Prolate spheroidal wave functions, fourier analysis and uncertainty — III: The dimension of the space of essentially time- and band-limited signals

The purpose of this paper is to examine the mathematical truth in the engineering intuition that there are approximately 2WT independent signals ϕ
i
of bandwidth W concentrated in an interval of length T. Roughly speaking, the result is true for the best choice of the ϕ
i
(prolate spheroidal wave functions), but not for sampling functions (of the form sin t/t). Some typical conclusions are: Let f(t), of total energy 1, be band-limited to bandwidth W, and let
$\int_{-t/2}^{t/2} \vert f^{2}(t)\vert dt = 1- \epsilon_{T}^{2}$
. Then
${\rm inf}\limits_{\{a_{i}\}} \int_{-\infty}^{\infty} \left\vert f(t)- \sum_{0}^{[2WT]+N]} a_{n}\varphi_{n}\right\vert^{2} dt \lt C_{\epsilon_{T}^{2}}$
is (a) true for all such f with N = 0, C = 12, if the ϕ
n
are the prolate spheroidal wave functions; (b) false for some such f for any finite constants N and C if the ϕ
n
are sampling functions.