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.