Download 1 Distributions or generalized functions.

MM Vercelli.
L4: Distributions.
Contents:
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Distributions or generalized functions.
Approximations to the identity.
Calculus with distributions.
The Schwartz class and tempered distributions.
Distributions or generalized functions.
We start we the following synthesis of the introduction of the classical book by GelfandShilov [GeSh64, page 1].
Physicist (and engineers ?) have long been using ’singular functions’. The simples
example is the delta function δ(x − x0 ).
These ’singular functions’ occur as a rule only in intermediate stages of a solution
of a problem and if some of them occurs in the final solution it is only in an integrand
where it is multiplied by some other sufficiently good function.
Thus given a ’singular function’ we know the result of its integration against a good
function ψ :
Z
δ(x − x0 ) ϕ(x)dx = ϕ(x0 )
R
This allows us to introduce this ’singular functions’ as linear functionals over the
space of good functions.
Let V be a vector space over R. Recall that the dual V∗ is by definition the set of
functions
λ:V →R
such that λ(av + bw) = aλ(v) + bλ(w) for all a, b ∈ R and v, w ∈ V.
It is also usual to write λ(v) = hλ, vi.
Let D be the vector space C0∞ of all C0∞ of functions ϕ : R → R with compact
support. That is to say, ϕ ∈ C0∞ if the set supp(ϕ) := {x : ϕ(x) 6= 0} is compact and ϕ
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1.1 Functions as distributions
MM Vercelli.
is infinitely derivable.
We say that the sequence (ϕn ), ϕn ∈ D converges to zero if all this functions vanish
outside a fixed interval I = [a, b], the same for all ϕn and converge to uniformly to zero
together with their derivatives of any order.
We say that a linear functional L ∈ D∗ is continuous if L(ϕn ) → 0 if ϕn converges
to zero.
The set of continuous linear functionals is denoted by D0 and its elements are called
distributions or generalized functions.
Example 1.1. Let δ : D → R be defined by
hδ, ϕi := ϕ(0) .
Then δ is continuous and so δ ∈ D0 .
In the literature we will find also the integral notation. Namely, if L ∈ D0 then:
Z
L(ϕ) =
Z
L(x) ϕ(x)dx = hL, ϕi
−∞
R
1.1
+∞
L(x) ϕ(x)dx =
Functions as distributions
Let f : R → R be a ordinary function. The integral
Z
hLf , ϕi =
f (x) ϕ(x)dx
R
gives rise a distribution 1 denoted by Lf . This is the reason why distributions are called
generalized functions.
Warning In many books the associated distribution Lf is also denoted by f .
Notice that the δ is not given by any function.
Here is another distribution P V which is not given by any function and which is also
different from the δ (i.e. P V 2 do not comes from a mesure):
1
R
Under suitable conditions on f , e.g. if I |f (x)|dx < K for any interval I .
2
PV comes form Principal Value. Actually, PV is the principal value of x1 .
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Mathematical Methods
1.2 Linear combinations of distributions
MM Vercelli.
Z
hP V, ϕi = lim
a→0
|ξ|>a
ϕ(ξ)
dξ
ξ
This distribution is related to the so called Hilbert transform 3 .
1.2
Linear combinations of distributions
As in an abstract vector space we do not define a multiplication between distributions.
But we do define the multiplication between C ∞ functions and distributions.
If f is a C ∞ function and L is a distribution we define its multiplication f.L ∈ D0
as follows
hf.L, ϕi = hL, f ϕi
Example 1.2.
f.δ = f (0)δ
Since we can add distributions we can construct linear combinations
f1 L1 + f2 L2 + · · · + fk Lk
of distributions with coefficients in the ring of C ∞ functions. This is an example of what
mathematician call a module over C ∞ (R).
Warning: Notice that for f (x) = x we have f.δ = 0 but neither f nor δ are zero.
→
−
−
This is different from what happens in a vector space, i.e. in a vector space r→
v = 0
−
imply either r = 0 or →
v = 0. This is one of the main difference between modules and
vector spaces.
Proposition 1.3. If L ∈ D0 satisfies x.L = 0 then there exists a constant c such that
L = cδ .
Proof. Fix ψ ∈ D a function such that ψ(0) = 1. We claim that L − cδ = 0 where
c = hL, ψi.
Indeed, any test ϕ can be written as
]
ϕ = ϕ(0)ψ + xϕ(x)
]∈D
where ϕ(x)
3
http://en.wikipedia.org/wiki/Hilbert_transform
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1.2 Linear combinations of distributions
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Then
]
hL − cδ, ϕi = hL − cδ, ϕ(0)ψ + xϕ(x)i
]
= hL − cδ, ϕ(0)ψi + hL − cδ, xϕ(x)i
= hL − cδ, ϕ(0)ψi
= h−cδ, ϕ(0)ψi + hL, ϕ(0)ψi
= −c ϕ(0) + ϕ(0)hL, ψi
=0
This shows L − cδ = 0 that is to say that L is a multiple of δ . 2
It is possible to prove a generalization of the above result for C ∞ -functions with
simple roots. If α is a root of C ∞ -function f (x) we say that it is a simple root if x−α
f (x)
is C ∞ near α.
