Fourier inversion theorem

In mathematics, Fourier inversion recovers a function from its Fourier transform. Several different Fourier inversion theorems exist.

Sometimes the following expression is used as the definition of the Fourier transform:

$(\mathcal{F}f)(t)=\int_{-\infty}^\infty f(x)\, e^{-itx}\,dx.$

Then it is asserted that

$f(x)=\frac{1}{2\pi}\int_{-\infty}^\infty (\mathcal{F}f)(t)\, e^{itx}\,dt.$

In this way, one recovers a function from its Fourier transform.

However, this way of stating a Fourier inversion theorem sweeps some more subtle issues under the carpet. One Fourier inversion theorem assumes that f is Lebesgue-integrable, i.e., the integral of its absolute value is finite:

$\int_{-\infty}^\infty\left|f(x)\right|\,dx<\infty.$

In that case, the Fourier transform is not necessarily Lebesgue-integrable. For example, the function f(x) = 1 if −a < x < a and f(x) = 0 otherwise has Fourier transform

2sin(at) / t.

In such a case, Fourier inversion theorems usually investigate the convergence of the integral

$\lim_{b\rightarrow\infty}\frac{1}{2\pi}\int_{-b}^b (\mathcal{F}f)(t) e^{itx}\,dt.$

By contrast, if we take f to be a tempered distribution -- a type of generalized function -- then its Fourier transform is another tempered distribution; and the Fourier inversion formula is then more simple to prove.

Proof of the inversion theorem

First we will consider Fourier transforms of functions in the Schwartz space; these are smooth functions $f : \mathbb{R}^N \to \mathbb{C}$ such that, for any multi-indices α and β,

$\sup_{x \in \mathbb{R}^N}|x^{\alpha}\partial^{\beta}f(x)| < \infty. \,$

These functions are clearly seen to be absolutely integrable, and the Fourier transform of a Schwarz function is also a Schwartz function. An example is the Gaussian function $g(x) = e^{-\pi|x|^2}$, which we will actually use in proving the inversion formula. We will use the convention that $\widehat{f}(\xi) = \int e^{-2\pi i x\cdot\xi}f(x)\,dx$, and the claim is that for a Schwartz function f,

$f(x) = \int_{\mathbb{R}^N}e^{2\pi i x\cdot\xi}\widehat{f}(\xi)\,d\xi.$

To do this, we will need a few facts.

1. For f and g Schwartz functions, Fubini's theorem implies that $\int f\widehat{g} = \int\widehat{f} g$.
2. If $\eta \in \mathbb{R}^N$ and $g(x) = e^{2\pi i x\cdot \eta}f(x)$, then $\widehat{g}(\xi) = \widehat{f}(\xi - \eta)$.
3. If $a \in \mathbb{R}$ and g(x) = f(ax), then $\widehat{g}(\xi) = \widehat{f}(\xi/a)/a^N.$
4. Define $\phi(x) = e^{-\pi|x|^2}$; then $\widehat{\phi} = \phi.$
5. Set $\phi_{\varepsilon}(x) = \frac{1}{\varepsilon^N}\phi\left(\frac{x}{\varepsilon}\right)$. Then with $\ast$ denoting convolution, ϕ_ε is an approximation to the identity: $\lim_{\varepsilon\to 0}\phi_{\varepsilon} * f \to f$, where the convergence is uniform on bounded sets for f bounded and uniformly continuous and the convergence is in the p-norm for $f \in L^p$.

