We now consider the vector space of all complex-valued periodic functions on the integers with period *b*. It turns out that every such function *a*(*n*) on the integers can be written as a polynomial in the *b*^{th} root of unity ξ^{n} := *e*^{2πin/b}. Such a representation for *a*(*n*) is called a **finite Fourier series**. Here we develop the finite Fourier theory using rational functions and their partial fraction decomposition. We then define the Fourier transform and the convolution of finite Fourier series, and show how one can use these ideas to prove identities on trigonometric functions, as well as find connections to the classical Dedekind sums.

## Keywords

Fourier Series Periodic Function Geometric Series Arithmetic Function Fourier Series Expansion
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© Springer Science+Business Media, LLC 2007