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Aspects Harmonic Analysis on Locally Compact Abelian Groups

by Gallier Jean

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Fourier analysis through the structure of groups

$201.82 USD
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Hardcover New English

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AlbakiReads Editorial

Fourier analysis through the structure of groups

Gallier develops harmonic analysis on locally compact abelian groups from foundations in measure theory and integration, connecting those tools to the Fourier transform. The 747-page text ranges from Haar measure, Bochner integrals, and Radon functionals to Fourier analysis on the circle and the discrete Fourier transform. Its mathematical route also points toward Peter–Weyl theorems and their stated connections to Gelfand pairs and equivariant convolutional neural networks. The description identifies linear algebra and introductory real analysis as the background for approaching the material, making this a substantial theory-focused text with applications in engineering and computer vision in view.

Good fit for readers who enjoy

  • Undergraduate students with a background in linear algebra and introductory real analysis
  • Readers seeking a structured study of harmonic analysis and Fourier methods
  • Those interested in theoretical foundations relevant to Gelfand pairs or equivariant convolutional neural networks

Themes

Harmonic analysisFourier transformsMeasure theory and integrationLocally compact abelian groupsMathematical foundations of applications

Based on publisher information and book metadata.

Book overview

The Fourier transform is a "tool" used in engineering and computer vision to model periodic phenomena. Starting with the basics of measure theory and integration, this book delves into the harmonic analysis of locally compact abelian groups. It provides an in-depth tour of the beautiful theory of the Fourier transform based on the results of Gelfand, Pontrjagin, and Andre Weil

in a manner accessible to an undergraduate student who has taken linear algebra and introductory real analysis.

 

Highlights of this book include the Bochner integral, the Haar measure, Radon functionals, the theory of Fourier analysis on the circle, and the theory of the discrete Fourier transform. After studying this book, the reader will have the preparation necessary for understanding the Peter-Weyl theorems for complete, separable Hilbert algebras, a key theoretical concept used in the construction of Gelfand pairs and equivariant convolutional neural networks.

 

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