Discrete Fourier Analysis [Paperback]

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.

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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 a section 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.

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Discrete Fourier 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.

Presenting the basic notions and techniques of Fourier analysis in discrete settings, this book opens up what is a key area of mathematics research to a much wider readership and covers the finite Fourier transform as well as Hilbert spaces and Fourier series.

Preface.- The Finite Fourier Transform.- Translation-Invariant Linear Operators.- Circulant Matrices.- Convolution Operators.- Fourier Multipliers.- Eigenvalues and Eigenfunctions.- The Fast Fourier Transform.- Time-Frequency Analysis.- Time-Frequency Localized Bases.- Wavelet Transforms and Filter Banks.- Haar Wavelets.- Daubechies Wavelets.- The Trace.- Hilbert Spaces.- Bounded Linear Operators.- Self-Adjoint Operators.- Compact Operators.- The Spectral Theorem.- Schattenvon Neumann Classes.- Fourier Series.- Fourier Multipliers on S¹.- Pseudo-Differential Operators on S¹