III.27 The Fourier Transform

Terence Tao

Let f be a function from to . Typically, there is not much that one can say about f, but certain functions have useful symmetry properties. For instance, f is called even if f (-x) = f(x) for every x, and it is called odd if f (-x) = - f(x) for every x. Furthermore, every function f can be written as a superposition of an even part, f_e, and an odd part, f₀. For instance, the function f(x) = x³ + 3x² + 3x + 1 is neither even nor odd, but it can be written as f_e(x) + f₀(x), where f_e(x) = 3x² + 1 and f₀(x) = x³ + 3x. For a general function f, the decomposition is unique and is given by the formulas

f_e(x) = (f (x) + f(-x))

and

f₀(x) = (f (x) -f(-x)).

What are the symmetry properties enjoyed by even and odd functions? A useful way to regard them is as follows. We have a group of two transformations of the real line: one is the identity map ι : x x and the other is the reflection ρ : x - x. Now any transfor mation of the real line gives rise to a transformation of the functions defined on the real line: given a function f, the transformed function is the function g(x) = f ( (x)). In the case at hand, if = ι then the transformed function is just f(x), while if = ρ then it is f (-x). If f is either even or odd, then both the transformed functions are scalar multiples of the original function f. In particular, when = ρ, the transformed function is f(x) when f is even (so the scalar multiple is 1) and -f(x) when f is odd (so the scalar multiple is -1).

The procedure just described can be thought of as a very simple prototype of the general notion of a Fourier transform. Very broadly speaking, a Fourier transform is a systematic way to decompose “generic” functions into a superposition of “symmetric” functions. These symmetric functions are usually quite explicitly defined: for instance, one of the most important examples is a decomposition into the TRIGONOMETRIC FUNCTIONS [III.92] sin(nx) and cos(nx). They are also often related to physical concepts such as frequency or energy. The symmetry will usually be associated with a GROUP [I.3 §2.1] G, which is usually Abelian. (In the case considered above, it is the two-element group.) Indeed, the Fourier transform is a fundamental tool in the study of groups, and more precisely in the REPRESENTATION THEORY [IV.9] Of groups, which concerns different ways in which a group can be regarded as a group of symmetries. It is also related to topics in linear algebra, such as the representation of a vector as linear combinations of an ORTHONORMAL BASIS [III.37], or as linear combinations of EIGENVECTORS [I.3 §4.3] of a matrix Or LINEAR OPERATOR [III.50].

For a more complicated example, let us fix a positive integer n and let us define a systematic way of decomposing functions from to , that is, complex-valued functions defined on the complex plane. If f is such a function and j is an integer between 0 and n - 1, then we say that f is a harmonic of order j if it has the following property. Let ω = e^2πi/n, so that ω is a primitive n th root of 1 (meaning that ωⁿ = 1 but no smaller positive power of ω gives 1). Then f (ω z) = ω^j f (z) for every z . Notice that if n = 2, then ω = -1, so when j = 0 we recover the definition of an even function and when j = 1 we recover the definition of an odd function. In fact, inspired by this, we can give a general formula for a decomposition of f into harmonics, which again turns out to be unique. If we define

then it is a simple exercise to prove that

for every z (use the fact that Σ_jω^{- jk} = n if k = 0 and 0 otherwise), and that f_j(ω z) = ω^j f_j(z) for every z. Thus, f can be decomposed as a sum of harmonics. The group associated with this Fourier transform is the multiplicative group of the n th roots of unity 1, ω, . . . , ωn^{- 1}, or the cyclic group of order n. The root ω^j is associated with the rotation of the complex plane through an angle of 2 π j/n.

Now let us consider infinite groups. Let f be a complex-valued function defined on the unit circle {z : |z| = 1}. To avoid technical issues we shall assume that f is smooth—that is, it is infinitely differentiable. Now if f is a function of the simple form f(z) = czⁿ for some integer n and some constant c, then f will have rotational symmetry of order n. That is, if ω = e^2πi/n again, then f(ωz) = f(z) for all complex numbers z. After our earlier examples, it should come as no surprise that an arbitrary smooth function f can be expressed as a superposition of such rotationally symmetric functions. Indeed, one can write

where the numbers (n), called the Fourier coefficients of f at the frequencies n, are given by the formula

This formula can be thought of as the limiting case n → ∞ of the previous decomposition, restricted to the unit circle. It can also be regarded as a generalization of the Taylor series expansion of a HOLOMORPHIC FUNCTION [I.3 §5.6]. If f is holomorphic on the closed unit disk {z : |z| ≤ 1}, then one can write

where the Taylor coefficient a_n is given by the formula

In general, there are very strong links between Fourier analysis and complex analysis.

