User talk:Metterklume

Hi, I saw your comment on my talk page about the Bessel function closure equation. I'm not sure if I ever found "closure" to explain what was going on, but here's what might be happening. Let's look at the Fourier Transform relation you wrote, that the integral of the exponential is the delta function. What certain IS true is that the Fourier Transform of the all one function (call this f, so f(x) = 1 for all x) is the delta function. But the integral representation of the Fourier transform is only valid for functions in $$L_1$$, ie. functions that are integrable. f is certainly not integrable. However, we can still speak about the fourier transform of f because we can use a density argument (e.g. let f_n(x) = 1 if abs(x) < n, and then look at the limit of the Fourier Transform of f_n). This is how we extend the domain of the Fourier Transform from $$L_1$$ functions to include $$L_2$$ functions.

So, the Fourier transform of f is the delta function, but that doesn't mean we can rigorously write the integral. It also depends on what kind of integral -- a Riemann integral, or a Lebesgue integral. I think the integral actually might make sense as a Riemman integral, where we define $$ \int_{-\infty}^\infty f(x)dx = \lim_{n\rightarrow \infty} \int_{-n}^{n} f(x)dx $$

but I don't think it works as a Lebesgue integral.

Hope that doesn't confuse you more. I'm certainly not the last word on the subject. If you're interested in investigating further, you might look for advanced functional analysis texts, e.g. stuff about tempered distributions, the Schwarz (sp?) space, and Paley-Wiener theory.

Good question! Lavaka (talk) 02:35, 12 March 2009 (UTC)