Pontryagin duality

In mathematics, in particular in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform. It places in a unified context a number of observations about functions on the real line or on finite abelian groups:

References

The following books (available in most university libraries) have chapters on locally compact abelian groups, duality and Fourier transform. The Dixmier reference (also available in English translation) has material on non-commutative harmonic analysis.

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• Jacques Dixmier, Les C*-algèbres et leurs Représentations, Gauthier-Villars,1969.
• Lynn H. Loomis, An Introduction to Abstract Harmonic Analysis, D. van Nostrand Co, 1953
• Walter Rudin, Fourier Analysis on Groups, 1962
• Hans Reiter, Classical Harmonic Analysis and Locally Compact Groups, 1968 (2nd ed produced by Jan D. Stegeman, 2000).
• Hewitt and Ross, Abstract Harmonic Analysis, vol 1, 1963.