Happy $\pi$ day!
(These notes are for an expository lecture on the Fourier Transform prepared and delivered by J. Colliander for the Astronomy club at the University of Toronto. The talk is aimed at students who’ve seen calculus and know a little linear algebra.)
[Source file in MultiMarkDown.]
[See also my PDE2 lecture notes for an expanded discussion.]
The Fourier Transform acts of functions $f$ 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 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 $\mathbb{A} = \mbox{diag}[\lambda_1, \lambda_2, \lambda_3, \lambda_4].$
$$g(x) = e^{- \pi x^2}$$ $$ \widehat{g}(\xi) = \int_{-\infty}^{+\infty} e^{-2 \pi i x \xi} e^{-\pi x^2}dt $$ $$ = e^{-\pi \xi^2} \int e^{-\pi [ x - i \xi]^2}dx $$ $$ = e^{-\pi \xi^2} \int e^{-\pi [ x ]^2}dx $$ $$ = e^{-\pi \xi^2}. $$
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.
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….