In these notes we lay out the basic theory of the Fourier transform, which is of course the most fundamental tool in harmonic analysis and also of major importance in related fields (functional analysis, complex analysis, PDE, number theory, additive combinatorics, representation theory, signal processing, etc.). The Fourier transform, in conjunction with the Fourier inversion formula, allows one to take essentially arbitrary (complex-valued) functions on a group (or more generally, a space
that
acts on, e.g. a homogeneous space
), and decompose them as a (discrete or continuous) superposition of much more symmetric functions on the domain, such as characters
; the precise superposition is given by Fourier coefficients
, which take values in some dual object such as the Pontryagin dual
of
. Characters behave in a very simple manner with respect to translation (indeed, they are eigenfunctions of the translation action), and so the Fourier transform tends to simplify any mathematical problem which enjoys a translation invariance symmetry (or an approximation to such a symmetry), and is somehow “linear” (i.e. it interacts nicely with superpositions). In particular, Fourier analytic methods are particularly useful for studying operations such as convolution
and set-theoretic addition
, or the closely related problem of counting solutions to additive problems such as
or
, where
are constrained to lie in specific sets
. The Fourier transform is also a particularly powerful tool for solving constant-coefficient linear ODE and PDE (because of the translation invariance), and can also approximately solve some variable-coefficient (or slightly non-linear) equations if the coefficients vary smoothly enough and the nonlinear terms are sufficiently tame.
The Fourier transform also provides an important new way of looking at a function
, as it highlights the distribution of
in frequency space (the domain of the frequency variable
) rather than physical space (the domain of the physical variable
). A given property of
in the physical domain may be transformed to a rather different-looking property of
in the frequency domain. For instance:
- Smoothness of
in the physical domain corresponds to decay of
in the Fourier domain, and conversely. (More generally, fine scale properties of
tend to manifest themselves as coarse scale properties of
, and conversely.)
- Convolution in the physical domain corresponds to pointwise multiplication in the Fourier domain, and conversely.
- Constant coefficient differential operators such as
in the physical domain corresponds to multiplication by polynomials such as
in the Fourier domain, and conversely.
- More generally, translation invariant operators in the physical domain correspond to multiplication by symbols in the Fourier domain, and conversely.
- Rescaling in the physical domain by an invertible linear transformation corresponds to an inverse (adjoint) rescaling in the Fourier domain.
- Restriction to a subspace (or subgroup) in the physical domain corresponds to projection to the dual quotient space (or quotient group) in the Fourier domain, and conversely.
- Frequency modulation in the physical domain corresponds to translation in the frequency domain, and conversely.
(We will make these statements more precise below.)
On the other hand, some operations in the physical domain remain essentially unchanged in the Fourier domain. Most importantly, the norm (or energy) of a function
is the same as that of its Fourier transform, and more generally the inner product
of two functions
is the same as that of their Fourier transforms. Indeed, the Fourier transform is a unitary operator on
(a fact which is variously known as the Plancherel theorem or the Parseval identity). This makes it easier to pass back and forth between the physical domain and frequency domain, so that one can combine techniques that are easy to execute in the physical domain with other techniques that are easy to execute in the frequency domain. (In fact, one can combine the physical and frequency domains together into a product domain known as phase space, and there are entire fields of mathematics (e.g. microlocal analysis, geometric quantisation, time-frequency analysis) devoted to performing analysis on these sorts of spaces directly, but this is beyond the scope of this course.)
In these notes, we briefly discuss the general theory of the Fourier transform, but will mainly focus on the two classical domains for Fourier analysis: the torus , and the Euclidean space
. For these domains one has the advantage of being able to perform very explicit algebraic calculations, involving concrete functions such as plane waves
or Gaussians
.
— 1. Generalities —
Let us begin with some generalities. An abelian topological group is an abelian group with a topological structure, such that the group operations of addition
and negation
are continuous. (One can of course also consider abelian multiplicative groups
, but to fix the notation we shall restrict attention to additive groups.) For technical reasons (and in particular, in order to apply many of the results from the previous quarter) it is convenient to restrict attention to abelian topological groups which are locally compact Hausdorff (LCH); these are known as locally compact abelian (LCA) groups.
Some basic examples of locally compact abelian groups are:
- Finite additive groups (with the discrete topology), such as cyclic groups
.
- Finitely generated additive groups (with the discrete topology), such as the standard lattice
.
- Tori, such as the standard
-dimensional torus
with the standard topology.
- Euclidean spaces, such the standard
-dimensional Euclidean space
(with the standard topology, of course).
- The rationals
are not locally compact with the usual topology; but if one uses the discrete topology instead, one recovers an LCA group.
- Another example of an LCA group, of importance in number theory, is the adele ring
, discussed in this earlier blog post.
Thus we see that locally compact abelian groups can be either discrete or continuous, and either compact or non-compact; all four combinations of these cases are of importance. The topology of course generates a Borel -algebra in the usual fashion, as well as a space
of continuous, compactly supported complex-valued functions. There is a translation action
of
on
, where for every
,
is the translation operation
LCA groups need not be -compact (think of the free abelian group on uncountably many generators, with the discrete topology), but one has the following useful substitute:
Exercise 1 Show that every LCA group
contains a
-compact open subgroup
, and in particular is the disjoint union of
-compact sets. (Hint: Take a compact symmetric neighbourhood
of the identity, and consider the group
generated by this neighbourhood.)
An important notion for us will be that of a Haar measure: a Radon measure on
which is translation-invariant (i.e.
for all Borel sets
and all
, where
is the translation of
by
). From this and the definition of integration we see that integration
against a Haar measure (an operation known as the Haar integral) is also translation-invariant, thus
for all and
. The trivial measure
is of course a Haar measure; all other Haar measures are called non-trivial.
Let us note some non-trivial Haar measures in the four basic examples of locally compact abelian groups:
- For a finite additive group
, one can take either counting measure
or normalised counting measure
as a Haar measure. (The former measure emphasises the discrete nature of
; the latter measure emphasises the compact nature of
.)
