Representation of real discrete Fourier transform in terms of a new set of functions based upon Möbius inversion

The trigonometric functions sin(2πn/N) and cos(2πn/N) are transformed into a new set of basis functions using Möbius inversion of certain types of series. The new basis functions are number theoretic series. They are used to represent the real discrete Fourier transform (RDFT) in terms of 2 matrices of factorization. The first matrix, with elements 1, -1 and 0 is obtained by replacing cos(2πk/N) and sin(2πk/N) by μ(k/N + 1/4) and μ(k/N), where μ(x) is the bipolar rectangular wave function. The second matrix is block-diagonal where each block is a circular correlation and consists of the new basis functions. Some applications of the new representation are discussed.

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