22
C. MOUHOT
3. Fourier decomposition of functions
3.1. The Fourier transform.
Definition ´3.1 (Fourier Transform on L1 (Rd )). Given f 2 L1 (Rd ) define its Fourier transform
F(f )(⇠) := Rd e 2i⇡x·⇠ f (x) dx for all ⇠ 2 Rd . F is linear and bounded from L1 (Rd ) to (Cb (Rd ), k·k1 ).
Proof. The application is well-defined since |e 2i⇡x·⇠ f (x)| = |f (x)| is integrable. Supremum bound
on F(f ) follows from Hölder’s inequality and give kF(f )kL1  kf kL1 . Continuity of ⇠ 7! F(f )(⇠)
follows from the dominated convergence theorem.
⇤
Proposition 3.2. Basic properties on the Fourier transform:
(i) Scaling: given f 2 L1 (Rd ) and 2 R \ {0}, F(f ( ·))(⇠) = | | d F(f )(⇠/ ).
(ii) Duality convolution/multiplication: given f, g 2 L1 (Rd ), the convolution f ⇤ g 2 L1 (Rd ) and
F(f ⇤ g) = F(f )F(g).
(iii) Duality translation/phase change: given f 2 L1 (R) and h 2 Rd , define ⌧h f := f (· h) 2
L1 (Rd ) then F(⌧h f )(⇠) = e2i⇡h·⇠ F(f )(⇠).
(iv) Duality regularity/decay: (1) if f 2 L1 (Rd ) and xj f 2 L1 (Rd ) then F(f ) is di↵erentiable
wrt ⇠j and @⇠j F(f ) = 2i⇡F(xj f ); (2) if f 2 L1 (Rd ) is C 1 with @xj f 2 L1 (Rd ) then F(@xj f )(⇠) =
2i⇡⇠j F(f )(⇠).
Remark 3.3. Last point (iv) is also a duality between derivation and multiplication, and motivates
Fourier theory for reducing di↵erential equations to algebraic equations. We have used in point (ii)
the Young inequality: given p, q, r 2 [1, +1] s.t. 1/p + 1/q = 1/r + 1 and f 2 Lp (R), g 2 Lq (Rd ), the
function f ⇤ g 2 Lr (Rd ) with kf ⇤ gkLr  kf kLp kgkLq (prove it with Hölder’s inequality).
Proof. Points (i) and (iii) follow by change of variable, point (ii) follow by Fubini’s theorem. To
prove (iv)-(1), observe that ⇠j 7! e 2i⇡x·⇠ f (x) is di↵erentiable for a.e x 2 Rd , and the derivative has
modulus less than 2⇡|xj f | 2 L1 (Rd ), hence the mean-value and dominated convergence theorem show
that the derivative of ⇠j 7! F(f )(⇠) exists and is @⇠j F(f ) = 2i⇡F(xj f ). To prove (iv)-(2) one uses
integration by part, the only thing to be careful with is to take a limit [ M, M ]d
for the domain of integration with |f | ! 0 on the boundary.
M !1
! ( 1, +1)d
⇤
Proposition 3.4 (Oscillation and decay). Given f 2 L1 (Rd ), the Fourier transform F(f )(⇠)
⇠!1
! 0.
Remark 3.5. It is a particular case of application of the more general Riemann-Lebesgue lemma.
Proof. We have proved that Cc1 (Rd ) is dense in L1 (Rd ). Assume f 2 Cc1 (Rd ), then define
:=
@x21 + · · · + @x2d and since @xi f, @x2i f 2 L1 (Rd ) from the previous proposition (1
)f 2 L1 (Rd ) and
F((1
)f )(⇠) = (1 + 4⇡ 2 |⇠|2 )F(f )(⇠). Therefore since F((1
)f )(⇠) is bounded, the initial Fourier
transform |F(f )(⇠)|  C(1 + 4⇡ 2 |⇠|2 ) 1 and goes to zero as ⇠ ! 1. Now for f 2 L1 (Rd ) and any
" > 0, take g 2 Cc1 (Rd ) s.t. kf gkL1  "/2 and use sup|⇠| M |F(g)(⇠)|  "/2 for M large enough
from above. Deduce sup|⇠| M |F(f )(⇠)|  sup|⇠| M |F(g)(⇠)| + kf gkL1 (Rd )  ".