- For finitely generated additive groups such as
, counting measure
is a Haar measure.
- For the standard torus
, one can obtain a Haar measure by identifying this torus with
in the usual manner and then taking Lebesgue measure on the latter space. This Haar measure is a probability measure.
- For the standard Euclidean space
, Lebesgue measure is a Haar measure.
Of course, any non-negative constant multiple of a Haar measure is again a Haar measure. The converse is also true:
Exercise 2 (Uniqueness of Haar measure up to scalars) Let
be two non-trivial Haar measures on a locally compact abelian group
. Show that
are scalar multiples of each other, i.e. there exists a constant
such that
. (Hint: for any
, compute the quantity
in two different ways.)
The above argument also implies a useful symmetry property of Haar measures:
Exercise 3 (Haar measures are symmetric) Let
be a Haar measure on a locally compact abelian group
. Show that
for all
. (Hint: expand
in two different ways.) Conclude that Haar measures on LCA groups are symmetric in the sense that
for all measurable
, where
is the reflection of
.
Exercise 4 (Open sets have positive measure) Let
be a non-trivial Haar measure on a locally compact abelian group
. Show that
for any non-empty open set
. Conclude that if
is non-negative and not identically zero, then
.
Exercise 5 Let
be an LCA group with non-trivial Haar measure
. Let
be the space of equivalence classes of functions
such that for every set
of finite measure, the restriction of
to
lies in
with a norm bounded uniformly in
, with two functions in
in the same equivalence class if they agree almost everywhere on every set
of finite measure, and with the
norm of
equal to the supremum of the
norms of the restrictions. (For
-finite groups
,
is identical to
, but the two spaces differ slightly in general.) Show that
is identifiable with
. (Unfortunately,
is not always
-finite, and so the standard duality theorem from Notes 3 of 245B does not directly apply. However, one can get around this using Exercise 1.)
It is a (not entirely trivial) theorem, due to André Weil, that all LCA groups have a non-trivial Haar measure. For discrete groups, one can of course take counting measure as a Haar measure. For compact groups, the result is due to Haar, and one can argue as follows:
Exercise 6 (Existence of Haar measure, compact case) Let
be a compact metrisable abelian group. For any real-valued
, and any Borel probability measure
on
, define the oscillation
of
with respect to
to be the quantity
.
- (a) Show that a Borel probability measure
is a Haar measure if and only if
for all
.
- (b) If a sequence
of Borel probability measures converges in the vague topology to another Borel probability measure
, show that
for all
.
- (c) If
is a Borel probability measure and
is such that
, show that there exists a Borel probability measure
such that
and
for all
. (Hint: take
to be the an average of certain translations of
.)
- (d) Given any finite number of functions
, show that there exists a Borel probability measure
such that
for all
. (Hint: Use Prokhorov’s theorem, see Corollary 13 of 245B Notes 12. Try the
case first.)
- (e) Show that there exists a unique Haar probability measure on
. (Hint: One can identify each probability measure
with the element
of the product space
, which is compact by Tychonoff’s theorem. Now use (d) and the finite intersection property.)
(The argument can be adapted to the case when
is not metrisable, but one has to replace the sequential compactness given by Prokhorov’s theorem by the topological compactness given by the Banach-Alaoglu theorem.)
For general LCA groups, the proof is more complicated:
Exercise 7 (Existence of Haar measure, general case) Let
be an LCA group. Let
denote the space of non-negative functions
that are not identically zero. Given two
, define a
-cover of
to be an expression of the form
that pointwise dominates
, where
are non-negative numbers and
. Let
denote the infimum of the quantity
for all
-covers of
.
- (a) (Finiteness) Show that
for all
.
- (b) Let
is a Haar measure on
. Show that
for all
. Conversely, for every
and
, and any neighbourhood
of the identity, show that there exists
supported in
such that
. (Hint:
is uniformly continuous. Take
to be an approximation to the identity.) Thus Haar integrals are related to certain renormalised versions of the functionals
; this observation underlies the strategy for construction of Haar measure in the rest of this exercise.
- (c) (Transitivity) Show that
for all
.
- (d) (Translation invariance) Show that
for all
and
.
- (e) (Sublinearity) Show that
and
for all
and
.
- (f) (Approximate superadditivity) If
and
, show that there exists a neighbourhood
of the identity such that
whenever
is supported in
. (Hint:
are all uniformly continuous. Take a
-cover of
and multiply the weight
at
by weights such as
and
.)
Next, fix a reference function
, and define the functional
for all
by the formula
.
- (g) Show that for any fixed
,
ranges in the compact interval
; thus
can be viewed as an element of the product space
, which is compact by Tychonoff’s theorem.
- (h) From (d), (e) we have the translation-invariance property
, the homogeneity property
, and the sub-additivity property
for all
,
, and
; we also have the normalisation
. Now show that for all
and
, there exists
such that
for all
.
- (i) Show that there exists a unique Haar measure
on
with
. (Hint: Use (h) and the finite intersection property to obtain a translation-invariant positive linear functional on
, then use the Riesz representation theorem.)
Now we come to a fundamental notion, that of a character.
Definition 8 (Characters) Let
be a LCA group. A multiplicative character
is a continuous function
to the unit circle
which is a homomorphism, i.e.
for all
. An additive character or frequency
is a continuous function
which is a homomorphism, thus
for all
. The set of all frequencies
is called the Pontryagin dual of
and is denoted
; it is clearly an abelian group. A multiplicative character is called non-trivial if it is not the constant function
; an additive character is called non-trivial if it is not the constant function
.
Multiplicative characters and additive characters are clearly related: if is an additive character, then the function
is a multiplicative character, and conversely every multiplicative character arises uniquely from an additive character in this fashion.
Exercise 9 Let
be an LCA group. We give
the topology of local uniform convergence on compact sets, thus the topology on
are generated by sets of the form
for compact
,
, and
. Show that this turns
into an LCA group. (Hint: Show that for any neighbourhood
of the identity in
, the sets
for
(say) are compact.) Furthermore, if
is discrete, show that
is compact.
The Pontryagin dual can be computed easily for various classical LCA groups:
Exercise 10 Let
be an integer.
- (a) Show that the Pontryagin dual
of
is identifiable as an LCA group with
, by identifying each
with the frequency
given by the dot product.
- (b) Show that the Pontryagin dual
of
is identifiable as an LCA group with
, by identifying each
with the frequency
given by the dot product.
- (c) Show that the Pontryagin dual
of
is identifiable as an LCA group with
, by identifying each
with the frequency
given by the dot product.
- (d) (Contravariant functoriality) If
is a continuous homomorphism between LCA groups, show that there is a continuous homomorphism
between their Pontryagin duals, defined by
for
and
.
- (e) If
is a closed subgroup of an LCA group
(and is thus also LCA), show that
is identifiable with
, where
is the space of all frequencies
which annihilate
(i.e.
for all
).
- (f) If
are LCA groups, show that
is identifiable as an LCA group with
.
- (g) Show that the Pontryagin dual of a finite abelian group
is identifiable with itself. (Hint: first do this for cyclic groups
, identifying
with the additive character
), then use the classification of finite abelian groups.) Note that this identification is not unique.
Exercise 11 Let
be an LCA group with non-trivial Haar measure
, and let
be a measurable function such that
for almost every
. Show that
is equal almost everywhere to a multiplicative character
of
. (Hint: on the one hand,
a.e. for almost every
. On the other hand,
depends continuously on
in, say, the local
topology.)
In the remainder of this section, is a fixed LCA group with a non-trivial Haar measure
.
Given an absolutely integrable function , we define the Fourier transform
by the formula
This is clearly a linear transformation, with the obvious bound
It converts translations into frequency modulations: indeed, one easily verifies that
for any ,
, and
. Conversely, it converts frequency modulations to translations: one has
for any and
, where
is the multiplicative character
.
Exercise 12 (Riemann-Lebesgue lemma) If
, show that
is continuous. Furthermore, show that
goes to zero at infinity in the sense that for every
there exists a compact subset
of
such that
for
. (Hint: First show that there exists a neighbourhood
of the identity in
such that
(say) for all
. Now take the Fourier transform of this fact.) Thus the Fourier transform maps
continuously to
, the space of continuous functions on
which go to zero at infinity; the decay at infinity is known as the Riemann-Lebesgue lemma.
Exercise 13 Let
be an LCA group with non-trivial Haar measure
. Show that the topology of
is the weakest topology such that
is continuous for every
.
Given two , recall that the convolution
is defined as
From Young’s inequality (Exercise 25 of Notes 1) we know that is defined a.e., and lies in
; indeed, we have
Exercise 14 Show that the operation
is a bilinear, continuous, commutative, and associative operation on
. As a consequence, the Banach space
with the convolution operation as “multiplication” operation becomes a commutative Banach algebra. If we also define
for all
, this turns
into a Banach
-algebra.
for all
; thus the Fourier transform converts convolution to pointwise product.
Exercise 15 Let
be LCA groups with non-trivial Haar measures
respectively, and let
,
. Show that the tensor product
(with product Haar measure
) has a Fourier transform of
, where we identify
with
as per Exercise 10(f). Informally, this exercise asserts that the Fourier transform commutes with tensor products. (Because of this fact, the tensor power trick is often available when proving results about the Fourier transform on general groups.)
Exercise 16 (Convolution and Fourier transform of measures) If
is a finite Radon measure on an LCA group
with non-trivial Haar measure
, define the Fourier-Stieltjes transform
by the formula
(thus for instance
for any
). Show that
is a bounded continuous function on
. Given any
, define the convolution
to be the function
and given any finite Radon measure
, let
be the measure
Show that
and
for all
, and similarly that
is a finite measure and
for all
. Thus the convolution and Fourier structure on
can be extended to the larger space
of finite Radon measures.
— 2. The Fourier transform on compact abelian groups —
In this section we specialise the Fourier transform to the case when the locally compact group is in fact compact, thus we now have a compact abelian group
with non-trivial Haar measure
. This case includes that of finite groups, together with that of the tori
.
As is a Radon measure, compact groups
have finite measure. It is then convenient to normalise the Haar measure
so that
, thus
is now a probability measure. For the remainder of this section, we will assume that
is a compact abelian group and
is its (unique) Haar probability measure, as given by Exercise 6.
A key advantage of working in the compact setting is that multiplicative characters now lie in
and
. In particular, they can be integrated:
Lemma 17 Let
be a multiplicative character. Then
equals
when
is trivial and
when
is non-trivial. Equivalently, for
, we have
, where
is the Kronecker delta function at
.
Proof: The claim is clear when is trivial. When
is non-trivial, there exists
such that
. If one then integrates the identity
using (2) one obtains the claim.
Exercise 18 Show that the Pontryagin dual
of a compact abelian group
is discrete (compare with Exercise 9).
Exercise 19 Show that the Fourier transform of the constant function
is the Kronecker delta function
at
. More generally, for any
, show that the Fourier transform of the multiplicative character
is the Kronecker delta function
at
.
Since the pointwise product of two multiplicative characters is again a multiplicative character, and the conjugate of a multiplicative character is also a multiplicative character, we obtain
Corollary 20 The space of multiplicative chararacters is an orthonormal set in the complex Hilbert space
.
Actually, one can say more:
Theorem 21 (Plancherel theorem for compact abelian groups) Let
be a compact abelian group with probability Haar measure
. Then the space of multiplicative characters is an orthonormal basis for the complex Hilbert space
.
The full proof of this theorem will have to wait until later notes, once we have developed the spectral theorem, though see Exercise 56 below. However, we can work out some important special cases here.
- When
is a torus
, the multiplicative characters
separate points (given any two
, there exists a character which takes different values at
and at
). The space of finite linear combinations of multiplicative characters (i.e. the space of trigonometric polynomials) is then an algebra closed under conjugation that separates points and contains the unit
, and thus by the Stone-Weierstrass theorem, is dense in
in the uniform (and hence in
) topology, and is thus dense in
(in the
topology) also.
- The same argument works when
is a cyclic group
, using the multiplicative characters
for
. As every finite abelian group is isomorphic to the product of cyclic groups, we also obtain the claim for finite abelian groups.
- Alternatively, when
is finite, one can argue by viewing the linear operators
as
unitary matrices (in fact, they are permutation matrices) for each
. The spectral theorem for unitary matrices allows each of these matrices to be diagonalised; as
is abelian, the matrices commute and so one can simultaneously diagonalise these matrices. It is not hard to see that each simultaneous eigenvector of these matrices is a multiple of a character, and so the characters span
, yielding the claim. (The same argument will in fact work for arbitrary compact abelian groups, once we obtain the spectral theorem for unitary operators.)
If , the inner product
of
with any multiplicative character
is just the Fourier coefficient
of
at the corresponding frequency. Applying the general theory of orthonormal bases (see Notes 5), we obtain the following consequences:
Corollary 22 (Plancherel theorem for compact abelian groups, again) Let
be a compact abelian group with probability Haar measure
.
- (Parseval identity) For any
, we have
.
- (Parseval identity, II) For any
, we have
.
- (Unitarity) Thus the Fourier transform is a unitary transformation from
to
.
- (Inversion formula) For any
, the series
converges unconditionally in
to
.
- (Inversion formula, II) For any sequence
in
, the series
converges unconditionally in
to a function
with
as its Fourier coefficients.
We can record here a textbook application of the Riesz-Thorin interpolation theorem from the previous notes. Observe that the Fourier transform map maps
to
with norm
, and also trivially maps
to
with norm
. Applying the interpolation theorem, we conclude the Hausdorff-Young inequality
for all and all
; in particular, the Fourier transform maps
to
, where
is the dual exponent of
, thus
. It is remarkably difficult (though not impossible) to establish the inequality (6) without the aid of the Riesz-Thorin theorem. (For instance, one could use the Marcinkiewicz interpolation theorem combined with the tensor power trick.) The constant
cannot be improved, as can be seen by testing (6) with the function
and using Exercise 19. By combining (6) with Hölder’s inequality, one concludes that
whenever and
. These are the optimal hypotheses on
for which (14) holds, though we will not establish this fact here.
Exercise 23 If
, show that the Fourier transform of
is given by the formula
Thus multiplication is converted via the Fourier transform to convolution; compare this with (5).
Exercise 24 (Hardy-Littlewood majorant property) Let
be an even integer. If
are such that
for all
(in particular,
is non-negative), show that
. (Hint: use Exercise 23 and the Plancherel identity.) The claim fails for all other values of
, a result of Fournier.
Exercise 25 In this exercise and the next two, we will work on the torus
with the probability Haar measure
. The Pontryagin dual
is identified with
in the usual manner, thus
for all
. For every integer
and
, define the partial Fourier series
to be the expression
- Show that
, where
is the Dirichlet kernel
.
- Show that
for some absolute constant
. Conclude that the operator norm of
on
(with the uniform norm) is at least
.
- Conclude that there exists a continuous function
such that the partial Fourier series
does not converge uniformly. (Hint: use the uniform boundedness principle.) This is despite the fact that
must converge to
in
norm, by the Plancherel theorem. (Another example of non-uniform convergence of
is given by the Gibbs phenomenon.)
Exercise 26 We continue the notational conventions of the preceding exercise. For every integer
and
, define the Césaro-summed partial Fourier series
to be the expression
- Show that
, where
is the Fejér kernel
.
- Show that
. (Hint: what is the Fourier coefficient of
at zero?)
- Show that
converges uniformly to
for every
. (Thus we see that Césaro averaging improves the convergence properties of Fourier series.)
Exercise 27 Carleson’s inequality asserts that for any
, one has the weak-type inequality
for some absolute constant
. Assuming this (deep) inequality, establish Carleson’s theorem that for any
, the partial Fourier series
converge for almost every
to
. (Conversely, a general principle of Stein, analogous to the uniform boundedness principle, allows one to deduce Carleson’s inequality from Carleson’s theorem. A later result of Hunt extends Carleson’s theorem to
for any
, but a famous example of Kolmogorov shows that almost everywhere convergence can fail for
functions; in fact the series may diverge pointwise everywhere.)
— 3. The Fourier transform on Euclidean spaces —
We now turn to the Fourier transform on the Euclidean space , where
is a fixed integer. From Exercise 10 we can identify the Pontryagin dual of
with itself, and then the Fourier transform
of a function
is given by the formula
Remark 28 One needs the Euclidean inner product structure on
in order to identify
with
. Without this structure, it is more natural to identify
with the dual space
of
. (In the language of physics, one should interpret frequency as a covector rather than a vector.) However, we will not need to consider such subtleties here. In other areas of mathematics than harmonic analysis, the normalisation of the Fourier transform (particularly with regard to the positioning of the sign
and the factor
) is sometimes slightly different from that presented here. For instance, in PDE, the factor of
is often omitted from the exponent in order to slightly simplify the behaviour of differential operators under the Fourier transform (at the cost of introducing factors of
in various identities, such as the Plancherel formula or inversion formula).
In Exercise 12 we saw that if was in
, then
was continuous and decayed to zero at infinity. One can improve both the regularity and decay on
by strengthening the hypotheses on
. We need two basic facts:
Exercise 29 (Decay transforms to regularity) Let
, and suppose that
both lie in
, where
is the
coordinate function. Show that
is continuously differentiable in the
variable, with
(Hint: The main difficulty is to justify differentiation under the integral sign. Use the fact that the function
has a derivative of magnitude
, and is hence Lipschitz by the fundamental theorem of calculus. Alternatively, one can show first that
is the indefinite integral of
and then use the fundamental theorem of calculus.)
Exercise 30 (Regularity transforms to decay) Let
, and suppose that
has a derivative
in
, for which one has the fundamental theorem of calculus
for almost every
. (This is equivalent to
being absolutely continuous in
for almost every
.) Show that
In particular, conclude that
goes to zero as
.
Remark 31 Exercise 30 shows that Fourier transforms diagonalise differentiation: (constant-coefficient) differential operators such as
, when viewed in frequency space, become nothing more than multiplication operators
. (Multiplication operators are the continuous analogue of diagonal matrices.) It is because of this fact that the Fourier transform is extremely useful in PDE, particularly in constant-coefficient linear PDE, or perturbations thereof.
It is now convenient to work with a class of functions which has an infinite amount of both regularity and decay.
Definition 32 (Schwartz class) A rapidly decreasing function is a measurable function
such that
is bounded for every non-negative integer
. A Schwartz function is a smooth function
such that all derivatives
are rapidly decreasing. The space of all Schwartz functions is denoted
.
Example 33 Any smooth, compactly supported function
is a Schwartz function. The gaussian functions
for
,
,
are also Schwartz functions.
Exercise 34 Show that the seminorms
for
, where we think of
as a
-dimensional vector (or, if one wishes, a rank
![]()
-dimensional tensor), give
the structure of a Fréchet space. In particular,
is a topological vector space.
Clearly, every Schwartz function is both smooth and rapidly decreasing. The following exercise explores the converse:
Exercise 35
- Give an example to show that not all smooth, rapidly decreasing functions are Schwartz.
- Show that if
is a smooth, rapidly decreasing function, and all derivatives of
are bounded, then
is Schwartz. (Hint: use Taylor’s theorem with remainder.)
One of the reasons why the Schwartz space is convenient to work with is that it is closed under a wide variety of operations. For instance, the derivative of a Schwartz function is again a Schwartz function, and that the product of a Schwartz function with a polynomial is again a Schwartz function. Here are some further such closure properties:
Exercise 36 Show that the product of two Schwartz functions is again a Schwartz function. Moreover, show that the product map
is continuous from
to
.
Exercise 37 Show that the convolution of two Schwartz functions is again a Schwartz function. Moreover, show that the convolution map
is continuous from
to
.
Exercise 38 Show that the Fourier transform of a Schwartz function is again a Schwartz function. Moreover, show that the Fourier transform map
is continuous from
to
.
The other important property of the Schwartz class is that it is dense in many other spaces:
Exercise 39 Show that
is dense in
for every
, and is also dense in
(with the uniform topology). (Hint: one can either use the Stone-Weierstrass theorem, or convolutions with approximations to the identity.)
Because of this density property, it becomes possible to establish various estimates and identities in spaces of rough functions (e.g. functions) by first establishing these estimates on Schwartz functions (where it is easy to justify operations such as differentiation under the integral sign) and then taking limits.
Having defined the Fourier transform , we now introduce the adjoint Fourier transform
by the formula
(note the sign change from (8)). We will shortly demonstrate that the adjoint Fourier transform is also the inverse Fourier transform: .
we see that obeys much the same propeties as
; for instance, it is also continuous from
to
. It is also the adjoint to
in the sense that
for all .
Now we show that inverts
. We begin with an easy preliminary result:
Next, we perform a computation:
Exercise 41 (Fourier transform of Gaussians) Let
. Show that the Fourier transform of the gaussian function
is
. (Hint: Reduce to the case
and
, then complete the square and use contour integration and the classical identity
.) Conclude that
.
Exercise 42 With
as in the previous exercise, show that
converges in the Schwartz space topology to
as
for all
. (Hint: First show convergence in the uniform topology, then use the identities
and
for
.)
From Exercises 40, 41 we see that
for all and
. Taking limits as
using Exercises 38, 42 we conclude that
for all , or in other words we have the Fourier inversion formula
for all . From (10) we also have
Taking inner products with another Schwartz function , we obtain Parseval’s identity
for all , and similarly for
. In particular, we obtain Plancherel’s identity
for all . We conclude that
Theorem 43 (Plancherel’s theorem for
) The Fourier transform operator
can be uniquely extended to a unitary transformation
.
Exercise 44 Show that the Fourier transform on
given by Plancherel’s theorem agrees with the Fourier transform on
given by (8) on the common domain
. Thus we may define
for
or
(or even
without any ambiguity (other than the usual identification of any two functions that agree almost everywhere).
Note that it is certainly possible for a function to lie in
but not in
(e.g. the function
). In such cases, the integrand in (8) is not absolutely integrable, and so this formula does not define the Fourier transform of
directly. Nevertheless, one can recover the Fourier transform via a limiting version of (8):
Exercise 45 Let
. Show that the partial Fourier integrals
converge in
to
as
.
Remark 46 It is a famous open question whether the partial Fourier integrals of an
function also converge pointwise almost everywhere for
. For
, this is essentially the celebrated theorem of Carleson mentioned in Exercise 27.
Exercise 47 (Heisenberg uncertainty principle) Let
. Define the position operator
and momentum operator
by the formulae
and the formal self-adjointness relationships
and then establish the inequality
(Hint: start with the obvious inequality
for real numbers
, and optimise in
and
.) If
, deduce the Heisenberg uncertainty principle
for any
. (Hint: one can use the translation and modulation symmetries (3), (4) of the Fourier transform to reduce to the case
.) Classify precisely the
for which equality occurs.
Remark 48 For
and
, define the gaussian wave packet
by the formula
These wave packets are normalised to have
norm one, and their Fourier transform is given by
Informally,
is localised to the region
in physical space, and to the region
in frequency space; observe that this is consistent with the uncertainty principle. These packets “almost diagonalise” the position and momentum operators
in the sense that (taking
for simplicity)
(where the errors terms are morally of the form
and
respectively). Of course, the non-commutativity of
and
as evidenced by the last equation in (12) shows that exact diagonalisation is impossible. Nevertheless it is useful, at an intuitive level at least, to view these wave-packets as a sort of (overdetermined) basis for
that approximately diagonalises
and
(as well as other formal combinations
of these operators, such as differential operators or pseudodifferential operators). Meanwhile, the Fourier transform morally maps the point
in phase space to
, as evidenced by (13) or (12); it is the model example of the more general class of Fourier integral operators, which morally move points in phase space around by canonical transformations. The study of these types of objects (which are of importance in linear PDE) is known as microlocal analysis, and is beyond the scope of this course.
The proof of the Hausdorff-Young inequality (6) carries over to the Euclidean space setting, and gives
for all and all
; in particular the Fourier transform is bounded from
to
. The constant of
on the right-hand side of (14) turns out to not be optimal in the Euclidean setting, in contrast to the compact setting; the sharp constant is in fact
, a result of Beckner. (The fact that this constant cannot be improved can be seen by using the gaussians from Exercise 41.)
Exercise 49 (Entropy uncertainty principle) For any
with
, show that
(Hint: differentiate (!) (14) in
at
, where one has equality in (14).) Using Beckner’s improvement to (6), improve the right-hand side to the optimal value of
.
Exercise 50 (Fourier transform under linear changes of variable) Let
be an invertible linear transformation. If
and
, show that the Fourier transform of
is given by the formula
where
is the adjoint operator to
. Verify that this transformation is consistent with (14), and indeed shows that the exponent
on the left-hand side cannot be replaced by any other exponent. (One can also establish this latter claim by dimensional analysis.)
Remark 51 As a corollary of Exercise 50, observe that if
is spherically symmetric (thus
for all rotation matrices
) then
is spherically symmetric also.
Exercise 52 (Fourier transform intertwines restriction and projection) Let
, and let
. We express
as
in the obvious manner.
- (Restriction becomes projection) If
is the restriction
of
to
, show that
for all
.
- (Projection becomes restriction) If
is the projection
of
to
, show that
for all
.
Exercise 53 (Fourier transform on large tori) Let
, and let
be the torus of length
with Lebesgue measure
(thus the total measure of this torus is
. We identify the Pontryagin dual of this torus with
in the usual manner, thus we have the Fourier coefficients
for all
and
.
- Show that for any
, the Fourier series
converges unconditionally in
.
- Use this to give an alternate proof of the Fourier inversion formula (11) in the case where
is smooth and compactly supported.
Exercise 54 (Poisson summation formula) Let
. Show that the function
defined by
has Fourier transform
for all
(note the two different Fourier transforms in play here). Conclude the Poisson summation formula
Exercise 55 Let
be a compactly supported, absolutely integrable function. Show that the function
is real-analytic. Conclude that it is not possible to find a non-trivial
such that
and
are both compactly supported.
— 4. The Fourier transform on general groups (optional) —
The field of abstract harmonic analysis is concerned, among other things, with extensions of the above theory to more general groups, for instance arbitrary LCA groups. One of the ways to proceed is via Gelfand theory, which for instance can be used to show that the Fourier transform is at least injective:
Exercise 56 (Fourier analysis via Gelfand theory) (Optional) In this exercise we use the Gelfand theory of commutative Banach *-algebras (see 245B Notes 12) to establish some basic facts of Fourier analysis in general groups. Let
be an LCA group. We view
as a commutative Banach *-algebra
(see Exercise 14).
- (a) If
is such that
, where
is the convolution of
copies of
, show that there exists a non-zero complex number
such that the map
is not invertible on
. (Hint: If
contains a unit, one can use Exercise 35 of 245B Notes 12; otherwise, adjoin a unit.)
- (b) If
and
are as in (a), show that there exists a character
(in the sense of Banach *-algebras, see Definition 16 of 245B Notes 12) such that
lies in the kernel of
for all
. Conclude in particular that
is non-zero.
- (c) If
is a character, show that there exists a multiplicative character
such that
for all
. (You will need Exercise 5 and Exercise 11.)
- (d) For any
and
, show that
, where
is the group identity and
is the conjugate of
. (Hint: the inner product
is positive semi-definite.)
- (e) Show that if
is not identically zero, then there exists
such that
. (Hint: first find
such that
and
, and conclude using (d) repeatedly that
. Then use (a), (b), (c).) Conclude that the Fourier transform is injective on
. (The image of
under the Fourier transform is then a Banach *-algebra known as the Wiener algebra, and is denoted
.)
- (f) Prove Theorem 21.
It is possible to use arguments similar to those in Exercise 56 to characterise positive measures on in terms of continuous functions on
, leading to Bochner’s theorem:
Theorem 57 (Bochner’s theorem) Let
be a continuous function on an LCA group
. Then the following are equivalent:
- (a)
for all
and
.
- (b) There exists a non-negative finite Radon measure
on
such that
.
Functions obeying either (a) or (b) are known as positive-definite functions. The space of such functions is denoted .
Exercise 58 Show that (b) implies (a) in Bochner’s theorem. (The converse implication is significantly harder, reprising much of the machinery in Exercise 56, but with
taking the place of
: see Rudin’s book for details.)
Using Bochner’s theorem, it is possible to show
Theorem 59 (Plancherel’s theorem for LCA groups) Let
be an LCA group with non-trivial Haar measure
. Then there exists a non-trivial Haar measure
on
such that the Fourier transform on
can be extended continuously to a unitary transformation from
to
. In particular we have the Plancherel identity
for all
, and the Parseval identity
for all
. Furthermore, the inversion formula
is valid for
in a dense subclass of
(in particular, it is valid for
).
Again, see Rudin’s book for details. A related result is that of Pontryagin duality: if is the Pontryagin dual of an LCA group
, then
is the Pontryagin dual of
. (Certainly, every element
defines a character
on
, thus embedding
into
via the Gelfand transform; the non-trivial fact is that this embedding is in fact surjective.) One can use Pontryagin duality to convert various properties of LCA groups into other properties on LCA groups. For instance, we have already seen that
is compact (resp. discrete) if
is discrete (resp. compact); with Pontryagin duality, the implications can now also be reversed. As another example, one can show that
is connected (resp. torsion-free) if and only if
is torsion-free (resp. connected). We will not prove these assertions here.
It is natural to ask what happens for non-abelian locally compact groups . One can still build non-trivial Haar measures (the proof sketched out in Exercise 7 extends without difficulty to the non-abelian setting), though one must now distinguish between left-invariant and right-invariant Haar measures. (The two notions are equivalent for some classes of groups, notably compact groups, but not in general. Groups for which the two notions of Haar measures coincide are called unimodular.) However, when
is non-abelian then there are not enough multiplicative characters
to have a satisfactory Fourier analysis. (Indeed, such characters must annihilate the commutator group
, and it is entirely possible for this commutator group to be all of
, e.g. if
is simple and non-abelian.) Instead, one must generalise the notion of a multiplicative character to that of a unitary representation
from
to the group of unitary transformations on a complex Hilbert space
; thus the Fourier coefficients
of a function will now be operators on thisl Hilbert space
, rather than complex numbers. When
is a compact group, it turns out to be possible to restrict attention to finite-dimensional representations (thus one can replace
by the matrix group
for some
). The analogue of the Pontryagin dual
is then the collection of (irreducible) finite-dimensional unitary representations of
, up to isomorphism. There is an analogue of the Plancherel theorem in this setting, closely related to the Peter-Weyl theorem in representation theory. We will not discuss these topics here, but refer the reader instead to any representation theory text.
The situation for non-compact non-abelian groups (e.g. ) is significantly more subtle, as one must now consider infinite-dimensional representations as well as finite-dimensional ones, and the inversion formula can become quite non-trivial (one has to decide what “weight” each representation should be assigned in that formula). At this point it seems unprofitable to work in the category of locally compact groups, and specialise to a more structured class of groups, e.g. algebraic groups. The representation theory of such groups is a massive subject and well beyond the scope of this course.
— 5. Relatives of the Fourier transform (optional) —
There are a number of other Fourier-like transforms used in mathematics, which we will briefly survey here. Firstly, there are some rather trivial modifications one can make to the definition of Fourier transform, for instance by replacing the complex exponential by trigonometric functions such as
and
, or moving around the various factors of
,
,
, etc. in the definition. In this spirit, we have the Laplace transform
of a measurable function with some reasonable growth at infinity, where
. Roughly speaking, the Laplace transform is “the Fourier transform without the
” (cf. Wick rotation), and so has the (mild) advantage of being definable in the realm of real-valued functions rather than complex-valued functions. It is particularly well suited for studying ODE on the half-line
(e.g. initial value problems for a finite-dimensional system). The Laplace transform and Fourier transform can be unified by allowing the
parameter in (15) to vary in the right-half plane
.
When the Fourier transform is applied to a spherically symmetric function on
, then the Fourier transform is also spherically symmetric, given by the formula
where
is the Fourier-Bessel transform (or Hankel transform)
where is the Bessel function of the first kind with index
. In practice, one can then analyse the Fourier-analytic behaviour of spherically symmetric functions in terms of one-dimensional Fourier-like integrals by using various asymptotic expansions of the Bessel function.
There is a relationship between the -dimensional Fourier transform and the one-dimensional Fourier transform, provided by the Radon transform, defined for
(say) by the formula
where ,
, and the integration is with respect to
-dimensional measure. Indeed one checks that the
-dimensional Fourier transform of
at
for some
and
is nothing more than the one-dimensional Fourier coefficient of the function
at
. The Radon transform is often used in scattering theory and related areas of analysis, geometry, and physics.
In analytic number theory, a multiplicative version of the Fourier-Laplace transform is often used, namely the Mellin transform
(Note that is a Haar measure for the multiplicative group
.) To see the relation with the Fourier-Laplace transform, write
, then the Mellin transform becomes
Many functions of importance in analytic number theory, such as the Gamma function or the zeta function, can be expressed neatly in terms of Mellin transforms.
In electrical engineering and signal processing, the z-transform is often used, transforming a sequence of complex numbers to a formal Laurent series
(some authors use instead of
here). If one makes the substitution
then this becomes a (formal) Fourier series expansion on the unit circle. If the sequence
is restricted to only be non-zero for non-negative
, and does not grow too quickly as
, then the
-transform becomes holomorphic on the unit disk, thus providing a link between Fourier analysis and complex analysis. For instance, the standard formula
for the Taylor coefficients of a holomorphic function at the origin can be viewed as a version of the Fourier inversion formula for the torus
. Just as the Fourier or Laplace transforms are useful for analysing differential equations in continuous settings, the
-transform is useful for analysing difference equations in discrete settings. The
-transform is of course also very similar to the method of generating functions in combinatorics and probability.
In probability theory one also considers the characteristic function of a real-valued random variable
; this is essentially the Fourier transform of the probability distribution of
. Just as the Fourier transform is useful for understanding convolutions
, the characteristic function is useful for understanding sums
of independent random variables.
We have briefly touched upon the role of Gelfand theory in the general theory of the Fourier transform. Indeed, one can view the Fourier transform as the special case of the Gelfand transform for Banach *-algebras, which we already discussed in 245B Notes 12.
The Fast Fourier Transform (FFT) is not, strictly speaking, a variant of the Fourier transform, but rather is an efficient algorithm for computing the Fourier transform
on a cyclic group , when
is large but composite. Note that a brute force computation of this transform for all
values of
would require about
addition and multiplication operations. The FFT algorithm, in contrast, takes only
operations, and is based on reducing the FFT for a large
to FFT for smaller
. For instance, suppose
is even, say
, then observe that
where are the functions
. Thus one can obtain the Fourier transform of the length
vector
from the Fourier transforms of the two length
vectors
after about
operations. Iterating this we see that we can indeed compute
in
operations, at least in the model case when
is a power of two; the general case has a similar but more complicated analysis.
In many situations (particularly in ergodic theory), it is desirable not to perform Fourier analysis on a group directly, but instead on another space
that
acts on. Suppose for instance that
is a compact abelian group, with probability Haar measure
, which acts in a measure-preserving (and measurable) fashion on a probability space
. Then one can decompose any
into Fourier components
, where
, where the series is unconditionally convergent in
. The reason for doing this is that each of the
behaves in a simple way with respect to the group action, indeed one has
for (almost) all
. This decomposition is closely related to the decomposition in representation theory of a given representation into irreducible components. Perhaps the most basic example of this type of operation is the decomposition of a function
into even and odd components
,
; here the underlying group is
, which acts on
by reflections,
.
The operation of converting a square matrix of numbers into eigenvalues
or singular values
can be viewed as a sort of non-commutative generalisation of the Fourier transform. (Note that the eigenvalues of a circulant matrix are essentially the Fourier coefficients of the first row of that matrix.) For instance, the identity
can be viewed as a variant of the Plancherel identity. More generally, there are close relationships between spectral theory and Fourier analysis (as one can already see from the connection to Gelfand theory). For instance, in
and
, one can view Fourier analysis as the spectral theory of the gradient operator
(note that the characters
are joint eigenfunctions of
). As the gradient operator is closely related to the Laplacian
, it is not surprising that Fourier analysis is also closely related to the spectral theory of the Laplacian, and in particular to various operators built using the Laplacian (e.g. resolvents, heat kernels, wave operators, Schrödinger operators, Littlewood-Paley projections, etc.). Indeed, the spectral theory of the Laplacian can serve as a partial substitute for the Fourier transform in situations in which there is not enough symmetry to exploit Fourier-analytic techniques (e.g. on a manifold with no translation symmetries).
Finally, there is an analogue of the Fourier duality relationship between an LCA group and its Pontryagin dual
in algebraic geometry, known as the Fourier-Mukai transform, which relates an abelian variety
to its dual
, and transforms coherent sheaves on the former to coherent sheaves on the latter. This transform obeys many of the algebraic identities that the Fourier transform does, although it does not seem to have much of the analytic structure.
183 comments
Comments feed for this article
26 April, 2020 at 1:52 pm
Anonymous
In Exercise 10, could you elaborate a bit what one needs to show in order to show “identifiable”? The solutions seem to be given, what else is one supposed to show?
27 April, 2020 at 9:54 am
Terence Tao
The exercise asks to show that the given identification is an isomorphism of LCA groups (so the identification need to be group isomorphisms and also topological homeomorphisms).
26 April, 2020 at 2:04 pm
Anonymous
Something is missing in Line 3 of Exercise 9.
[Corrected, thanks – T.]
23 January, 2021 at 12:56 pm
246B, Notes 2: Some connections with the Fourier transform | What's new
[…] the space of tempered distributions, but we will not pursue this direction here; see for instance these lecture notes of mine for a […]
18 February, 2021 at 9:10 am
N is a number
Is there a way to see (or understand the reason behind coining) these terms “frequency space”, “physical space” ?
[See https://en.wikipedia.org/wiki/Frequency_domain and https://en.wikipedia.org/wiki/Time-domain in signal processing (in image processing one uses spatial domains instead of time domains). -T]
18 June, 2021 at 1:34 pm
Anonymous
Let
be the characteristic function of the interval
. Suppose
. This is an example given in Stein-Shakarchi’s Fourier Analysis that one has a Fourier series that converges for every
but does not converge absolutely for any
. The absolute convergence boils down to the series
where
. How can one show that
uniformly for many
so that the series above is divergent?
18 June, 2021 at 4:42 pm
Anonymous
19 June, 2021 at 11:01 am
Terence Tao
When
is a rational multiple of
this follows from the periodicity of
. For irrational
one can either use the equidistribution theorem, or lower bound the sum by
and use either summation by parts or Fourier analysis to show that the second sum on the RHS converges while the first sum diverges.
25 June, 2021 at 4:15 pm
J.
The sawtooth function defined by
on the interval
with
and extended by periodicity to all of
is used to demonstrate the Gibbs phenomenon in Stein-Shakarchi. The Fourier series is given by
. Working with the Riemann sum, one can show that the partial sum
as the page linked at the end of Exercise 25 shows.
Alternatively, if one works with the Dirichlet kernel, then

) bound for


?
How can one give a uniform (in
to show
as
[One can use the upper bound
for
, and the lower bound
for small
and some positive constant
(for instance one has
for
). -T]
21 September, 2021 at 7:58 am
J
If one considers the Fourier series of a periodic function
on a closed interval, one can approximate the function by its first
modes. This looks similar to the low-rank matrix approximation by SVD: https://en.wikipedia.org/wiki/Singular_value_decomposition#Separable_models
Are there any connections between these two notions? Can the matrix approximation be viewed as Fourier series in some sense?
27 September, 2021 at 9:29 am
Terence Tao
Yes; in the case when a matrix is a circulant matrix, the eigenvalues are the Fourier coefficients of the first row, and low rank approximations of the matrix correspond to partial Fourier series of that row.
15 January, 2022 at 10:12 pm
Anonymous
Dear Prof. Tao:
You mentioned in the beginning of the notes that “Characters behave in a very simple manner with respect to translation (indeed, they are eigenfunctions of the translation action)”. How to see characters as the eigenfunctions of the translation?
27 November, 2022 at 10:11 am
J
Can the condition of being $\latex L^1$ be relaxed to improperly integrable (e.g., the limit
exists) in the Riemann-Lebesgue lemma?
28 November, 2022 at 9:38 am
Anonymous
Don’t see why not. If the improper integral exists, then it also belongs to
29 November, 2022 at 1:45 pm
Anonymous
The existence of
is NOT the same as that of
.