The Fourier Transform on the Real Line

Part 13, Chapter 13: Fourier Series and Transforms

Learning objectives

Write the Fourier transform $\hat{f}(\xi)$ and the inverse Fourier transform
Apply the convolution theorem and the time-shift property
State and use Plancherel's theorem (the continuous analog of Parseval)
Recognize the Gaussian as the canonical fixed point $\hat{f} = f$

What replaces a Fourier series when the function is not periodic but lives on all of $\mathbb{R}$ ? The discrete index $n$ becomes a continuous variable $\xi$ , the sum becomes an integral, and the formulas are nearly identical, with one immensely useful new property. Convolution in space (which describes blurring, smoothing, the action of any linear time-invariant filter) becomes pointwise multiplication in frequency. That single fact, the convolution theorem, is why the Fourier transform is the central tool of signal processing, image analysis, quantum mechanics, and partial differential equations.

Definition and inversion

For a function $f: \mathbb{R} \to \mathbb{C}$ in $L^1$ (so the integrals exist), define the Fourier transform:

$\hat{f}(\xi) = \displaystyle\int_{-\infty}^{\infty} f(x) e^{-2\pi i \xi x} \, dx$ −inftyinftyf(x)e−2piixix,dx

The inverse is symmetric:

$f(x) = \displaystyle\int_{-\infty}^{\infty} \hat{f}(\xi) e^{2\pi i \xi x} \, d\xi$ −inftyinftyhatf(xi)e2piixix,dxi

(Caveat: different books use different normalisations. The convention here uses $2\pi$ in the exponent and no leading factors; some texts put $1/\sqrt{2\pi}$ in front of both formulas, others put $1/(2\pi)$ in front of one. Any consistent choice works; just stick to one.)

The core properties

Linearity: $\widehat{\alpha f + \beta g} = \alpha \hat{f} + \beta \hat{g}$ . The Fourier transform is a linear operator on $L^1$ (and extends to other spaces).

Time shift: if $g(x) = f(x - a)$ , then $\hat{g}(\xi) = e^{-2\pi i a \xi} \hat{f}(\xi)$ . Shifting in time multiplies the transform by a phase factor, the magnitude is unchanged.

Modulation: if $g(x) = e^{2\pi i a x} f(x)$ , then $\hat{g}(\xi) = \hat{f}(\xi - a)$ . Multiplying by a complex exponential in time shifts the spectrum, this is how AM radio works.

Scaling: if $g(x) = f(\alpha x)$ with $\alpha > 0$ , then $\hat{g}(\xi) = (1/\alpha) \hat{f}(\xi / \alpha)$ . Compressing in time stretches in frequency, and vice versa, the uncertainty principle in disguise.

The convolution theorem

Define the convolution of $f$ and $g$ :

$(f * g)(x) = \displaystyle\int_{-\infty}^{\infty} f(t) g(x - t) \, dt$ −inftyinftyf(t)g(x−t),dt

The convolution theorem states:

$\widehat{f * g} = \hat{f} \cdot \hat{g}$

Convolution (the messy, expensive integral) becomes pointwise multiplication (cheap and parallel) in the frequency domain. That is the whole reason the Fast Fourier Transform exists in production code: you transform, multiply, and inverse-transform, an $O(N \log N)$ procedure that would otherwise cost $O(N^2)$ .

Plancherel's theorem

The continuous analog of Parseval:

$\displaystyle\int_{-\infty}^{\infty} |f(x)|^2 \, dx = \displaystyle\int_{-\infty}^{\infty} |\hat{f}(\xi)|^2 \, d\xi$ hatf(xi)∣2,dxi

The Fourier transform preserves the $L^2$ norm, so it extends from $L^1 \cap L^2$ to a unitary operator on all of $L^2(\mathbb{R})$ .

