Fourier Analysis on Finite Groups
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We start with some Fourier analysis on cyclic groups as the base case. This theory comes along with some basic notions about character theory. So, we develop some properties of the additive characters of Z/qZ. Formulas like the matrix version of the DFT, its inverse formula and Plancherel’s theorem are proved for this case. Then, we give a constructive generalization to any finite abelian group. We also present some work on properties of characters in Z/qZ*. All these tools are used to define and develop multiplicative characters. In particular, we mention Dirichlet characters and Gauss and Jacobi sums. The most important result of this work is to note that Fourier transform in Z/qZ gives us a method to write a Dirichlet character in terms of additive characters. Finally, we apply some of this theory to Cayley graphs.
Rivera Montes De Oca, Jose Manuel (2017). Fourier Analysis on Finite Groups. Master's thesis, Texas A & M University. Available electronically from