If f is smooth, then its Fourier coefficients decay to zero very quickly and it is easy to show that the Fourier series (n)zⁿ converges. The issue becomes more subtle if f is not smooth (for instance, if it is merely continuous). Then one must be careful to specify the precise sense in which the series converges. In fact, a significant portion of HARMONIC ANALYSIS [IV.11] is devoted to questions of this kind, and to developing tools for answering them.

The group of symmetries associated with this version of Fourier analysis is the circle group . (Notice that we can think of the number e^iθ both as a point in the circle and as a rotation through an angle of θ. Thus, the circle can be identified with its own group of rotational symmetries.) But there is a second group that is important here as well, namely the additive group of all integers. If we take two of our basic symmetric functions, Z^m andzⁿ, and multiply them together, then we obtain the function z^m+n, so the map n → zⁿ is an isomorphism from to the set of all these functions under multiplication. The group is known as the Pontryagin dual of .

In the theory of partial differential equations and in related areas of harmonic analysis, the most important Fourier transform is defined on the Euclidean space ^d. Among all functions f : ^d → , the ones considered to be “basic” are the plane waves f (x) = c_ξe^2πix.ξ, where ξ ^d is a vector (known as the frequency of the plane wave), x · ξ is the dot product between the position x and the frequency ξ, and c_ξ is a complex number (whose magnitude is the amplitude of the plane wave). Notice that sets of the form H = {x : x · ξ = } are (hyper)planes orthogonal to ξ and on each such set the value of f(x) is constant. Moreover, the value taken by f on H is always equal to the value taken on H+2π· This explains the name “plane waves.” It turns out that if a function f is sufficiently “nice” (e.g., smooth and rapidly decreasing as x gets large), then it can be represented uniquely as the superposition of plane waves, where a “superposition” is now interpreted as an integral rather than a summation. More precisely, we have the formulas¹

where

The function (ξ) is known as the Fourier transform of f, and the second formula is known as the Fourier inversion formula. These two formulas show how to determine the Fourier-transformed function from the original function and vice versa. One can view the quantity (ξ) as the extent to which the function f contains a component that oscillates at frequency ξ. As it turns out, there is no difficulty in justifying the convergence of these integrals when f is sufficiently nice, though the issue again becomes more subtle for functions that are somewhat rough or slowly decaying. In this case, the underlying group is the Euclidean group ^d (which can also be thought of as the group of d-dimensional translations); note that both the position variable x and the frequency variable ξ are contained in ^d, so ^d is also the Pontryagin dual group in this setting.²

One major application of the Fourier transform lies in understanding various linear operations on functions, such as, for instance, the Laplacian on ^d. Given a function f : ^d →., its Laplacian Δf is defined by the formula

where we think of the vector x in coordinate form, x = (x1,. . . , x_d), and of f as a function f (x1,. . . ,x_d) of d real variables. To avoid technicalities let us consider only those functions that are smooth enough for the above formula to make sense without any difficulty.

In general, there is no obvious relationship between a function f and its Laplacian Δf. But when f is a plane wave such as f(x) = e^2πix·ξ, there is a very simple relationship:

Δe^2πix·ξ= -4π²|ξ|²e^2πix·ξ.

That is, the effect of the Laplacian on the plane wave e^2πix·ξ is to multiply it by the scalar -4π²|ξ|². In other words, the plane wave is an eigenfunction³ for the Laplacian Δ, with eigenvalue -4π²|ξ|². (More generally, plane waves will be eigenfunctions for any linear operation that commutes with translations.) Therefore, the Laplacian, when viewed through the lens of the

Fourier transform, is very simple: the Fourier transform lets one write an arbitrary function as a superposition of plane waves, and the Laplacien has a very simple effect on each plane wave. To be explicit about it,

which gives us a formula for the Laplacian of a general function. Here we have interchanged the Laplacian Δ with an integral; this can be rigorously justified for suitably nice f, but we omit the details.

This formula represents Δf as a superposition of plane waves. But any such representation is unique, and the Fourier inversion formula tells us that

Therefore,

a fact that can also be derived directly from the definition of the Fourier transform using integration by parts. This identity shows that the Fourier transform diagonalizes the Laplacian: the operation of taking the Laplacian, when viewed using the Fourier transform, is nothing more than multiplication of a function F(ξ) by the multiplier -4π²|ξ|². The quantity -4π²|ξ|² can be interpreted as the energy level associated⁴ with the frequency ξ. In other words, the Laplacian can be viewed as a Fourier multiplier, meaning that to calculate the Laplacian you take the Fourier transform, multiply by the multiplier, and then take the inverse Fourier transform again. This viewpoint allows one to manipulate the Laplacian very easily. For instance, we can iterate the above formula to compute higher powers of the Laplacian:

Indeed, we are now in a position to develop more general functions of the Laplacian. For instance, we can take a square root as follows:

This leads to the theory of fractional differential operators (which are in turn a special case of pseudodifferenticil operators), as well as the more general theory

Of FUNCTIONAL CALCULUS [IV.15 §3.1], in which one starts with a given operator (such as the Laplacian) and then studies various functions of that operator, such as square roots, exponentials, inverses, and so forth.

