On deriving the (inverse) Fourier transform from Fourier series

[psie]

TL;DR Summary

I have seen in quite a few texts a derivation of the inverse Fourier transform via Fourier series. It is often claimed this derivation is informal and not rigorous, however, I don't understand what is the issue. And if there's an issue, then I don't understand why one presents the argument in the first place.

Here's the standard argument made in some books. I'm using the notation as used in Vretblad's Fourier Analysis and its Applications.

For the complex Fourier series of $f$ we have \begin{align} f(t)&\sim\sum_{n=-\infty}^\infty c_n e^{in\pi t/P}, \tag1 \\ \text{where} \quad c_n&=\frac1{2P}\int_{-P}^P f(t)e^{-in\pi t/P} \ dt. \tag2 \end{align} [...] We define, provisionally, $$\hat{f}(P,\omega)=\int_{-P}^P f(t)e^{-i\omega t} \ dt, \quad \omega\in\mathbb R,\tag3$$ so that $c_n=\frac{1}{2P}\hat{f}(P,n\pi/P)$. The formula $(1)$ is translated into $$f(t)\sim\frac1{2P}\sum_{n=-\infty}^\infty \hat{f}(P,\omega_n)e^{i\omega_nt}=\frac1{2\pi}\sum_{n=-\infty}^\infty \hat{f}(P,\omega_n)e^{i\omega_nt}\cdot\frac{\pi}{P},\quad \omega_n=\frac{n\pi}{P}.\tag4$$ Because of $\Delta\omega_n=\omega_{n+1}-\omega_n=\frac\pi{P}$, this last sum looks rather like a Riemann sum. Now we let $P\to\infty$ in $(3)$ and define $$\hat{f}(\omega)=\lim_{P\to\infty} \hat{f}(P,\omega)=\int_{-\infty}^\infty f(t)e^{i\omega t} \ dt\quad \omega\in\mathbb R.\tag5$$ (at this point we disregard all details concerning convergence). If $(4)$ had contained $\hat{f}(\omega_n)$ instead of $\hat{f}(P,\omega_n)$, the limiting process $P\to\infty$ would have resulted in $$f(t)\sim \frac1{2\pi}\int_{-\infty}^\infty\hat{f}(\omega)e^{i\omega t} \ d\omega.\tag6$$

What is the problem with having $\hat{f}(P,\omega_n)$ instead of $\hat{f}(\omega_n)$ in $(4)$? What is the point of presenting this argument if it doesn't actually derive the inverse Fourier transform?

 

[anuttarasammyak]

The argument is about from Fourier series to Fourier integral extending the method from periodic function to non periodic one. I am not sure of inverse you say.

 

[psie]

Hmm, I still do not see the connection between Fourier series and Fourier transform. They seem not so related as I first thought. Some say it is taking the period to infinity, but the above derivation shows that it doesn't work. We don't really get a representation of the function we started with. It seems like $f(t)$ in $(4)$ and $(6)$ above are two different functions, because, pretty much out of the blue, we swap $\hat{f}(P,\omega_n)$ for $\hat{f}(\omega_n)$.

 

[anuttarasammyak]

Periodic function of period 2P is expressed as Fourier series, sum of discrete terms. Non periodic function is reagarded as periodic function with infininite period. Periodic function of infinite period is expressed as Fourier integral, integral of continuous function.

 

[hutchphd]

Some say it is taking the period to infinity, but the above derivation shows that it doesn't work.

I prefer the outlook that the finite Fourier series is a special case of the continuous (integral) Fourier Transform. The discreteness can be imposed to match either experimental bandwidth (sampling) rerstrictions or for computational (e.g.FFT) reasons. I have no bedrock idea what "taking the period to infinity" means and so I ignore such descriptions. There are good and sufficient arguments for Transform but I don't dwell on them or the other vaguely disquieting activities too often required for normalization, (re)normalization or just sanity.