Eigenvalues and Eigenvectors of the Fourier Transform
Bao Luong · Applied and numerical harmonic analysis · 2009
Having reduced the general theory of the FT on finite Abelian groups to the theory of the FT on finite cyclic groups, it suffices to study the FT of functions defined on the cyclic group Z n for an arbitrary value of n , where n > 1. Our next goals are the following: Determine the form of eigenvectors of the FT (Section 7.2). Find the spectral decomposition of the FT on Z n or, equivalently, the decomposition of the space V Z n as a direct sum of its invariant subspaces (Section 7.3). Determine the multiplicity of the eigenvalues of the FT (Section 7.5). We will use symmetric and antisymmetric functions on Z n to We will use symmetric and antisymmetric functions on Z n to achieve these goals. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.