6.5 Proofs of FOT Counterparts of Some Results on ACS and GACS Signals
In this section, proofs of some results presented in Section 1.3 in the stochastic approach are carried out in the functional approach. Moreover, when possible, results are derived in the more general case of GACS signals.
6.5.1 Proof of FOT Counterpart of (1.114)
By substituting (1.112) into (1.113) we have
(6.79) 
where, in the last equality, the variable changes s1 = t − u1 and s2 = t − u2 are made. Thus by assuming that the order of sum, integral, and almost-periodic component extraction operations can be interchanged, and observing that almost-periodic functions can be led outside the almost-periodic component extraction operator, we get
where, in the last equality, the variable change s2 − s1 = u is made and, according to (1.95) (with the replacements 
and 
), we put
(6.81) 
By substituting (1.115) into
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