Fourier analysis on finite abelian groups

A character is a homomorphism from the finite abelian group to the complex unit circle. Distinct characters are orthogonal under uniform averaging: a nontrivial character has mean zero because translating its average multiplies it by a character value different from one while leaving that average unchanged. The character group has exactly as many elements as the original group, and its characters form an orthonormal basis for complex functions on G. Completeness follows, for example, by decomposing G into cyclic factors and using the elementary roots-of-unity basis on each factor.

The coefficient of a function is its normalized inner product with a character, so complex conjugation belongs in the definition. Inversion sums those coefficients times the characters without a further division by N. Parseval likewise has an average on the spatial side and a sum on the frequency side. The trivial-character coefficient is the mean. For an indicator it is the density; subtracting that density deletes exactly this coefficient and leaves all other coefficients unchanged.

Define normalized convolution by averaging the product f(y)g(x−y) over y. Substitution into the Fourier definition and a change of variables show that its transform is the product of the two transforms. The indicator convolution therefore equals the sum representation count divided by N. Squaring and averaging it gives counting energy divided by N cubed. Those three factors of N come from the convolution twice and the final spatial average once.

Parseval applied to this convolution yields the fourth-moment formula for one set and a product of squared magnitudes for two sets. The trivial character contributes alpha to the fourth power in the one-set case. Every other contribution is nonnegative, giving an exact description of energy above that baseline. This formula is not identical to simply replacing every indicator in an energy expression by its balanced version without checking the multilinear expansion. The frequency identity provides the clean way to identify which terms remain.

A large nontrivial coefficient says the set correlates with a character, so membership is biased toward some phases rather than evenly distributed around the character circle. It need not mean that the set is an interval, a subgroup, or a single level set. In a finite vector space, characters are constant on affine hyperplanes, allowing a useful density increment. In an integer interval, comparable localization involves long progressions or Bohr-type neighborhoods and requires more work to handle boundary effects.

Translation changes a Fourier coefficient by a unit complex factor and therefore preserves its magnitude and all additive energy. This explains why energy cannot distinguish a subgroup from one of its cosets. The phase is still meaningful when reconstructing the actual function, so discarding phases is not harmless for every task. Fourier analysis gives exact algebraic coordinates here; a picture of sine waves illustrates interference but does not automatically compute a finite-group transform or certify a set’s pseudorandomness.