(These notes are for an expository lecture on the Fourier Transform by J. Colliander for teachers participating in the Classroom Adventures in Mathematics: Summer Institute. The talk is aimed at people who’ve seen calculus and know a little linear algebra.)
http://www.math.toronto.edu/colliand/notes/2011-08-16_Fourier_Transform.html
[Source file in MultiMarkDown.]
[See also my PDE2 lecture notes for an expanded discussion.]
The Fourier Transform acts of functions $f: \mathbb{R} \longmapsto \mathbb{C}$ and produces a related function $\widehat{f}$ called the Fourier transform of the function $f$. So, we have a mapping $\mathcal{F}(f) = \widehat{f}$ and $\mathcal{F}: f \longmapsto \widehat{f}$. Here is how the transformation is defined: $$\widehat{f}(\xi) = \int_{\mathbb{R}^d} e^{- 2\pi i x \cdot \xi} f(x)\ \,dx .$$ This is a linear transformation so $\widehat{f+g} = \widehat{f} + \widehat{g}$, etc. The inverse Fourier transform is the reverse operation, ${\mathcal{F}}^{-1}: g \longmapsto \check{g}$. The inverse Fourier transform is defined via the formula $$ {\check{g}} (x) = \int_{{\mathbb{R}}^d} e^{+ 2 \pi i x \cdot \xi} g(\xi)\ \, d\xi. $$ We have the remarkable formula ${\mathcal{F}}^{-1} \circ {\mathcal{F}} = Identity$ and this allows us to think of functions in a new way: $$ f(x) = \int e^{+ 2 \pi i x \cdot \xi} \widehat{f} (\xi) d\xi. $$ This formula allows us to view any function as a superposition of pure exponentials $e^{2 \pi i x \cdot \xi}$.
This is a good idea because the exponentials are the eignefunctions of the derivative operator.
Consider a 4x4 matrix $\mathbb{A}$.
Now, suppose the matrix was in diagonal form $\mathbb{A} = \mbox{diag}[\lambda_1, \lambda_2, \lambda_3, \lambda_4].$
$$g(x) = e^{- \pi x^2}$$ $$ \widehat{g}(\xi) = \int_{-\infty}^{+\infty} e^{-2 \pi i x \xi} e^{-\pi x^2}dt $$ $$ = e^{-\pi \xi^2} \int e^{-\pi [ x - i \xi]^2}dx $$ $$ = e^{-\pi \xi^2} \int e^{-\pi [ x ]^2}dx $$ $$ = e^{-\pi \xi^2}. $$
Remark: The shift in contour appearing above can be justified with a separate argument involving basic complex variables.
Summary: We observe that translation and modulation appear as dual operations under Fourier transform. The influence of a spatial translation on the output of Fourier Transformation is the appearance of a modulation factor linked to the translation parameter. The influence of a spatial modulation on the output of Fourier Transformation is the appearance of a translation.
Summary: We observe that spatial rescaling induces a dual rescaling on the Fourier Transform side.
Summary: We observe that conjugation on the physical side induces a conjugation and a reflection on the frequency space side.
Summary: Pointwise multiplication on the spatial side corresponds with convolution on the frequency side. Similarly, convolution on the physical side corresponds to pointwise multiplication on the frequency side.
Remark: The preceding formal calculations involve assertions of equality between expressions involving improper integrals. The interpretation and understanding of these formulae is one launching point for the field of Harmonic Analysis.
We can now jack up our understanding by considering the actions of these different operations on the Gaussian! In this way, we build up what are called Gaussian Wave Packets: $$ f(x) = (M_{\xi_0} \circ T_{x_0} \circ \rho_\lambda ) (g) (x) $$ Furthermore, by using the linearity, we can calculate the Fourier Transform of sums of these wave packets. If we allow for infinite sums and extreme values of the parameters, we can pass to a limit and calculate the transform of basically any function we can imagine using these ideas. In fact, the theory goes beyond what I could imagine when I was an undergraduate….
Imagine an infinitely long metal rod with an initial temperature distribution $u_0 (x)$. How does the heat in the rod move with time? Imagine the surface ${ z = u(t,x) }$ describing the temperature at location $x$ along the rod at time $t>0$ in the future. What does this surface look like?
Careful thinking along these lines (performed by Fourier) leads to a partial differential equation describing the evolution of the temperature function $u(t,x)$: $$ \partial_t u = \partial_x^2 u. $$ The slope in the $t$ direction equals the “curvature” in the $x$ direction. At a cool spot, the temperature is a minimum where the graph is turning upward so the curvature is positive. The temperature should increase at such a point. At a maximum of the temperature, the graph is curving downwards so the temperature is going to decrease from such a point. How can we solve the heat equation? With the Fourier Transform, the solution is almost obvious.
Suppose first that $u_0 (x) = e^{i \xi x}$. We then inspect that $u(t,x) = e^{i \xi x} e^{-it |\xi|^2}$ satisfies the heat equation. (Just check it!)
Now suppose more generally that $$u_0 (x) = \int e^{i x \xi} \widehat{u_0}(\xi) d\xi. $$ We can use superposition to build the solution! We can then inspect (formally, by differentiating under the integral sign) that $$ u(t,x) = \int e^{i x \xi} e^{-it |\xi|^2} \widehat{u_0}(\xi) d\xi. $$ satisfies the heat equation.