Prolate spheroidal spectral estimates

An estimate of the spectral density of a stationary time series can be obtained by taking the finite Fourier transform of an observed sequence x0,x1,…,xN−1 of sample size N with taper a discrete prolate spheroidal sequence and computing its square modulus. It is typical to take the average K of several such estimates corresponding to different prolate spheroidal sequences with the same bandwidth W(N) as the final computed estimate. For the mean square error of such an estimate to converge to zero as N→∞, it is shown that it is necessary to have W(N)↓0 with NW(N)→∞ as N→∞ and significantly have K(N)≤2NW(N) but K=K(N)→∞ as N→∞.