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Zh. Mat. Fiz. Anal. Geom., 2013, Volume 9, Number 4, Pages 476–495 (Mi jmag577)  

Eigenfunctions of the Cosine and Sine Transforms

V.  Katsnelson

The Weizmann Institute of Science, Rehovot 76100, Israel

Abstract: A description of the eigensubspaces of the cosine and sine operators is given. The spectrum of each of these two operators consists of two eigenvalues $(1,-1)$ and their eigensubspaces are infinite-dimensional. There are many possible bases for these subspaces, but most popular are the ones constructed from the 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 to be a continuation and further development of the results obtained by Hardy and Titchmarsh: “Self-reciprocal functions” (Quart. J. Math., Oxford, Ser. 1 (1930)).

Key words and phrases: Fourier transform, cosine-sine transforms, eigenfunctions, Melline transform.

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Bibliographic databases:
MSC: Primary 47A38; Secondary 47B35, 47B06, 47A10
Received: 29.10.2012
Revised: 17.12.2012
Language:

Citation: V.  Katsnelson, “Eigenfunctions of the Cosine and Sine Transforms”, Zh. Mat. Fiz. Anal. Geom., 9:4 (2013), 476–495

Citation in format AMSBIB
\Bibitem{Kat13}
\by V.~~Katsnelson
\paper Eigenfunctions of the Cosine and Sine Transforms
\jour Zh. Mat. Fiz. Anal. Geom.
\yr 2013
\vol 9
\issue 4
\pages 476--495
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\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3155139}
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