As the above discussion shows, the Fourier transform can be used to develop a number of interesting operations, which have particular importance in the theory of differential equations. To analyze these operations effectively, one needs various estimates on the Fourier transform. For instance, it is often important to know how the size of a function f, as measured by some norm, relates to the size of its Fourier transform, as measured by a possibly different norm. For a further discussion of this point, see FUNCTION SPACES [III.29]. One particularly important and striking estimate of this type is the Plcincherel identity,

which shows that the L₂-norm of a Fourier transform is actually equal to the L₂-norm of the original function. The Fourier transform is therefore a unitary operation, so one can view the frequency-space representation of a function as being in some sense a “rotation” of the physical-space representation.

Developing further estimates related to the Fourier transform and associated operators is a major component of harmonic analysis. A variant of the Plancherel identity is the convolution formula:

This formula allows one to analyze the convolution

of two functions f and g in terms of their Fourier transforms; in particular, if the Fourier coefficients of f or g are small, then we expect the convolution f * g to be small as well. This relationship means that the Fourier transform controls certain correlations of a function with itself and with other functions, which makes the Fourier transform an important tool in understanding the randomness and uniform distribution properties of various objects in probability theory, harmonic analysis, and number theory. For instance, one can pursue the above ideas to establish the central limit theorem, which asserts that the sum of many independent random variables will eventually resemble a GAUSSIAN DISTRIBUTION [III.71 §5]; one can even use such methods to establish VINOGRADOV’S THEOREM [V.27], that every sufficiently large odd number is the sum of three primes.

There are many directions in which to generalize the above set of ideas. For instance, one can replace the Laplacian by a more general operator and the plane waves by (generalized) eigenfunctions of that operator. This leads to the subject Of SPECTRAL THEORY [III.86] and functional calculus; one can also study the algebra of Fourier multipliers (and of convolution) more abstractly, which leads to the theory of C^*-ALGEBRAS [IV.15 §3]. One can also go beyond the theory of linear operators and study bilinear, multilinear, or even fully nonlinear operators. This leads in particular to the theory of paraproducts, which are generalizations of the pointwise product operation (f (x) , g (x)) fg(x) that are of importance in differential equations. In another direction, one can replace Euclidean space ^d by a more general group, in which case the notion of a plane wave is replaced by the notion of a character (if the group is Abelian) or a representation (if the group is non-Abelian). There are other variants of the Fourier transform, such as the Laplace transform or the Mellin transform (for more about other transforms, see the article TRANSFORMS [III.91]), which are very similar algebraically to the Fourier transform and play similar roles (for instance, the Laplace transform is also useful in analyzing differential equations). We have already seen that Fourier transforms are connected to Taylor series; there is also a connection to some other important series expansions, notably Dirichlet series, as well as expansions of functions in terms of SPECIAL POLYNOMIALS [III.85] such as orthogonal polynomials Or SPHERICAL HARMONICS [III.87].

The Fourier transform decomposes a function exactly into many components, each of which has a precise frequency. In some applications it is more useful to adopt a “fuzzier” approach, in which a function is decomposed into fewer components but each component has a range of frequencies rather than consisting purely of a single frequency. Such decompositions can have the advantage of being less constrained by the uncertainty principle, which asserts that it is impossible for both a function and its Fourier transform to be concentrated in very small regions of ^d. This leads to some variants of the Fourier transform, such as WAVELET TRANSFORMS [VII.3], which are better suited to a number of problems in applied and computational mathematics, and also to certain questions in harmonic analysis and differential equations. The uncertainty principle, being fundamental to quantum mechanics, also connects the Fourier transform to mathematical physics, and in particular to the connections between classical and quantum physics, which can be studied rigorously using the methods of geometric quantization and microlocal analysis.

1. In some texts, the Fourier transform is defined slightly differently, with factors such as 2π and -1 being moved to other places. These notational differences have some minor benefits and drawbacks, but they are all equivalent to each other.

2. This is because of our reliance on the dot product; if one did not want to use this dot product, the Pontryagin dual would instead be (^d)^*, the dual vector space to ^d. But this subtlety is not too important in most applications.

3. Strictly speaking, this is a general?zed eigenfunction, as plane waves are not square-integrable on ^d.

4. When taking this view, it is customary to replace Δ by -Δ in order to make the energies positive.