Abstract
The class of all symmetric unimodal probability densities on the line whose Fourier transforms have support in an interval [− T, T] is studied. It is shown that there exists a unique density with minimal variance in this class. The Fourier transform of the minimizing density is everywhere non-negative. These results complement earlier work by Bohman who solved the analogous problem with no unimodality restriction. Motivation for studying this problem is given in the context of estimating the spectral density of a second-order stationary continuous-time stochastic process. Some comments are made regarding analogous problems for measures on the unit circle and discrete-time processes.
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