There are multiple ways to think of this. Additive For any define where (the Characters). Let and . Then the Discrete Fourier Transform of is We will write instead of . Subsets Identify each with . Then . Thus take the group and the Discrete Fourier Transform of is: Multiplicative Take the group and Characters of the form where . We will write this as . Note that the character is actually the function , but we may write both for the function and the value at . The Discrete Fourier Transform is then