We can now prove the inversion formula. First, note that by the dominated convergence theorem

$\int e^{2\pi i x\cdot\xi}\widehat{f}(\xi)\,d\xi = \lim_{\varepsilon \to 0}\int e^{-\pi\varepsilon^2|\xi|^2 + 2\pi i x\cdot\xi}\widehat{f}(\xi)\,d\xi.$

Define $g(\xi) = e^{-\pi\varepsilon^2|\xi|^2 + 2\pi i x\cdot\xi}$. Applying the second and then third fact from above, $\widehat{g}(y) = \frac{1}{\varepsilon^N}e^{-\frac{\pi}{\varepsilon^2}|x - y|^2}.$ With ϕ_ε as before, we can push the Fourier transform onto g in the last integral to get

$\int e^{-\pi\varepsilon^2|\xi|^2 + 2\pi i x\cdot\xi}\widehat{f}(\xi)\,d\xi = \int \widehat{g}(y)f(y)\,dy = \int \frac{1}{\varepsilon^N}e^{-\frac{\pi}{\varepsilon^2}|x - y|^2}f(y)\,dy = (\phi_{\varepsilon} * f)(x),$

the convolution of ƒ with an approximate identity. Hence by the last fact

$\lim_{\varepsilon\to 0}\int e^{2\pi i x\cdot\xi}\widehat{f}(\xi)e^{-\pi\varepsilon^2|\xi|^2}d\xi = \lim_{\varepsilon\to 0}\int \frac{1}{\varepsilon^N}e^{-\frac{\pi}{\varepsilon^2}|y - x|^2}f(y)\,dy = \lim_{\varepsilon\to 0}\phi_{\varepsilon} * f (x) = f(x).$

This establishes that the Fourier transform is an invertible map of the Schwartz space to itself. In particular, it is an isometry in the L² norm, and Schwartz functions are dense in L². The Fourier transform and its inverse then extend to unitary operators $\mathcal{F}, \mathcal{F}^{-1}$ on all of L² for which $\mathcal{F}\mathcal{F}^{-1} = \mathcal{F}^{-1}\mathcal{F} = I$, with I the identity map.

While the integral defining the Fourier transform or its inverse may not make sense for general L² functions, one can always integrate over a symmetric rectangle and take the limits as its length tends to infinity. What one is really doing here is taking an increasing sequence E_n of relatively compact sets growing to $\mathbb{R}^N$, and taking the limit of $\mathcal{F}(\chi_{E_n}f)$, where χ denotes the indicator function of a set. Since $\chi_{E_n}f$ is compactly supported, the integral defining its Fourier transform exists. But clearly $\chi_{E_n}f \to f$ in L², hence $\mathcal{F}(\chi_{E_n}f) \to \mathcal{F}f$ as well.

Fourier transforms of square-integrable functions

Via the Plancherel theorem, one can also define the Fourier transform of a square-integrable function, i.e., one satisfying

$\int_{-\infty}^\infty\left|f(x)\right|^2\,dx<\infty.$

Then the Fourier transform is another quadratically integrable function.

In case f is a square-integrable periodic function on the interval [ − π,π], it has a Fourier series whose coefficients are

$\widehat{f}(n)=\frac{1}{2\pi}\int_{-\pi}^\pi f(x)\,e^{-inx}\,dx.$

The Fourier inversion theorem might then say that

$\sum_{n=-\infty}^{\infty} \widehat{f}(n)\,e^{inx}=f(x).$

What kind of convergence is right? "Convergence in mean square" can be proved fairly easily:

$\lim_{N\rightarrow\infty}\int_{-\pi}^\pi\left|f(x)-\sum_{n=-N}^{N} \widehat{f}(n)\,e^{inx}\right|^2\,dx=0.$

What about convergence almost everywhere? That would say that if f is square-integrable, then for "almost every" value of x between 0 and 2π we have

$f(x)=\lim_{N\rightarrow\infty}\sum_{n=-N}^{N} \widehat{f}(n)\,e^{inx}.$

This was not proved until 1966 in (Carleson, 1966).

For strictly finitary discrete Fourier transforms, these delicate questions of convergence are avoided.

See also

• Convergence of Fourier series

References

• G.B. Folland, Introduction to Partial Differential Equations, 2nd ed, Princeton University Press, 1995.
• Lennart Carleson (1966). On the convergence and growth of partial sums of Fourier series. Acta Math. 116, 135–157.