Characteristic functions, Bochner's Thm.

[KrugalSausage]

Hi,

I am taking an online course on functional integration and the professor is introducing measure sets and characteristic functions.

He introduced properties of characteristic functions and then gave us Bochner's Thm. which basically says that if a function satisfies the properties he listed, then it is a Fourier transform of a measure.

As an example he showed that the function c(t) = 1 is a function that satisfies the properties listed, so 1 is a Fourier transform of some measure H(x)

He then said that it is discrete (as opposed to a gaussian measure which is absolutely continuous), and that it is related to the Helmholtz function. Then he moved on to another topic.

My question is, what is H(x)? What function can you take the Fourier transform of and get the value 1? I think the hint was that it is discrete, but I can't think of what it might be.

I tried searching Helmholtz function but the search results give me the differential equation resulting from the Fourier transform of another differential equation that also has time dependence. I only mention this because I know that somewhere there is a link that I am not seeing as these transformations have been mentioned in the context of probability measures.

Thanks a lot!
-k

 

[KrugalSausage]

Thinking about it more,

It might be possible that me misspoke, and H(x) is just the heavy side step function, and not some Helmholtz function.

So that dH(x) is the delta function, so that the Fourier transform of the delta function would be 1.

Does this sound okay?