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 ϕiof bandwidth W concentrated in an interval of length T. Roughly speaking, the result is true for the best choice of the ϕi(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 ϕnare the prolate spheroidal wave functions; (b) false for some such f for any finite constants N and C if the ϕnare sampling functions.