Abstract: We now look at the orthogonal complement of 0 L 2(Г\G) and prove a spectral decomposition theorem following Godemenťs paper [Go 2], using the Poisson summation formula. The method works for arithmetic subgroups Г, and has the advantage of being rapid and easy. It fails for more general discrete subgroups, and the question is reconsidered by other methods in the next and last chapter. The spectral decomposition is achieved by the Eisenstein transform, which maps the orthogonal complement of 0 L 2(Г\G) and the constant functions on the L 2 space of a positive real line—with our normalization, the upper half of the imaginary line Re s = 1/2.

Abstract: The performance assessments of array antennas with N antenna elements is generally undertaken by using N-dimensional linear matrix algebra irrespective of the number K of received signals. However, the fundamental dimension (rank) of these problems does not exceed the minimum of K and N. Therefore, for antennas with a large number of antenna elements and in situations when K ? N, there is a great advantage to reduce the problem to K dimensions. Yet, the success of this approach rests on having both explicit and simple linear operators (matrices) to partition the problem into its principal domain and orthogonal complement. In this paper, we will employ this approach to both steady state and transient analysis of adaptive array antennas and show its power to provide a complete solution to these problems. The approach has general applicability and can easily be extended to arbitrary large number of antenna elements including, in the limit, continuous antenna apertures that may appear in optical systems.