Fourier analysis on finite groups

Fourier analysis on finite groups expands functions on a finite group into components indexed by its irreducible representations. It generalizes the discrete Fourier transform from cyclic groups to arbitrary finite groups.

Group representation: A homomorphism from a group to the invertible linear transformations of a vector space. Representations supply the matrix-valued frequencies used to decompose functions on nonabelian groups.
Finite group: A group with finitely many elements. The transform's domain is a finite group, whose multiplication governs its algebraic structure.
Fast Fourier transform: An algorithm that computes a discrete Fourier transform using fewer operations than direct evaluation. Fast transform algorithms extend Fourier computation to suitable finite groups.
Discrete Fourier transform: The linear transform expressing a finite sequence as a sum of complex exponential frequencies. It is the familiar cyclic-group case of the broader finite-group theory.
Irreducible representation: A nonzero representation with no proper nonzero invariant subspaces. Irreducible representations are the indivisible frequency components in the finite-group transform.
Character theory: The study of traces of representations and their properties. Characters summarize representations and are especially useful for class functions.
Fast Fourier transform on finite groups: Algorithms for computing Fourier transforms of functions on finite groups efficiently. This is the computational counterpart of the general transform, with complexity shaped by group structure.
Fourier analysis on finite abelian groups: The study of functions on finite abelian groups using their one-dimensional characters. Commutativity makes every irreducible representation one-dimensional, recovering scalar Fourier coefficients.
Group Fourier transform: A transform that maps a function on a group to matrix coefficients indexed by its irreducible representations. It is the central operation that converts pointwise data into representation-indexed coefficients.
Group character: The trace-valued function associated with a finite-dimensional representation of a group. Characters provide scalar frequency data when the function is constant on conjugacy classes.

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