN: 978-3-032-42559-1
Verlag: Birkhäuser

Buch, Englisch, 228 Seiten, Format (B × H): 168 mm x 240 mm

Reihe: Pseudo-Differential Operators

This textbook presents basic notions and techniques of Fourier analysis in discrete settings. Written in a concise style, it is interlaced with remarks, discussions and motivations from signal analysis.

The first part is dedicated to topics related to the Fourier transform, including discrete time-frequency analysis and discrete wavelet analysis. Basic knowledge of linear algebra and calculus is the only prerequisite. The second part is built on Hilbert spaces and Fourier series and culminates in six chapters on pseudo-differential operators, providing a lucid introduction to this advanced topic in analysis. Some measure theory language is used, although most of this part is accessible to students familiar with an undergraduate course in real analysis.

is aimed at advanced undergraduate and graduate students in mathematics and applied mathematics. Enhanced with exercises, it will be an excellent resource for the classroom as well as for self-study.

This second edition is enriched by several new chapters on Hilbert–Schmidt and trace class pseudo-differential operators, constituting an additional module on discrete Fourier analysis.

Zielgruppe

Graduate

Autoren/Hrsg.

Wong, M. W.

Fachgebiete

• Mathematik | Informatik Mathematik Mathematische Analysis Harmonische Analysis, Fourier-Mathematik
• Mathematik | Informatik Mathematik Mathematische Analysis Differentialrechnungen und -gleichungen

Inhaltsverzeichnis

Chapter 1 The Finite Fourier Transform.- Chapter 2 Translation-Invariant Linear Operators.- Chapter 3 Circulant Matrices.- Chapter 4 Convolution Operators.- Chapter 5 Fourier Multipliers.- Chapter 6 Eigenvalues and Eigenfunctions.- Chapter 7 The Fast Fourier Transform.- Chapter 8 Time-Frequency Analysis.- Chapter 9 Time-Frequency Localized Bases.- Chapter 10 Wavelet Transforms and Filter Banks.- Chapter 11 Haar Wavelets.- Chapter 12 Daubechies Wavelets.- Chapter13 The Trace.- Chapter 14 Hilbert Spaces.- Chapter 15 Bounded Linear Operators.- Chapter 16 Self-Adjoint Operators.- Chapter 17 Compact Operators.- Chapter 18 The Spectral Theorem.- Chapter 19 Schatten–von Neumann Classes.- Chapter 20 A Canonical Form for Compact Operators.- Chapter 21 A Factorization Formula for Trace Class Operators.- Chapter 22 An Integral Representation of Hilbert–Schmidt Operators.- Chapter 23 Fourier Series.- Chapter 24 Fourier Multipliers on S1.- Chapter 25 Pseudo-Differential Operators on S1.- Chapter 26 Pseudo-Differential Operators on Z.- Chapter 27 Pseudo-Differential Operators on the Infinite Cylindrical Surface C.- Chapter 28 Products and Adjoints of Pseudo-Differential Operators on L2(C).- Chapter 29 Trace Class Pseudo-Differential Operators on L2(C).- Chapter 30 The H*-Algebra of Pseudo-Differential Operators on L2(C).

Über Autor(innen)

M. W. Wong is currently Chair (2006-12) of Department of Mathematics and Statistics at York University, Toronto (Canada), past-President (2005-07 and 2007-09) of the International Society for Analysis, its Applications and Computation (ISAAC). Since 2003, he has been the Director of the ISAAC Special Interest Group in Pseudo-Differential Operators (IGPDO).