Abstract
Integral operators having a kernel of convolution type on a finite interval play an important part in noise theory, in the representation of band-limited signals and in other parts of applied mathematics [1]. As such, it would seem important to investigate the properties they possess beyond those that they inherit from the Fredholm and Hilbert-Schmidt theories. This note arose from an (unpublished) observation of H. 0. Pollak in his studies of the Dirichlet kernel occurring in the theory of band-limited functions. The observation was, in effect, that if the operator-theoretic square of a finite convolution operator (defined below) were again one, then the eigenfunctions are exponentials. He asked whether a similar result held for more general functions of an operator, or When can two such operators commute? We prove a theorem below which gives a partial answer. A finite convolution operator is an integral operator of the form
Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
