Eigenfunctions of the Cosine and Sine Transforms

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)).