⇤
Remark 3.6. The map F : L1 (Rd ) ! C0 (Rd ) continuously where C0 (Rd ) denotes the continuous
functions going to zero at infinity (endowed with the supremum norm). Prove that is not surjective.
p
2
⇡2 2
Proposition 3.7 (“Fixed point when a = ⇡”). Given a > 0: F(e ax )(⇠) = ⇡a e a ⇠ in R.
2
Proof. Check that f (x) = e ax on R satisfies f 0 (x) = 2axf (x) and use to establish the di↵erential
equation g 0 (⇠) = 2⇡ 2 a 1 g(⇠) on g(⇠) := F(f )(⇠) and solve it.
⇤
Theorem
3.8 (Inversion in L1 (Rd )). Consider f 2 L1 (Rd ) s.t. F(f ) 2 L1 (Rd ) then g(y) :=
´
2i⇡y·⇠
e
F(f )(⇠) d⇠ is equal to f (y) almost everywhere, i.e. F F(f ) = f or F 2 (f ) = fˇ := f ( ·).
Rd
P
´
Proof. This relies on an important calculation:
define H(⇠) := e 2⇡(´ |⇠j |) and hk (y) := H(⇠/k)e2i⇡y·⇠ d⇠.
´
Prove by explicit calculation that H = ⇡ d and hk
0 and hk = 1 and moreover for any ⌘,
ANALYSIS OF FUNCTIONS
´
23
hk ! 1 as k ! 1: the sequence hk is an approximation of the unit. By Fubini’s theorem
´
f ⇤ hk (y) = H(⇠/k)e2i⇡y·⇠ F(f )(⇠) d⇠ and use here F(f ) 2 L1 (Rd ) to pass to the limit as k ! 1. ⇤
|y|⌘
Corollary 3.9. The map F : L1 (Rd ) ! C0 (Rd ) is injective.
Proof. By linearity consider f 2 L1 (Rd ) s.t. F(f ) = 0 2 L1 (Rd ) and apply the previous inversion
result to get f = F(0) = 0.
⇤
Theorem 3.10 (Plancherel). Given f 2 L1 (Rd ) \ L2 (Rd ) then the Fourier transform F(f ) 2 L2 (Rd )
and kF(f )kL2 (Rd ) = kf kL2 (Rd ) .
Proof. Define f˜(x) := f ( x). Since f 2 L1 \ L2 check that f˜ 2 L1 \ L2 by change of variable,
and g := f ⇤ f˜ 2 L1 \ L1 ; moveover it is uniformly continuous: for h 2 Rd \ {0}, k⌧h g gkL1 
k⌧h f f kL2 kf kL2 ! 0 as h ! 0 (continuity of the translation in L2 ).´ Hence g ⇤ hk ! g uniformly
as k ! 1, and thus (g ⇤ hk )(0) ! g(0) = kf k2L2 . But (g ⇤ hk )(0) = H(⇠/k)F(g)(⇠) d⇠ and from
property (ii) above F(g) = |F(f )|2 . ´Finally H(⇠/k)
0 and converges increasingly to 1 hence
(monotone convergence) (g ⇤ hk )(0) ! |F(f )|2 and F(f ) 2 L2 with the equality of L2 norms.
⇤
Corollary 3.11. There is unique a isometry from L2 (Rd ) to L2 (Rd ) denoted f 7! fˆ s.t. for any
f 2 L1 \ L2 , fˆ = F(f ), we call it Fourier-Plancherel transform.
Proof. The application F : L1 \ L2 ! L2 is linear isometric for the L2 norm, hence uniformly
continuous. Since Cc1 ⇢ L1 \ L2 and Cc1 dense in L2 , L1 \ L2 is also dense in L2 . Finally L2 is
complete normed space, hence there is a unique continuous extension to L2 , it is isometric by limit. ⇤
Remark 3.12. Be careful that the usual formula defining F in L1 does not always make sense in L2
ˇ
and results must be proved by density. Prove for instance by density that fˆˇ = fˆ.
Proposition 3.13 (Inversion in L2 (Rd )). The Fourier-Plancherel transform is a bijective isometry
ˆ
ˆ: L2 (Rd ) ! L2 (Rd ) and fˆ = fˇ.
Proof. First consider f 2 L1 \ L2 and the approximation of the unit (for instance) hk 2 L1 \ L2
constructed above. Then f ⇤ hk 2 L1 \ L2 and F(f ⇤ hk ) = F(f )F(hk ) = (L2 )(L2 ) 2 L1 . Hence by
ˆ
inversion in L1 : FF(f ⇤ hk ) = (f ⇤ hk )ˇ. But FF(f ⇤ hk ) = (f ⇤ hk )ˆ and passing to the limit gives
ˆ
by convergence in L2 (using the isometry property) fˆ = fˇ. Finally both ˆˆ and ˇ are isometric hence
2
continuous and the result extends to L by density.
⇤
Finally it is natural to search for another “fixed space” instead of L2 that has regularity and decay.
Proposition 3.14 (The Schwartz space). The Schwartz space or space of rapidly decreasing functions
S(Rd ) on Rd is the space of smooth functions s.t. all their derivatives decay faster than any polynomial
at infinity: f 2 S(Rd ) i↵ 8 ↵ 2 Nd , 2 N, |x| @ ↵ f (x) ! 0 as |x| ! +1. The Fourier transform F
maps S(Rd ) to itself bijectively.
Proof. The Fourier transform is well-defined on S since S ⇢ L1 , check F(S) ⇢ S thanks to the duality
regularity/decay and the bound on F : L1 ! L1 . The inversion then implies bijectivity.
⇤
3.2. Fourier series and orthonormal basis. The Fourier transform decomposes a function on
Rd into its “oscillatory modes”, which form a continuum. On a bounded domain the corresponding
decomposition involves “harmonics”, i.e. discrete sets of possible frequencies5.
Definition 3.15. Consider a Hilbert space6 H. A family (fi )i2I of elements of H (not necessarily
countable) is said orthonormal if hfi , fj i = ij for any i, j 2 I ( ij = 1 if i = j and zero otherwise).
5This relates to continuum spectrum vs. discrete spectrum.
6A pre-Hilbertian space would be enough here in fact.
24
C. MOUHOT
Proposition 3.16. There is a unique isometric linear application I : `2 (I) ! H s.t. for any 2 `2 (I)
P
with finite support, I( ) = i2I (i)fi 2 H. Its range is F := Span(fi , i 2 I) (closure of the vector
subspace spanned by the family) and its inverse on it is given by the application ˆ : F ! `2 (I) s.t.
fˆ(i) = hfi , f i for any i 2 I. Finally I( )ˆ = on 2 `2 (I) and I(fˆ) = PF (f ) on f 2 H where PF is
the orthogonal projection on F (hence kfˆk`2 (I)  kf kH ).
Remark 3.17. We only consider countable orthonormal families for the exam. We however mention
that when I is uncountable the space `2 (I) is the space of functions f : I ! C s.t. f is non-zero only
for a countable subset of I, with square-summability on this subset. Note also that PF is the Hilbertian
projection on a non-empty convex closed set that we have seen in the previous chapter.
Proof. (We restrict to the case where I is countable) Observe that Pythagoras’s theorem implies that
an orthogonal family is linearly independent and that I is isometric on the set D ⇢ `2 (I) of functions
f : I ! C with finite support, i.e. non-zero only on a finite subset of I, and that ˆ is isometric on
I(`2 (I)). Moreover D is dense in `2 (I) by considering the limits of partial sums. There is thus a unique
isometric extension I on `2 (I). The identity I( )ˆ = is satisfied on 2 D ⇢ `2 (I) and thus extends to
`2 (I) by density-continuity. Since I(D) ⇢ Span(fi , i 2 I) and I(`2 (I)) = I(D) by density-continuity,
we deduce that I(`2 (I)) ⇢ F and the whole F is attained by explicit construction of a pre-image.
