Distributions: Some Motivations
1. The wave equation:
Consider the initial value problem
Solution due to d’Alembert:
1
1
u(x, t) = (f (x + t) + f (x − t)) +
2
2
Z
x+t
g(s) ds.
x−t
If f is of class C2 and g is of class C1, we can easily check that u satisfies the wave
equation and initial conditions above. But the expression for u makes perfect sense
if, for instance, f is piecewise continuous and g is integrable. We are tempted to call
u a “generalized” solution in that case. But what does this mean in more complicated
situations where we do not have an explicit formula?
2. Consider the ODE problem
u0 (t) = f² (t) =
½1
, if 1 < t < 1 + ²,
0, otherwise.
²
The solution is
u² (t) =
(
if t ≤ 1,
t−1
, if 1 ≤ t ≤ 1 + ²,
²
1,
if t ≥ 1 + ².
0,
For ² 0, the limit of u² is a step function which jumps from 0 to 1. But the limit of
f² does not exist in a traditional sense.
3. Laplace’s equation
Consider the ODE boundary value problem
−u00 (x) = f (x),
0 < x < 1,
u(0) = u(1) = 0.
The solution exists and is unique and given explicitly by u(x) = −
Z
0
x
(x − s)f (s) ds + x
Z
1
0
(1 − s)f (s) ds.
For any continuous f, the solution u is of class C2.
The Dirichlet problem for Laplace’s equation is the multidimensional analogue:
−∆u = f (x),
u(x) = 0,
x ∈ Ω,
x ∈ ∂Ω.
Now we have no formula for the solution. Moreover, if f is continuous, it only follows
that ∆u is continuous. Counterexamples show that it does not follow from this that
u is of class C2.
How might we go about proving existence of a solution? There are several methods,
the most important one is based on Dirichlet’s principle. Theorem from linear algebra: A symmetric strictly positive definite matrix has an inverse.
Proof: The solution of Au=f is the unique minimizer of 1
u · Au − u · f.
2
Formally, we have
Z
(−∆u)v dx =
Ω
Z
Ω
∇u · ∇v dx
for any function v which is zero on the boundary . So “in some sense,”
‐∆ is like a positive definite infinite dimensional matrix. How can this be made precise?
4. The Fourier transform:
The Fourier transform of a function on the real line is given by
1
ˆ
f (ξ) = √
2π
Z
∞
e−iξx f (x) dx.
−∞
As it stands, this is defined if f is absolutely integrable. Under suitable assumptions,
the inverse transform is given by
1
f (x) = √
2π
Z
∞
−∞
eiξx fˆ(ξ) dξ.
Fourier transforms are very useful for solving differential equations with constant coefficients. For instance, consider the equation
−u00 (x) + u(x) = f (x).
Of course, we can solve this using variation of constants, but keep in mind that
what we really want to do is generalize this to PDEs.
If we take the Fourier transform on both sides, we get
(ξ 2 + 1)û(ξ) = fˆ(ξ).
The solution is
fˆ(ξ)
.
û(ξ) = 2
ξ +1
This is great if, say f(x)=exp(‐x2). But what if, for instance, f(x)=sin(x)? Then the
Fourier transform as introduced above is undefined.
Test Functions and Distributions
The basic idea behind distributions is identifying a function f with the linear map
φ 7→
Z
f (x)φ(x) dx.
Ω
In the Hilbert space L2() , the Riesz representation theorem says that for every f∈ L2()
there is a continuous linear map like this and vice versa. If we narrow the space of admissible
φ’s, we can allow a broader class of f’s. For instance, the mapping
φ 7→ φ(0)
is not defined on L2(), but it is defined for all continuous functions φ. If we require φ
to be of class C1, we can also define the mapping ∂φ
φ 7→
(0).
∂xi
The more restrictions we put on φ, the broader the class of permissible f’s.
Distribution theory takes this idea to the extreme. We require φ to be very nice, and then
we can allow lots of f’s.
Definition:
A function φ defined on  is called a test function if φ∈ C∞(), and there is a compact subset K of  such that the support of φ lies in K. The set of all test functions is called
D().
Along with putting a lot of restrictions on test functions goes a very strong notion of
convergence.
Definition:
We say that φn converges to φ in D() if all derivatives of the φn converge uniformly to those of φ and, moreover, the supports of the φn lie in a common compact set K.
Example:
φy,² (x) =
½
²2
exp(− ²2 −|x−y|2 ),
0,
if |x − y| < ²,
otherwise.
This is a very important function, because it is used in all kinds of approximation
arguments. For instance:
Lemma:
Let K be a compact subset of , and let f∈ C() have support contained in K. For
²>0, let Z
1
C(²)
Z
C(²) =
f² (x) =
φy,² (x)f (y) dy,
K
φy,² (x) dy.
Rm
If ²<d(K,), then f²∈ D(), moreover f² f uniformly as ² 0.
Now let K be a compact subset of , let ²<d(K,), and let K1 be the ²/3 neighborhood
of K. In the previous lemma, replace f by
3
f (x) = 1 − min(1, d(x, K1 )).
²
Now consider the function as given by the lemma. This function is in D(), equals
f²/3
1 on K, and vanishes outside an ²‐neighborhood of K.
A refinement of this argument leads to partitions of unity.
Theorem:
Let Ui, i∈ N, be a sequence of bounded open subsets of  such that
1. The closure of each Ui is contained in .
2. Every compact subset of  intersects only a finite number of the Ui.
3. The union of the Ui is all of .
Then there exists a collection of test functions φi , called a partition of unity subordinate
to the covering Ui, with the properties
0 ≤ φi ≤ 1,
supp(φi ) ⊂ Ui ,
X
i∈N
φi = 1.
Definition:
A distribution or generalized function is a mapping φ (f,φ) from D() to R, which is
continuous in the following sense: If φn φ in D(), then (f,φn) (f,φ). The set of all
distributions is called D’().
Examples:
1. Any continuous function on 
Z can be identified with the map
φ 7→
f (x)φ(x) dx.
Ω
2. If 0∈, we can define
(δ, φ) = φ(0).
3. Same thing for derivatives: Multiindex notation:
φ 7→ Dα φ(0).
α = (α1 , α2 , ..., αm ),
∂ α1 +...+αm
D =
αm .
1
∂xα
...∂x
m
1
α
4. Let S be a two‐dimensional surface in R3, and let q:S R be integrable. We define
(qδS , φ) =
Z
q(x)φ(x) dSx .
S
5. The function 1/x has a nonintegrable singularity at 0. Nevertheless we can define the
principal value
(1/x, φ) =
Z
|x|≥²
φ(x)
dx +
x
Z
This definition is actually independent of ².
²
−²
φ(x) − φ(0)
dx.
x
Convergence of distributions:
We say that fn f in D’() if (fn,φ) (f,φ) for every φ∈ D().
Example:
Z
∞
1
(sin(nx), φ) =
sin(nx)φ(x) dx =
n
−∞
Z
∞
−∞
cos(nx)φ0 (x) dx → 0.
But:
1
1
(1
−
cos(2nx))
→
.
sin (nx) =
2
2
2
So the limit of the product is in general not the product of the limits!
Derivatives of Distributions
Definition: ∂f
∂φ
(
, φ) = −(f,
).
∂xi
∂xi
Examples:
1. Let H(x) =
½
1, if x > 0,
.
0, if x < 0
(H 0 , φ) = −(H, φ0 ) = −
Z
∞
0
φ0 (x) dx = φ(0) = (δ, φ).
2. Let
xλ+
=
½
xλ , if x > 0,
0,
if x < 0,
and assume ‐1<λ<0. Then
((xλ+ )0 , φ) = −(xλ+ , φ0 ) = −
= − lim
²→0
= lim
²→0
=
Z
0
Z
²
∞
∞
Z
∞
Z
∞
xλ φ0 (x) dx
0
xλ φ0 (x) dx
²
λxλ−1 (φ(x) − φ(²)) dx
λxλ−1 (φ(x) − φ(0)) dx.
λxλ−1
We say that the derivative of is a regularization of
xλ+
+ .
3. In three dimensions, we consider ∆(1/r). Here r is the radius in polar coordinates.
Note that 1/r is integrable in three dimensions, so it is well defined as a distribution.
We calculate
(∆(1/r), φ) = (1/r, ∆φ) = lim
²→0
Z
|x|≥²
∆φ
dx.
r
Now we integrate the last expression by parts to find
Z
r≥²
∆φ
dx =
r
Z
∆
r≥²
³1´
r
φ dx −
Z
r=²
∂φ 1
dS +
∂r r
Z
r=²
φ
∂ 1
( ) dS.
∂r r
The first term is zero, the second is of order ² for ² 0. The last term is equal to
−²−2
Z
r=²
φ dS → −4πφ(0).
Hence we have found ∆(1/r)=‐4πδ. This is an example of a fundamental solution.
Fundamental solutions play an important role in the theory of PDEs.
4. To prove that a sequence fn(x) converges to δ(x) in the sense of distributions, it suffices to show that the primitives of fn converge to the Heaviside function H(x).
The following conditions are sufficient for this:
a) For any ²>0, we have
lim
n→∞
Z
−a
fn (x) dx = 0,
−∞
lim
n→∞
uniformly for a∈[²,∞);
b)
lim
n→∞
c) |
Z
Z
∞
Z
∞
fn (x) dx = 0,
a
fn (x) dx = 1;
−∞
a
−∞
fn (x) dx|
is bounded by a constant independent of a∈ R and n∈ N.
Examples of sequences for which these criteria work are
²
,
2
2
π(x + ² )
³ x2 ´
1
ft (x) = √ exp −
,
4t
2 πt
f² (x) =
fn (x) =
sin(nx)
,
πx
² → 0;
t → 0+ ;
n → ∞.
Fourier Transform and Tempered Distributions
Definition of the Fourier transform:
fˆ(ξ) = F[f ](ξ) = (2π)−m/2
Z
e−iξ·x f (x) dx.
Rm
If both f and F[f] are in D(Rm), then f=0. For this reason, a different class of test
functions is introduced.
Definition:
We say φ belongs to the Schwartz class if |x|kDαφ is bounded for every nonnegative integer k and every multiindex α. The set of all Schwartz class functions
is denoted by S(Rm). A sequence φn converges in S if all the |x|kDαφn have uniform bounds and all the Dαφn converge uniformly.
Laurent Schwartz (French, 1915‐2002): developed theory of distributions
Hermann Schwarz (German, 1843‐1921): Cauchy‐Schwarz inequality, Schwarz lemma
Jacob Schwartz (American, 1930‐2009): coauthor of Dunford and Schwartz
Some elementary properties of the Fourier transform:
D α fˆ(ξ) = F[(−ix)α f ](ξ),
(iξ)α fˆ(ξ) = F[D α f ](ξ).
Notation:
xα = xα1 1 ...xαmm .
It is an immediate consequence of these identities that F is a continuous map from
S(Rm) to itself.
Inversion formula:
f (x) = (2π)−m/2
Z
eiξ·x fˆ(ξ) dξ.
Rm
Preservation of inner product:
(f, φ) =
Z
Rm
f (x)φ(x) dx = (fˆ, φ̂).
Definition:
A tempered distribution is a continuous linear map from S(Rm) to C. The set of all
tempered distributions is denoted by S’(Rm). A sequence fn in S’ converges to f if
(fn,φ) converges to (f,φ) for every φ∈ S.
We can now use the preservation of the inner product to define the Fourier transform
as a mapping from S’ to itself.
For f∈ S’, the Fourier transform F[f] is the linear functional given by
(F[f ], φ) = (f, F −1 [φ]).
Example: What is the Fourier transform of the constant function 1?
(F[1], φ) = (1, F −1 [φ]) =
Z
Rm
F −1 [φ](x) dx
= (2π)m/2 F[F −1 [φ]](0) = (2π)m/2 φ(0).
That is, F[1]=(2π)m/2δ.