Time and Band Limiting: From the Early Days to the Present

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.