Eigenfunctions of the Fourier-Plancherel Operator

Details Eigenfunctions of the Fourier-Plancherel Operator Eigenfunctions of the Fourier-Plancherel Operator

2012-03-12

arxiv.org/abs/1203.2427v2

Mathematics

Classical Analysis and ODEs

24 pages

Scientific paper

A description of eigensubspaces of the Fourier-Plancherel operator is presented. The spectrum of this operator consists of four eigenvalues 1, -1, i, -i and their eigensubspaces are infinite-dimensional. There are many possible bases for these subspaces, but most popular are bases constructed from Hermite functions. We present other "bases" which are not discrete orthogonal sequences of vectors, but continuous orthogonal chains of vectors. Our work can be considered a continuation and further development of results in "Self-reciprocal functions" by Hardy and Titchmarsh: Quarterly Journ. of Math.(Oxford Ser.) 1 (1930).

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