Fourier analysis of sub-stationary processes with a finite moment

Abstract

E I x(O) I P finite. x(t) will not in general approach 0 at oo and its Fourier tralnsform will not usually be a function, but can be defined as a tempered distribution in Schwartz's theory [8]. The main result is that if p > 1 the Fourier transform of x(t) is a integral, i.e. it is obtained from a locally integrable function by differentiating once in the sense of distributions. Since local properties of the Fourier transform have little to do with local properties of x(t), it is not surprising that the same thing is true if x(t) is replaced by a distribution, i.e. a process possibly so irregular that values x(to) at points to are not defined, but with averages fx(t)f(t) dt for smooth functions f defined; we require pth moments of such averages bounded under translation of f. It is also possible to replace the real line as domain of values for t by a Euclidean space of arbitrary dimension k; here a Stieltjes integral is of the form akF /1t1 ... tkwith F locally integrable, and this is the form of the Fourier transforms, again for p > 1. The results are stated in detail in ?2. The positive assertions for p > 1 are proved in ??4-6, and the counter-examples for p ? 1 are given in ?7. ??1 and 3 deal with random distributions in general. ?8 shows that the results are easily adapted to the case of processes with nonnegative or discrete parameter or stationary increments. For stationary Gaussian processes, the results appeared in my Princeton thesis, written with the helpful advice of G. A. Hunt and Edward Nelson. There the methods were based on Ito's spectral measure, requiring both second moments and strict stationarity of the process or its increments ([4], [5]); here using other methods, essentially the Hausdorff-Young inequality, almost as much is proved from much more general hypotheses.