Fourier transform on class functions

Definition
Let $$G$$ be a finite group. Then the Fourier transform on $$G$$ is a unitary map that takes a class function expressed in the standard basis (that is, with basis elements being the indicator functions of conjugacy classes) and outputs the class function in the character basis (that is, with basis elements being the characters). Note that finding the Fourier transform is just as hard as finding its inverse, because since it is a unitary operator, its inverse is simply its conjugate transpose.

Check out the Fourier transform computation problem to understand various issues related to computing the Fourier transform.

Also refer Fourier transform on all functions.

Properties of the Fourier transform
Note that we can define the Fourier transform of a function which is not a class function as well -- this is effectively the Fourier transform of the class function obtained by averaging over the conjugates.

Fourier transform of a characteristic function of a subgroup
The Fourier transform of the characteristic function of a subgroup is a uniform superposition of all the characters that are trivial on this subgroup. Note thus that this Fourier transform is a uniform superposition of one-dimensional characters.

Fourier transform of a characteristic function of a coset
The Fourier transform of the characteristic function of a coset can again be computed using the same idea, but only for the one-dimensional characters. In fact, we have the following. If $$gH$$ is a coset of $$H$$, and $$\chi$$ is a one-dimensional character, then the coefficient of $$\chi$$ is nonzero only when the restriction of $$\chi$$ to $$H$$ is trivial, and in that case, the coefficient is $$\chi(g^{-1})$$.

Fourier transform on direct products
The Fourier transform on the direct product of two groups equals the tensor product of the Fourier transforms for both groups. This means, in particular, that the Fourier transform on a direct power of a group equals the corresponding tensor power of its Fourier transform.