Fourier Transform of Finite Abelian Groups

The standard definition of the multidimensional Fourier transform (MDFT) assumes a fixed coordinate system representation of the indexing set. In this chapter, we will define and explore the MDFT in a more abstract setting, one that removes the dependence on coordinates and solely references the additive abelian group structure of the indexing set. This approach highlights the fundamental role played by the duality between periodization and decimation in MDFT algorithm design. This duality lies at the heart of all standard and recently discovered divide and conquer MDFT algorithms. Emphasizing the unity underlying these algorithms permits a deeper understanding of their differences and how these differences can be exploited in implementation. This is especially true in the design of massively parallel algorithms. Algorithm design is reduced to relatively few basic principles without having to account for the details of specific coordinates.