Proposition 1.4. Let f ∈ C ∞ be a function with simple roots and let Z be the set of
zeros of f . If L ∈ D0 satisfies
f (x).L = 0
then
L=
X
cα δα
α∈Z
The above is false without the hypothesis on the simple zeroes. Indeed, we will see
that the solutions of x2 .L = 0 are more distributions than the multiples of the a single
delta, i.e. L = cδ + dδ 0 .
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Mathematical Methods
1.3 The Heaviside step function H.
1.3
MM Vercelli.
The Heaviside step function H.
The function H is used in the mathematics of control theory and signal processing to
represent a signal that switches on at a specified time and stays switched on indefinitely.
It was named after the English polymath Oliver Heaviside4 (1850-1925).
To be precise H ∈ D0 and acts on tests as follows:
Z ∞
hH, ϕi =
ϕ(x)dx
0
The word ’step’ comes from the fact that H is the distribution associated to the step
function H(x) defined as follows:
(
0 if x < 0 ,
H(x) =
1 if x ≥ 0 .
That is to say
Z
hH, ϕi =
H(x) ϕ(x)dx
R
We can translate the Heaviside distribution at x0 and usually people write H(x − x0 )
to indicate such a translation.
A rectangular pulse of magnitude a of duration t is given by a function whose value
is a in an interval of length t. Such rectangular pulse Ra,t (x) can be obtained by using
the Heaviside function as follows:
Ra,t (x) = aH(x) − aH(x − t)
It is also called rectangle function, the gate function, pulse function, or window
function.
It is useful to denote rec(x) = H(x + 12 ) − H(x + 12 − 1) the special case of rectangular
pulse of magnitude and duration 1 starting at x = − 21 .
1.4
Translations and scaling
If L is a distribution and a ∈ R we denote by La its translation by a defined as
hLa , ϕi = hL, τa ϕi
4
http://www-history.mcs.st-andrews.ac.uk/Biographies/Heaviside.html
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1.4 Translations and scaling
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where τa ϕ(x) = ϕ(x + a).
Warning: Recall that τa shift the picture to the a units to the left. Moreover, if Lf
is the distribution associated to f then
(Lf )a = Lτ−a f
The scaling of sr L of a distribution is defined as
hsr L, ϕi = hL,
sr ϕ
i
r
where sr ϕ(x) = ϕ( xr ).
The motivation was of course given by the following:
Lsr f = sr Lf
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2
Approximation to the identity
2.1
Weak convergence
Let Ln ∈ D0 be a sequence of distributions. We say that Ln converge weakly to L ∈ D0
if for all test ϕ ∈ D
lim hLn , ϕi = hL, ϕi
n→∞
Example 2.1. Let ϕn (x) := ϕ(x−n) be the sequence of functions obtained by translating
a test function ϕ. Then the distributions ϕn converge weakly to the zero distribution.
Example 2.2. Let fn (x) := H(x) − H(x − n) be a sequence of functions. Then the
associated distributions Lfn converge weakly to the Heaviside function H.
The properties of weak limits are similar to those of ordinary limits.
2.2
Approximation to the identity
A sequence (fn ) of functions is called a approximation of the identity or δ -convergent if
the following two conditions holds:
R
R
(i) R fn (x)dx = 1 and R |fn (x)|dx ≤ K ,
R
(ii) for any r > 0 limn→∞ R\[−r,r] |fn (x)|dx = 0
Here is the important theorem
Theorem 2.3. Under the above hypothesis
lim Lfn = δ
n→∞
The proof is given in the Appendix A.
Example 2.4. The following are δ -convergent sequences:
-1- fn (x) = Rn, 1 (x +
n
-2- fn (x) =
n
π(1+n2 x2 )
-3- fn (x) =
pn
-4- fn (x) =
sin(nx)
πx
π
2
e−nx
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)
2n
,
,
,
.
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2.2 Approximation to the identity
MM Vercelli.
Actually, all the above examples are constructed by the following simple idea. A
function K(x) is called a kernel if the following condition hold:
Z
Z
K(x)dx = 1 ,
|K(x)|dx ≤ M .
R
R
Then the sequence fn (x) := K(nx)n is a delta convergent sequence.
Example −1 − corresponds to the kernel K(x) = R1,1 (x + 21 ).
Example −2 − corresponds to the so called Cauchy-Poisson kernel K(x) =
1
.
π(1+x2 )
2
1
K(x) = √π e−x .
Example −3 − corresponds to the so called Gauss-Weierstrass kernel
√
Notice that in this case n was replaced by n.
Finally, example −4 − corresponds to the Dirichlet kernel K(x) = sin(x)
.
πx
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3
The calculus of distributions
Here is the definition of the derivative L0 of a distribution L.
dϕ
i
dx
Notice that L0 ∈ D0 and that the derivative is linear. Motivation for the above
definition is provided by the ’integration by parts’.
hL0 , ϕi := −hL,
Example 3.1. hδ 0 , ϕi = − ϕ0 (0) and in general
hδ (n) , ϕi = (−1)n
dn ϕ
(0)
dxn
Theorem 3.2.
