Fourier transform on finite groups

In mathematics, the Fourier transform on finite groups is a generalization of the discrete Fourier transform from cyclic to arbitrary finite groups. In mathematics, the Fourier transform on finite groups is a generalization of the discrete Fourier transform from cyclic to arbitrary finite groups. The Fourier transform of a function f : G → C {displaystyle f:G ightarrow mathbb {C} ,} at a representation ϱ : G → G L ( d ϱ , C ) {displaystyle varrho :G ightarrow GL(d_{varrho },mathbb {C} ),} of G {displaystyle G,} is For each representation ϱ {displaystyle varrho ,} of G {displaystyle G,} , f ^ ( ϱ ) {displaystyle {widehat {f}}(varrho ),} is a d ϱ × d ϱ {displaystyle d_{varrho } imes d_{varrho },} matrix, where d ϱ {displaystyle d_{varrho },} is the degree of ϱ {displaystyle varrho ,} . The inverse Fourier transform at an element a {displaystyle a,} of G {displaystyle G,} is given by The convolution of two functions f , g : G → C {displaystyle f,g:G ightarrow mathbb {C} ,} is defined as The Fourier transform of a convolution at any representation ϱ {displaystyle varrho ,} of G {displaystyle G,} is given by For functions f , g : G → C {displaystyle f,g:G ightarrow mathbb {C} ,} , the Plancherel formula states where ϱ i {displaystyle varrho _{i},} are the irreducible representations of G . {displaystyle G.,} If the group G is a finite abelian group, the situation simplifies considerably: