The fourier transform

1 Princeton Companion to Mathematics Proof The Fourier Transform By Terence Tao Let f be a function from R to R 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, fe , and an odd part, fo For instance, the function f (x) = x3 + 3x2 + 3x + is neither even nor odd, but it can be written as fe (x) + fo (x), where fe (x) = 3x2 + and fo (x) = x3 + 3x For a general function f , the decomposition is unique and is given by the formulas fe (x) = 12 (f (x) + f (−x)) and fo (x) = 12 (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 transformation φ 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 (φ(x)) In the case at hand, if φ = ι then the transformed function is just f (x), while if φ = 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 φ = 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 sin(nx) and cos(nx)—and they are often associated with physical concepts such as frequency or energy The symmetry will usually be associated with a group G, which will usually be 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 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, or as linear combinations of eigenvectors of a matrix or linear operator For a more complicated example, let us fix a positive integer n and let us define a systematic way of decomposing functions from C to C, that is, complex-valued functions defined on the complex plane If f is such a function and j is an integer between and n − 1, then we say that f is a harmonic of order j if it has the following property Let ω = e2πi/n , so that ω is a primitive nth root of (meaning that ω n = but no smaller positive power of ω gives 1) Then f (ωz) = ω j f (z) for every z ∈ C Notice that if n = 2, then ω = −1, so when j = we recover the definition of an even function and when j = 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 fj (z) = n n−1 f (ω k z)ω −jk , k=0 then it is a simple exercise to prove that n−1 fj (z) f (z) = j=0 for every z (use the fact that j ω −jk = n if k = and otherwise), and that fj (ωz) = ω j f (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 nth roots of unity 1, ω, , ω n−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 2πj/n Now let us consider infinite groups Let f be a complex-valued function defined on the unit circle T := {z ∈ C : |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) = cz n for some integer n and some constant c, then f will have rotational symmetry of order n That is, if ω = e2πi/n again, then Princeton Companion to Mathematics Proof 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 ∞ fˆ(n)z n , f (z) = n=−∞ where the numbers fˆ(n), called the Fourier coefficients of f at the frequencies n, are given by the formula fˆ(n) := 2π 2π f (eiθ )e−iθ dθ 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 an analytic function If f is analytic on the closed unit disk {z ∈ C : |z| 1}, then one can write ∞ an z n , f (z) = n=0 where the Taylor coefficient an is given by the formula f (z) an = dz 2πi |z|=1 z n+1 In general, there are very strong links between Fourier analysis and complex analysis When f is smooth, then its Fourier coefficients decay to zero very quickly and it is easy to show ∞ that the Fourier series n=−∞ fˆ(n)z n 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 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 T (Notice that we can think of the number eiθ 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 Z of all integers If we take two of our basic symmetric functions, z m and z n , and multiply them together, then we obtain the function z m+n , so the map n → z n is an isomorphism from Z to the set of all these functions under multiplication The group Z is known as the Pontryagin dual to T 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 Rd Among all functions f : Rd → C, the ones considered to be “basic” are the plane waves f (x) = cξ e2πix·ξ , where ξ ∈ Rd 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 cξ is a complex number (whose magnitude is the amplitude of the plane wave) Notice that sets of the form Hλ = {x : x · ξ = λ} 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 Hλ is always equal to the value taken on Hλ+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 fˆ(ξ)e2πix·ξ dξ, f (x) = Rd where f (x)e−2πix·ξ dx fˆ(ξ) = Rd The function fˆ(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 fˆ(ξ) as the extent to which the function f contains a component which oscillates at frequency ξ As it turns out, there is no difficulty justifying the convergence of these integrals when f is sufficiently nice, though the issue again becomes more subtle for functions that are 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 3 Princeton Companion to Mathematics Proof somewhat rough or slowly decaying In this case, effect on each plane wave To be explicit about it, the underlying group is the Euclidean group Rd ∆f (x) = ∆ fˆ(ξ)e2πix·ξ dξ (which can also be thought of as the group of dRd dimensional translations); note that both the position variable x and the frequency variable ξ are = fˆ(ξ)∆e2πix·ξ dξ d d d R contained in R , so R is also the Pontryagin dual group in this setting.2 = (−4π|ξ|2 )fˆ(ξ)e2πix·ξ dξ, One major application of the Fourier transform lies in understanding various linear operations on functions, such as, for instance, the Laplacian on Rd Given a function f : Rd → C, its Laplacian ∆f is defined by the formula d ∆f (x) = j=1 ∂2f , ∂x2j Rd which gives us a formula for the Laplacian of a general function Here we have interchanged the Laplacian ∆ with an integral; this can be rigorously justified for suitably nice f , but we omit the details This formula represents ∆f as a superposition of plane waves But any such representation is unique, and the Fourier inversion formula tells us that ∆f (x) = ∆f (ξ)e2πix·ξ dξ where we think of the vector x in coordinate d R form, x = (x1 , , xd ), and of f as a function Therefore, f (x1 , , xd ) of d real variables To avoid technicalities let us consider only those functions that ∆f (ξ) = (−4π|ξ|2 )fˆ(ξ), are smooth enough for the above formula to make a fact that can also be derived directly from the sense without any difficulty definition of the Fourier transform using integraIn general, there is no obvious relationship tion by parts This identity shows that the Fourier between a function f and its Laplacian ∆f But transform diagonalizes the Laplacian: the operawhen f is a plane wave such as f (x) = e2πix·ξ , tion of taking the Laplacian, when viewed using then there is a very simple relationship: the the Fourier transform, is nothing more than multiplication of a function F (ξ) by the multiplier −4π|ξ|2 Such operators are called multiplica∆e2πix·ξ = −4π|ξ|2 e2πix·ξ tion operators The quantity −4π|ξ|2 can be interpreted as the energy level associated4 with the That is, the effect of the Laplacian on the plane frequency ξ In other words, the Laplacian can be wave e2πix·ξ is to multiply it by the scalar −4π|ξ|2 viewed as a Fourier multiplier, meaning that to In other words, the plane wave is an eigenfunc- calculate the Laplacian you take the Fourier transtion3 for the Laplacian ∆, with eigenvalue form, multiply by the multiplier, and then take the −4π|ξ|2 (More generally, plane waves will be inverse Fourier transform again This viewpoint eigenfunctions for any linear operation that com- allows one to manipulate the Laplacian very easily mutes with translations.) Therefore, the Laplacian, For instance, we can iterate the above formula to when viewed through the lens of the Fourier trans- compute higher powers of the Laplacian: form, is very simple: the Fourier transform lets one ∆n f (ξ) = (−4π|ξ|2 )n fˆ(ξ) for n = 0, 1, 2, write an arbitrary function as a superposition of plane waves, and the Laplacian has a very simple Indeed, we are now in a position to develop more general functions of the Laplacian For instance, This is because of our reliance on the dot product; if we can take a square root as follows: one did not want to use this dot product, the Pontryagin dual would instead be (Rd )∗ , the dual vector space to Rd But this subtlety is not too important in most applications Strictly speaking, this is a generalized eigenfunction, as plane waves are not square-integrable on Rd √ −∆f (ξ) = 4π|ξ|2 fˆ(ξ) When taking this view, it is customary to replace ∆ by −∆ in order to make the energies positive 4 Princeton Companion to Mathematics Proof This leads to the theory of fractional differential operators (which are in turn a special case of pseudodifferential operators), as well as the more general theory of functional calculus, 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 One particularly important and striking estimate of this type is the Plancherel identity, Rd |f (x)|2 dx = Rd |fˆ(ξ)|2 dξ, 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 frequencyspace 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: Rd f (y)g(x − y) dy = fˆ(ξ)ˆ g (ξ)e2πix·ξ dξ Rd This formula allows one to analyze the convolution f ∗ g(x) := Rd f (y)g(x − y) dy of two functions f , g in terms of their Fourier transform; 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 (also known as the law of large numbers), which asserts that the sum of many independent random variables will eventually resemble a Gaussian distribution; one can even use such methods to establish Vinogradov’s theorem, 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 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 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)) → f g(x) that are of importance in differential equations In another direction, one can replace Euclidean space Rd 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 in Part III), 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 such as orthogonal polynomials or spherical harmonics 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 Princeton Companion to Mathematics Proof 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 Rd This leads to some variants of the Fourier transform, such as wavelet transforms, 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 ... fˆ(ξ) = Rd The function fˆ(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... transform, such as the Laplace transform or the Mellin transform (for more about other transforms, see the article transforms in Part III), which are very similar algebraically to the Fourier transform. .. type is the Plancherel identity, Rd |f (x)|2 dx = Rd |fˆ(ξ)|2 dξ, which shows that the L2 -norm of a Fourier transform is actually equal to the L2 -norm of the original function The Fourier transform