The Fourier Transform
1. Definition and Basic Properties
Definition
For any improperly Riemann integrable function \(f\) on \({\mathbb R}^n\), the Fourier transform \(\hat{f}\) is the function on \({\mathbb R}^n\) given by
\[ \hat{f}(\xi) := \int e^{-2 \pi i x \cdot \xi} f(x) dx. \]
Note
If the function \(f\) is improperly Riemann integrable or is a Schwartz function and so is \(f(x) e^{-2\pi i x \cdot \xi}\) for each \(\xi \in {\mathbb R}^d\). Thus the integral defining the Fourier transform always exists in the improper Riemann integral sense.
2. Fourier Transform Symmetries
Proposition (Symmetries Involving Fourier Transforms and Differentiation)
Suppose \(f\) is a Schwartz function on \({\mathbb R}^d\).
- If \(e_j\) is the unit vector pointing in the \(j\)-th coordinate direction,\[{}\lim_{h \rightarrow 0} \frac{\hat{f}(\xi + h e_j) - \hat{f}(\xi)}{h}{}\]\[{}= \int e^{- 2 \pi i x \cdot \xi} (-2 \pi i x \cdot e_j) f(x) dx{}\]
- Similarly,\[ (2 \pi i \xi \cdot e_j) \hat{f}(\xi) = \int e^{-2 \pi i x \cdot \xi} \frac{\partial f}{\partial x_j}(x) dx. \]
Proof
Corollary (The Fourier Transform Maps Schwartz Space to Itself)
The Fourier transform \(\hat{f}\) is also a Schwartz function.
Proof
Example (The Gaussian)
The funciton \(e^{- \pi ||x||^2}\) (where \(||\cdot||\) is the Euclidean norm on \({\mathbb R}^d\)) is its own Fourier transform.
Proof
3. Illustrations of the Action of the Fourier Transform
The plots below illustrate the behavior of the Fourier transform in a number of important situations. The first few plots demonstrate the effect of dilations, translations, and modulations:
The first plot is simply \(e^{-\pi x^2}\), which is its Fourier transform. The Fourier transform exhibits dilation symmetry: if \(f_a (x) := f(a^{-1} x)\) for some constant \(a > 0\), then \(\widehat{f_a}(\xi) = a \widehat{f} (a \xi)\). In other words, dilations on the “physical side” correspond to inverse dilations (and an overall multiplicative factor) on the “frequency” side. The second plot above is an example: If we dilate \(f\) to become \(e^{-\pi (3 x)^2}\), the Fourier transform gets an inverse dilation to become \(\frac{1}{3} e^{-\pi (\xi/3)^2}\). Other important symmetries are translation and modulation: if \(\tau_h f(x) := f(x-h)\) and \(m_a f(x) = e^{2 \pi i a x} f(x)\), then \(\widehat{\tau_h f} = m_{-h} (\xi) \widehat{f}(\xi)\) and \(\widehat{m_a f} = \tau_a \widehat{f}\). We see above that multiplying \(e^{-\pi x^2}\) by \(\cos(4 \pi x)\) makes the Fourier transform split into two lumps at frequencies \(\xi = \pm 2\). Similarly, multiplying by \(\sin 4 \pi x\) also makes two lumps on the Fourier side at frequencies \(\pm 2\), but one shows up with a minus sign (and there's an overall factor of \(i\)). Multiplying by higher frequencies would move the lumps further away from the origin (note that we consider \(e^{4 \pi i x}\) to be a “pure” frequency, which is why \(\cos (4 \pi x) = (e^{4 \pi i x} + e^{-4 \pi i x})/2\) looks like a sum of two pure frequencies).
Below we have more examples, this time illustrating the relationship between regularity and decay. Roughly speaking, smoother functions have Fourier transforms which decay more rapidly and vice-versa. This can be understood rigorously via the identities above for differentiation and multiplication by monomials.
In the first example, the function \(f\) is the indicator function of \([-1/2,1/2]\); its Fourier transform is \(\frac{\sin \pi \xi}{\pi \xi}\), which decays like \(O(|\xi|^{-1})\). The next function is a tent, which happens to be the convolution of the indicator function with itself. On the Fourier side, convolutions become products, so the Fourier transform is \(\left(\frac{\sin \pi \xi}{\pi \xi}\right)^2\), which has decay \(O(|\xi|^{-2})\) as \(\xi \rightarrow \pm \infty\). In the latter cases, we have another function with a discontinuity; its Fourier transform also decays like \(O(|\xi|^{-1})\). The last function is \(e^{-\pi |x|}\) which is \(C^0\) but not \(C^1\). Its Fourier transform is \(\frac{2}{\pi (1 + 4 \xi^2)}\).