Pontryagin duality
compactness-discreteness duality
time-frequency considerations
Haar measure
Bochner-Khinchin theorem, etc.
Fourier inversion theorem
Plencherel theorem
decay-smoothness duality
Paley-Wiener-Schwartz theorem
uncertainty principle
maximal ideals of
Banach-algebra considerations
Reference: Walter Rudin, Fourier analysis on groups.

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