in the book of Frisch’s Turbulence (http://users.uoa.gr/~pjioannou/mechgrad/Frisch_Turbulence.pdf), at chapter of 4.5 The spectrum of stationary random functions, as following picture shows, i have a question about the "cumulative energy spectrum": the author said "It is easily shown, using Parseval's theorem, that the cumulative energy spectrum is a nondecreasing function of the cutoff frequency F", but i don't know how to using Parseval's theorem in this case, because the theorem needs a time integral in (4.47) but there is only ensemble average (which "<>" means) in (4.47), and i thought two ways to create time integral in (4.47), but both failed:
① using ergodic theorem as (4.39) shows, change ensemble average to time average, but at this time the formula has an additional T that tends to infinity, but according Parseval's theorem the time integral is finite. i don't know where is wrong. ② do a time integral directly to (4.47), still the problem, we have a infinite T causing cumulative energy spectrum to divergence.
above is my failed thought which maybe close to correct answer, to conclude by reiterating the question: how to using Parseval's theorem to proof the cumulative energy spectrum is a nondecreasing function of F?
(ps i very love this book, but i also have a few questions within, if you read and understand most of the book and if you don't mind, can we exchange contact, in a more instant communication
or perhaps you know some material about the notes/explanations of this book)