Where this shows up

MRI k-space reconstruction: An MR scanner measures the Fourier transform of the spin-density map of the slice being imaged. The acquired raw data live in k-space, and the image is reconstructed by the inverse Fourier transform, this is literally the inversion formula at industrial scale.
Heisenberg uncertainty principle: If $f$ is well-concentrated in $x$ , then $\hat{f}$ must be spread out in $\xi$ . The product of the two spreads is bounded below by a constant, the scaling property is the elementary version, and the rigorous statement is the variance inequality $\sigma_x \sigma_\xi \geq 1/(4\pi)\text{ (with our }e^{-2\pi i\xi x}\text{ convention; equals }\hbar/2\text{ in physics notation)}$ xigeq1/(4pi)text(withoure−2piixixtextconvention;equalshbar/2textinphysicsnotation).
Spectral analysis of seismic data: Earthquake records are non-periodic. To estimate the dominant frequencies (and hence subsurface structure), seismologists compute $|\hat{f}(\xi)|^2$ , the power spectrum, via FFT.
Heat equation on the line: The heat equation $u_t = u_{xx}$ on $\mathbb{R}$ is solved by Fourier transforming in $x$ . The PDE becomes the ODE $\hat{u}_t = -4\pi^2 \xi^2 \hat{u}$ t=−4pi2xi2hatu, whose solution is $\hat{u}(\xi, t) = e^{-4\pi^2 \xi^2 t} \hat{u}_0(\xi)$ , a Gaussian multiplier in the frequency domain.

Pause and think: The Gaussian $f(x) = e^{-\pi x^2}$ satisfies $\hat{f} = f$ (the standard Gaussian is its own Fourier transform). What is $\hat{f}(0)$ , and what classical integral does it equal?

Try it

Compute $\hat{f}(\xi)$ for the rectangular pulse $f(x) = 1$ on $|x| < 1/2$ , zero elsewhere. (Answer: $\text{sinc}(\xi) = \sin(\pi \xi)/(\pi \xi)$ .)
If $\hat{f}(\xi) = e^{-|\xi|}$ , find $\hat{f}(0)$ . What does it represent? (Hint: it equals $\int f(x) \, dx$ .)
Apply the time-shift property to compute the Fourier transform of $f(x) = \text{rect}(x - 3)$ , where $\text{rect}$ is the standard rectangular pulse from the first exercise.
If $\int |f|^2 \, dx = 7$ , what is $\int |\hat{f}|^2 \, d\xi$ ? (Plancherel.)
Use the convolution theorem: if $\hat{f}(\xi) = e^{-|\xi|}$ and $\hat{g}(\xi) = 1/(1 + \xi^2)$ , write down $\widehat{f * g}$ .

A trap to watch for

The most common Fourier-transform error is using the wrong convention. The formulas above use $e^{-2\pi i \xi x}$ in the forward transform, but many physics books use $e^{-i \omega x}$ instead. That switch changes the inversion constant (you pick up a $1/(2\pi)$ ) and changes the form of Plancherel's constant. Always check the convention printed on page 1 of whatever reference you are using, and never copy formulas between books without verifying their conventions match.

What you now know

You can compute Fourier transforms of basic signals, apply the convolution theorem to convert a hard convolution into easy multiplication, and use Plancherel to track energy across the transform. The next chapter applies all of this Fourier machinery (plus separation-of-variables techniques) to solve the three canonical PDEs: Laplace, heat, and wave.

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References

Garrity, T. (2002). All the Mathematics You Missed: But Need to Know for Graduate School. Cambridge University Press, ch. 13.
Stein, E. M., Shakarchi, R. (2003). Fourier Analysis: An Introduction. Princeton University Press, ch. 5.
Folland, G. B. (1992). Fourier Analysis and Its Applications. Wadsworth & Brooks/Cole, ch. 7-8.
Bracewell, R. N. (1999). The Fourier Transform and Its Applications (3rd ed.). McGraw-Hill, ch. 6-9.
Strichartz, R. S. (2003). A Guide to Distribution Theory and Fourier Transforms. World Scientific, ch. 2.