Eigenvalues and Eigenvectors of the Fourier Transform
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 Zn 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 Zn or, equivalently, the decomposition of the space VZn 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 Zn to We will use symmetric and antisymmetric functions on Zn to achieve these goals.
KeywordsOrthonormal Basis Column Vector Symmetric Function Vector Space Versus Spectral Theorem
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