Nice try, but I find these interpretations of "finite Fourier transform" do not capture the most common uses that I find in the literature, which is that the finite Fourier transform is the fourier series of a function truncated to the interval [0, T]. If we provide one or more definitions, we should at least cite a source or two for each. Here's a place to look: books. Dicklyon 21:06, 26 February 2007 (UTC)
First, the Google test: searching for "finite Fourier transform", in quotes, I find that every usage in the first page of links corresponds to the DFT, except for one which corresponds to the Fourier transform of a function with compact support .
Second, when I first noticed that Cleve Moler was using "finite Fourier transform" as a synonym for the DFT , I looked it up. If I remember correctly, this usage dates at least to the 1940's. I don't have that reference on hand now, but here is one from 1969: J. Cooley, P. Lewis, and P. Welch, "The finite Fourier transform," IEEE Trans. Audio Electroacoustics17 (2), 77-85 (1969). (This article cites another article from 1950.)
Third, you didn't provide a specific citation for your proposed definition, and it's not clear to me what you mean. Looking through the books in the Google-book search you suggest, several of them didn't clearly define the term "finite Fourier transform". Several of them defined it as the DFT. And at least one of them (Bachman) defined it a synonym for the Fourier series coefficients, so I've added this definition too (and I've come across this in several other references, although the DFT definition still seems the most common in recent sources.)
I'm going to remove your disputed tag pending more evidence that there is something wrong with the article.
Thanks for working on it. I'm not really familiar with these usages, but I find usually that searching books gets you more reliable stuff than a web search. My impression is that the fourier series coefficients of the signal limited to the interval 0 to T is a common use. Given the interval, the fourier series and the fourier tranform of the signal rectangularly windowed to that interval are entirely equivalent information; that is, the series completely determines the whole transform at intermediate frequencies. So maybe these two uses should be described in a more unified way? My impression is that the x(t) is not of finite support, but rather that the integral is finite, equivalent to applying to rectangular window before a fourier transform, and hence also closely related to the STFT. Dicklyon 07:49, 27 February 2007 (UTC)
I see you're just ahead of me. But what you said about "equivalent" is not quite. The windowing is equivalent to taking the integral over a finite time interval. But that's not equivalent to starting with an x(t) of finite support. Dicklyon 07:56, 27 February 2007 (UTC)
In what way is it not equivalent? You end up with exactly the same formula in both cases. The difference is merely one of interpretation. —Steven G. Johnson 16:21, 27 February 2007 (UTC)
In one case you assume the original x(t) has finite support; in the other you don't. Same formula, different conditions on what it applies to. Dicklyon 19:44, 27 February 2007 (UTC)
This is just interpretation since the formula does not depend on the values of x(t) outside [0,T]. —Steven G. Johnson 02:27, 28 February 2007 (UTC)
Please see how I fixed it and let me know if you agree. Dicklyon 04:54, 28 February 2007 (UTC)
I don't think it has any real difference from the previous text, but I'm fine with it if it makes you happier. —Steven G. Johnson 05:48, 28 February 2007 (UTC)