Finally we have for any f 2 H the decomposition f = PF (f ) + PF ? (f ) (with PF ? = 1 PF ) and
since hf, fi i = hPF (f ), fi i for any i 2 I, fˆ = PF (f )ˆ for f 2 H and thus I(fˆ) = PF (f ).
⇤
Definition 3.18. Given H Hilbert space and an orthonormal family (fi )i2I , this family is called
Hilbert basis if F := Span(fi , i 2 I) = H. When so H is isometrically isomorphic to `2 (I).
Exercise 45. Compare the definition of a Hilbert basis with a standard (algebraic basis) of a vector
space. Prove with Zorn’s lemma that every Hilbert space has a Hilbert basis. Prove (without Zorn’s
lemma) that a Hilbert space is separable i↵ it has a countable Hilbert basis. Prove that if a Hilbert
space has infinite dimension any algebraic basis is uncountable.
Proposition 3.19 (Practical criterion for a Hilbert basis). Given H Hilbert space, an orthonormal
family (fi )i2I is a Hilbert basis i↵ one of the following holds: (i) Span(fi , i 2 I)? = {0}, (ii) the
family is a maximal orthonormal system, (iii) for any f 2 H, kf kH = kfˆk`2 (I) , (iv) for any f, g 2 H,
P
hf, gi =
fˆ(i)ĝ(i).
i2I
Proof. The point (i) follows from proving that the orthogonal A? := {g 2 H | 8 f 2 A, hf, gi = 0} of
?
a set A ⇢ H is equal to A (orthogonal of the closure of the set), which follows from the continuity
of the inner product. To prove points (ii) and (iii): if H strictly larger than F then F ? 6= {0} and
one can find g 2 F ? with norm 1 and add it to the family (fi ) which would not be maximal, and this
g also satisfies 1 = kgkH 6= kĝk`2 (I) = 0. Finally (iv) is equivalent to (iii) by relating the norms and
inner products.
⇤
Example 1. Fourier series. Consider the Hilbert space H = L2 ([0, 1]).
⌘
PN ⇣Pk=n ˆ
2i⇡kx
Theorem 3.20 (Fejér). Given f 2 C 0 ([0, 1]), the sequence N1+1 n=0
converges
k= n f (k)e
´1
uniformly to f , where fˆ(k) :=
f (x)e 2i⇡kx dx denotes the k-th Fourier mode of f .
0
Proof. Prove by direct calculation that
N
1 X
N + 1 n=0
with the Fejér kernel
KN (y) :=
k=n
X
fˆ(k)e
2i⇡kx
k= n
!
1
=
2⇡
ˆ
2⇡
f (x
y)KN (y) dy
0
N k=+n
1 X X 2i⇡kx
1 sin2 (⇡(N + 1)y)
e
=
N + 1 n=0
N +1
sin2 (⇡y)
k= n
ANALYSIS OF FUNCTIONS
25
(that is computed by summing the geometric series). This kernel is a continuous 1-periodic function
´1
on R that satisfies KN 0 and 0 KN = 1 (it is easier
to check this last property on the first formula
´
for KN ). Moreover for any 2 (0, 1/2) one has |z| KN (z) dz ! 0 as N ! +1. This shows that
KN is an approximation of the unit and implies that KN ⇤ f converges uniformly to f as N ! 1. ⇤
Corollary 3.21. (i) The countable family (fn := e2i⇡kx ), k 2 Z is a Hilbert basis with fˆ(k) =
´1
f (x)e 2i⇡kx dx. (ii) Polynomials are dense in C 0 ([0, 1]) (Weierstrass theorem in R).
0
Proof. (i) The previous theorem proves that trigonometrical polynomials are dense in C 0 ([0, 1]) and
since C 0 ([0, 1]) is dense in L2 ([0, 1]) this proves that Span(fk , k 2 Z) = L2 ([0, 1]). (ii) Each fn can be
approximated by polynomials uniformly on [0, 1] by Taylor expansion.
⇤
P
ˆ 2
Remark 3.22. Because of (i) one has (Plancherel) kf kL2 ([0,1]) =
k2Z |f (k)| and moreover f
can be approximated by a finite number of its Fourier modes. However be careful that in general
Pk=n
f = limn!+1 k= n fˆ(n)ein· only holds in the sense of an L2 convergence, not necessarily pointwise.
The next theorem addresses this question.
The next two theorems are non-examinable:
Theorem 3.23 (Dirichlet). Consider f : R ! C measurable 1-periodic and integrable on [0, 1].
At any point x0 2 R where (1) f admits left limit f (x0 ) and right limit f (x+
0 ), (2) the functions
+
|f (x0 + t) f (x+
)|/t
and
|f
(x
t)
f
(x
)|/t
are
integrable
at
0
,
then
the
symmetric
Fourier sum
0
0
0
+
Pn
2i⇡kx0 n!+1 f (x0 )+f (x0 )
ˆ
at x0 converges as k= n f (k)e
!
.
2
Pn
Proof. The proof is based on writing the symmetric sum Sn (f )(x) := k= n fˆ(k)e 2i⇡kx as a convoPn
lution (Dn ⇤ f )(x) with the Dirichlet kernel Dn (x) = k= n e2i⇡kx = sin(⇡(2n+1)x)
, and then using a
sin(⇡x)
variant of the Riemann-Lebesgue lemma:

ˆ 1/2
f (x+
f (x0 + t) + f (x0 t) f (x+
f (x0 )
0 ) + f (x0 )
0)
Sn (f )(x0 )
=
sin(⇡(2n + 1)y)
dt
2
sin(⇡t)
0
which goes to zero as n ! 1 since the function under brackets is integrable.
⇤
Theorem 3.24 (Poisson summation). Consider a continuous function
P f : R ! C s.t. there are
↵
C
>
0,
↵
>
1
with
|f
(x)|

C(1
+
|x|)
for
all
x
2
R,
and
with
k2Z |F(f )(k)| < +1. Then
P
P
f
(n)
=
F(f
)(k)
(equality
of
two
absolutely
convergent
series).
n2Z
k2Z
P
Proof. Consider the function '(t) := n2Z f (t + n) which is well-defined, 1-periodic, and C 1 using
the uniform convergence of the series and of the series of derivatives. Its Fourier coefficients are
'(k)
ˆ
= F(f )(k) (the switching of summation and integration to compute this is justified because
of the bound we have
Since the series of Fourier coefficients is absolutely converging
P on |f (x)|).2i⇡kt
the function g(t) := k2Z '(k)e
ˆ
is well-defined and continuous and 1-periodic on R. The two
functions ' and g have the same Fourier coefficients hence are equal as L2 functions thus (continuity)
everywhere, and the formula is given at t = 0.
⇤
n
P+1 1
P
2
+1
(
1)
Exercise 46. Use the Poisson summation formula to prove that (1) n=1 n2 = ⇡6 , (2) n=1 n4 =
P+1
P+1
2
1
7⇡ 4
⇡n2 s
= p1s n= 1 e ⇡n s .
n= 1 e
720 , (3)
Example 2. Hermite polynomials. Consider the following polynomials for k 0 integer: Hk (x) :=
p
2
2
dk
dk
x2 /2
x2
( 1)k ex /2 dx
and H̃k (x) := ( 1)k ex dx
. Prove that (1) H̃k (x) = 2k/2 Hk (x 2), (2) Hk
ke
ke
and H̃k are polynomials with degree k, (3) each of the families (Hk )k 0 and (H̃k )k 0 is orthogonal
2
in the Hilbert space H = L2 (R, dµ) with the measure µ = e x /2 . Find induction relations on these
polynomials.
