At this point a very important theorem due to #iener [1930] 
is used. The theorem states that the power spectrum of the 
stationary process is given by the Fourier cosine transformation 
of the autocorrelation function. The theorem is much more general 
than needed here, and its proof will be given here only for the 
specific cases under study. 
The proof of equation (10.27) follows in equation (10.28). 
The infinite integral is replaced by the limit as M approaches 
infinity of the integral from 0 to M in the first expression on 
the right. The integral form for 9(p) given in equation (10.26) 
is also substituted for Q(p). The steps thereafter are straight- 
oenanch and upon the application of the lemma given in Chapter 
8, the result is obtained immediately. The second term in the 
next to the last expression is zero because the range of inte- 
gration does not cover a equal to zero. 
Tukey's formulas _ 
The problem of numerical analysis was stated in equation 
(10.25), but then it was necessary to carry out some preliminary 
theoretical derivations before continuation of the description 
of the numerical methods. Equation (10.29) states that it is 
more convenient to take the points of the record at equally spaced 
intervals of time, designated by At. The three basic formulas 
for the numerical estimation of the power spectrum as presented 
by Tukey [1949] and Tukey and Hamming [1949] are given by equations 
(10530) 5 MGi0.31) andi(l@.32)\. 
Equation (10.30) is the finite difference analogue of equation 
(10.26). It describes a procedure for finding an estimate of the 
= aie 
