WSEAS Transactions on Systems
Print ISSN: 1109-2777, E-ISSN: 2224-2678
Volume 24, 2025
Time and Band Limiting: From the Early Days to the Present
Author:
Abstract: There are many situations in communications theory, medical imaging, geophysics, signal processing,
and mathematics where one has an optimization problem whose solution is only rendered practical by some
kind of mathematical miracle. A good and canonical example of this is the computation of the singular value
decomposition for either a huge matrix or an integral operator. In particular these problems are typically extremely
ill-posed. The work of D. Slepian, H. Landau and H. Pollak at Bell Labs 1960-1965 gives a remedy to this situation.
Inspired in part by questions posed by Claude Shannon they found and exploited a miracle that allows for the
effective computation of the so called ”prolate spheroidal wave functions” which are defined as the eigenfunctions
of an integral operator but turn out to be computable since they are also the eigenfunctions of a second order
differential operator. The numerical computation of these functions has in this fashion become a stable problem,
while the initial one was a very ill-posed one.
I will try to give an account of these developments and indicate at least one open problem inspired by this
remarkable work at Bell Labs.
We will see that the original work started around 1960 has been extended in a few directions, and that the
mathematical miracle underlying this work has influenced many other areas of mathematics ranging from the
study of the Riemann zeta function to very recent work that is inspired by the same effort to find numerically
stable ways to compute quantities of interest.
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Keywords: Time and band limiting, Limited angle tomography, Korteweg-deVries equation, Commuting
integral and differential operators, Meixner-Pollaczek polynomials, Harmonic analysis
Pages: 359-366
DOI: 10.37394/23202.2025.24.31