Example 3. Haar polynomials. Another interesting example will be considered more in details
in the example sheet: the Haar basis which was the first example of wavelets: H1 (x) := [0,1] (t) and
26
C. MOUHOT
0 and 1  l  2k : H2k +l (x) := [ 2l 2 , 2l 1 ] (x)
2l
[ 22lk+21 , 2k+1
] (x). The family (Hm )m 1 is a
2k+2 2k+1
2
p
Hilbert basis of L ([0, 1]) and more generally a Schauder basis of L ([0, 1]) for 1  p < +1 (see the
example sheet for the later).
for k
3.3. Solving equations with Fourier theory. The Fourier theory was invented to reduce partial
di↵erential equations (PDEs) to algebraic equations or at least ordinary di↵erential equations (ODEs).
More precisely di↵erential operators with constant coefficients are transformed into multiplication operators in the Fourier world. ThisP
motivates the following more general definition: given a di↵erential
↵
operator P of order m: P f :=
|↵|m a↵ (x)@x f with a↵ (x) smooth functions, the symbol is deP
fined as (x, ⇠) := |↵|m a↵ (x)i↵ ⇠ ↵ where ⇠ ↵ := ⇠1↵1 . . . ⇠`↵` and the principal symbol is defined as
P
↵
p (x, ⇠) :=
|↵|=m a↵ (x)⇠ . The symbol corresponds to the action of the operator on plane waves
(x, ⇠) = e ix·⇠ P (eix⇠ ). The names elliptic-parabolic-hyperbolic are related to the conics described
by the symbol. They relate however to much deeper mathematical and physical meaning. Here are a
few standard examples:
3.3.1. Elliptic equations. Consider the Laplace-Poisson equation: f = g on Rd for g 2 S(Rd ). A
solution f 2 S(Rd ) would need to satisfy F(f )(⇠) = (4⇡ 2 ) 1 (⇠12 + ⇠22 + · · · + ⇠d2 ) 1 F(g)(⇠). Hence one
can construct a solution in S(Rd ) with the help of the Fourier transform if F(g) vanishes like k⇠k2 at
⇠ = 0. This is a sort of generalised zero-mass condition. The modified equation (1
)f = g would not
have this problem: in Fourier this new equation is (4⇡ 2 )(1 + ⇠12 + ⇠22 + · · · + ⇠d2 )F(f )(⇠) = F(g)(⇠) and
the so-called symbol of the operator (1 + ⇠12 + ⇠22 + · · · + ⇠d2 ) has no cancellation. The Laplace operator
gave its name to a class of PDEs along the type of conic described by its symbol, the elliptic PDEs:
no non-zero cancellation of the symbol, infinite regularisation, evolution problem overdetermined and
unstable, Dirichlet problem. . .
3.3.2. Hyperbolic equations. Consider the wave equation: @t2 f = @x2 f for f = f (t, x) defined on R+ ⇥R,
and with f (0, ·) = F and @t f (0, ·) = G. Assuming that F, G 2 S(R), one can construct a solution
with the help of the Fourier transform in x by reducing this equation to a second-order ODE. The
symbol agrees with the principal symbol, it is (denoting ⌧ for the Fourier variable of t and ⇠ for
the Fourier variable of x) (t, x, ⌧, ⇠) = (⌧, ⇠) = ⌧ 2 ⇠ 2 , and the hyperbola conic described by the
symbol gives its name to the class of equations that share key properties with the wave equation: “as
many” cancellations of the symbol as possible while preserving causality, propagation of singularity,
finite-speed of propagation of information (fluid mechanics, kinetic theory, general relativity. . . ).
3.3.3. Parabolic equations. Consider the heat equation: @t f = @x2 f for f = f (t, x) defined on R+ ⇥ R,
and with f (0, ·) = F 2 S(R). One can construct a solution with the help of the Fourier transform in
x by reducing this equation to a first-order ODE. The symbol is (t, x, ⌧, ⇠) = (⌧, ⇠) = i⌧ + ⇠ 2 while
the principal symbol is P (⌧, ⇠) = ⇠ 2 . The parabola conic described by the symbol gives its name to
the class of equations that share key properties with the heat equation: “degenerate” cancellations of
the symbol, one-sided causality: evolution problem ill-posed backward in time, infinite regularisation
forward in time, infinite-speed of propagation of information.
However some important equations of physics are transversal to this classification, e.g. the Schrödinger
equation is parabolic-hyperbolic.