Exponential Fourier densities and optimal estimation and detection on the circle

A new representation, called an exponential Fourier density, of a probability density on a circle,
S^{1}
is introduced. It is shown that a density of bounded variation on
S^{1}
can be approximated as closely as desired by such a representation in the space of square-integrable functions on
S^{1}
. The exponential Fourier densities have the desirable feature of being closed under the operation of taking conditional distributions. Facilitated by the use of these densities, finite-dimensional, recursive, and optimal estimation and detection schemes are derived for some simple models including a PSK communication system. A deficiency of the exponential Fourier densities is that they are not closed under convolution. How to circumvent this deficiency is still an open question.