Eigenfunctions of the Cosine and Sine Transforms

TL;DR

This paper explores the eigensubspaces of the cosine and sine transforms, revealing their spectra, infinite-dimensional eigenspaces, and introducing new continuous bases beyond Hermite functions.

Contribution

It extends previous work by Hardy and Titchmarsh, providing new continuous orthogonal bases for the eigenspaces of these transforms.

Findings

Eigenvalues are 1 and -1 for both transforms.

Eigenspaces are infinite-dimensional.

Introduces continuous orthogonal bases.

Abstract

A description of eigensubspaces of the cosine and sine operators is presented. The spectrum of each of these two operator consists of two eigenvalues (1,\,-1) and their eigensubspaces are infinite--dimensional. There are many possible bases for these subspaces, but most popular are bases 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 a continuation and further development of results in \textit{Self-reciprocal functions} by Hardy and Titchmarsh: Quarterly Journ. of Math. (Oxford Ser.) \textbf{1} (1930).