Transition from Fourier Series to Fourier Integral

We now extend the domain of definition of a linear system from a finite interval, say $(-c,c)$ , to the infinite interval $(-\infty ,\infty)$ . We shall do this by means of a line of arguments which is heuristic (``serving to discover or stimulate investigation''). Even though it pretty much ignores issues of convergence, it has the advantage of being physically precise. It highlights the relation between the discrete Fourier spectrum of a finite system with finitely separated walls and its limiting form as the system becomes arbitrarily large where the walls have arbitrarily large separation. The process of arriving at this limit will be revisited in the more general context of the spectral representation of the unit impulse response (``Green's function'' in Section 4.10.3 on page ) for an infinite string as the limit of a finite one (Section 4.11 on page ).

By contrast, the advantage of formulating Fourier analysis in mathematically more precise terms lies in that it highlights unambiguously the nature of the functions that lend themselves to being Fourier analyzed.

We start with the complete set of basis functions orthonormal on the interval $[-c,c]$ ,

$$\left\{ \frac{e^{\displaystyle in\pi x/c}}{\sqrt{2c}}\colon n=0,\pm 1,\dots\right\}\,.$$

The Fourier series for $f\in L^2(-c,c)$ is

$$\frac{1}{2} [ f(x^+)+f(x^-)]=\lim_{N\to\infty}\sum^N_{n=-N} \unde... ...\right]}_{\displaystyle c_n} \frac{e^{\displaystyle in\pi x/c}}{\sqrt{2c}}\,.$$

If $f$ is continuous at $x$ , then

$$\frac{1}{2} [f(x^+)+f(x^-)]=f(x)\,.$$

Second, we let

$$\Delta k=\frac{\pi}{c}$$

and, after rearranging some factors, obtain

$$f(x) = \sum^\infty_{n=-\infty} \Delta k \frac{e^{\displaystyle in \trian... ...i}} \int^c_{-c} \frac{e^{\displaystyle -in\triangle k t}}{\sqrt{2\pi}}f(t)dt~~.$$ (234)

Third, by introducing the points

$$k_n=n\triangle k, \quad n=0,\pm1,\pm2,\cdots \quad ,$$

we partition the real $k$ -axis into equally spaced subintervals of size $\triangle k=\pi / c$ . We introduce these points into the Fourier sum, Eq.(2.34),

$$f(x) = \lim_{N\to\infty}\sum^N_{n=-N}\Delta k \frac{e^{\displaystyle ik_n x}}{\sqrt{2\pi}} \int^c_{-c} \frac{e^{\displaystyle -ik_n t}}{\sqrt{2\pi}}f(t)dt$$ (235)

$$\equiv \sum^\infty _{n=-\infty} \Delta k g_c(k_n,x)~~$$ (236)

Note that this Fourier sum is, in fact, a Riemann sum, a precursor (i.e. approximation) to the definite integral

$$f(x) \approx \lim_{R\to\infty}\int^R_{-R}d k \frac{e^{\displaysty... ... x}}{\sqrt{2\pi}} \int^c_{-c} \frac{e^{\displaystyle -ik t}}{\sqrt{2\pi}}f(t)dt$$ (237)

over the limit of the interval $[-R,R]$ as

$$R \equiv k_N \equiv N\triangle k \to \infty~~.$$

The fourth and final step is to let $c\rightarrow \infty$ in order to obtain the result

$$f(x) = \int^\infty_{-\infty} d k \frac{e^{ikx}}{\sqrt{2\pi}} \int^\infty _{-\infty} \frac{e^{-ikt}}{\sqrt{2\pi}}f(t)~dt ~~.$$ (238)

This result is Fourier's Integral Theorem.