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 -ₓ/₂^t/2 f^2 (t) dt = 1- ₓ^2. Then inf\₀_{₈\} -^ f (t) - ₀^2WT+N] a₍₍^2 dt C_ₓ^{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.