In these notes we lay out the basic theory of the Fourier transform, which is of course the most fundamental tool in harmonic analysis and also of major importance in related fields (functional analysis, complex analysis, PDE, number theory, additive combinatorics, representation theory, signal processing, etc.). The Fourier transform, in conjunction with the Fourier inversion formula, allows one to take essentially arbitrary (complex-valued) functions on a group $latex {G}&fg=000000$ (or more generally, a space $latex {X}&fg=000000$ that $latex {G}&fg=000000$ acts on, e.g. a homogeneous space $latex {G/H}&fg=000000$), and decompose them as a (discrete or continuous) superposition of much more symmetric functions on the domain, such as characters $latex {\chi: G \rightarrow S^1}&fg=000000$; the precise superposition is given by *Fourier coefficients* $latex {\hat f(\xi)}&fg=000000$, which take values in some dual object such as the Pontryagin dual $latex {\hat G}&fg=000000$ of $latex {G}&fg=000000$. Characters behave in a very simple manner…

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