16. April 2015:
ARI QUEST TALK
ARI Seminar Room, Wohllebengasse 12-14 / 1st Floor
Slepian Functions and Their Use in Signal Estimation and Spectral Analysis: Theory and Applications - Frederik J Simons Signals that are timelimited (or spacelimited) cannot be simultaneously bandlimited (in frequency). Yet the finite precision of measurement and computation unavoidably bandlimits our observation and modeling scientific data, and we often only have access to, or are only interested in, a study area that is temporally or spatially bounded. In the sciences, we may be interested in spectrally modeling a time series defined only on a certain interval, or we may want to characterize a specific region of space observed using an effectively bandlimited measurement device. It is clear that analyzing and representing data of this kind will be facilitated if a basis of functions can be found that are “spatiospectrally” concentrated, i.e., “localized” in both domains at the same time. I give a theoretical overview of one particular approach to this “concentration” problem, as originally proposed for time series by Slepian, Landau and Pollak, in the 1960s. Their framework leads to practical algorithms and statistically performant methods for the analysis of signals and their power spectra in one and two dimensions, and on the surface of a sphere. New theoretical and numerical results have led to a myriad of applications in the geosciences, planetary sciences and cosmology, but also in medical imaging, computer science, and acoustics (head-related transfer functions and ambisonics). I will touch upon the applications that I have helped develop, and am eager to hear about the ones I don't know about.