H0 = δ
Proof. Let ϕ ∈ D be a test function. We have to show that
hH0 , ϕi = hδ, ϕi .
The RHS of the above equation is ϕ(0) whilst the LHS is
Z ∞
dϕ
dϕ
0
i=−
dx = ϕ(0)
hH , ϕi = −hH,
dx
dx
0
due to the fundamental theorem of calculus. 2
Example 3.3. Here is the derivative of the rectangular impulse Ra,t :
0
Ra,t
(x) = aδ(x) − aδ(x − t)
If f is right continuous and left continuous at a point s we write J(f, s) := f (s+ ) −
f (s− ) for the ’jump’ at s.
Here is an important property of the derivative of piecewise derivable function with
a discrete set of jumps.
Theorem 3.4. Let f (x) be a function derivable outside of a discrete set S = {s1 , s2 , · · · }
where f jumps. Then f 0 is the distribution
df X
+
J(f, s)δ(x − s)
dx s∈S
Proof. See Appendix B.
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3.1 Derivative of a product.
3.1
MM Vercelli.
Derivative of a product.
Then we have the following result.
Theorem 3.5.
(f.L)0 =
df
.L + f.L0
dx
Here is an interesting example:
Example 3.6.
(x.H)0 = H + x.δ = H
3.2
Graphical derivation
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The Schwartz class S and tempered distributions
4
We say that a function f (x) is rapidly decreasing if for all n:
lim |xn f (x)| = 0
x→∞
The Schwartz 5 class S consists of the C ∞ rapidly decreasing functions f all of whose
derivatives are also rapidly decreasing.
Notice D ⊂ S .
2
Example 4.1. f (x) = e−x is in S but not in D .
Notice that if L ∈ S ∗ then L ∈ D∗ . So if L ∈ S ∗ and L ∈ D0 the we will said that L
is a tempered distribution. It is common to write S 0 for the set of tempered distributions.
So a distribution is tempered if it can be also evaluated at functions of the Schwartz
2
class. For example the distribution Lf associated to the function f (x) = ex is not
2
tempered since it value at e−x is infinite.
Let SI(R) be the space of slowly increasing functions. Namely, the C ∞ functions g
such that gS ∈ S . Then S 0 is a SI(R)-module.
As we will see later tempered distributions are the natural model for the signals and
their spectral analysis via Fourier transforms.
5
Laurent Schwartz (1915-2002) won a Fields medal in 1950 for his work on the theory of distributions.http://www-history.mcs.st-andrews.ac.uk/Biographies/Schwartz.html
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Appendix A.
We have to show that for a given ϕ ∈ D :
lim hfn , ϕi = ϕ(0)
n→∞
which is equivalent, by property (i), to showing:
Z
lim
fn (x)(ϕ(x) − ϕ(0))dx = 0
n→∞
R
Let > 0 be arbitrarily small. Since ϕ has compact support and is continuous there
exists r such that | ϕ(x) − ϕ(0)| < 2K
if |x| < r and M such that | ϕ(x)| ≤ M2 . Then
Z
Z
|
R
fn (x)(ϕ(x) − ϕ(0))dx| = |
Z
≤
Z
fn (x)(ϕ(x) − ϕ(0))dx +
fn (x)(ϕ(x) − ϕ(0))dx|
[−r,r]
R\[−r,r]
Z
|fn (x)||(ϕ(x) − ϕ(0))|dx +
|fn (x)||(ϕ(x) − ϕ(0))|dx
[−r,r]
Z
Z
|fn (x)|dx
≤M
|fn (x)|dx +
2K [−r,r]
R\[−r,r]
Z
Z
≤M
|fn (x)|dx +
|fn (x)|dx
2K R
R\[−r,r]
Z
≤M
|fn (x)|dx +
2
R\[−r,r]
R\[−r,r]
where we have used property (i). Then by using property (ii) there exists n0 such
that
Z
|fn (x)|dx ≤
2M
R\[−r,r]
if n ≥ n0 . Then for n ≥ n0 it follows that
Z
| fn (x)(ϕ(x) − ϕ(0))dx| ≤ + = 2 2
R
This proves the theorem. 2
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REFERENCES
MM Vercelli.
Appendix B.
It is enough to show the theorem when S has just one point s. Actually, we can assume
w.l.o.g that s = 0. Then
Z
dϕ
0
hf , ϕi = − f (x) dx
dx
Z
ZR
dϕ
dϕ
f (x) dx
=−
f (x) dx −
dx
dx
[−r,r]
R\[−r,r]
since f is bounded near zero we have that
Z
dϕ
0
hf , ϕi = − lim
f (x) dx
r→0 R\[−r,r]
dx
Z −r
Z ∞
dϕ
dϕ
f (x) dx +
= − lim
f (x) dx
r→0
dx
dx
−∞
r
Z −r
Z ∞
dϕ
df
−r
ϕ(x)dx + (f. ϕ)|−∞ +
f (x) dx
= − lim −
r→0
dx
r
−∞ dx
Z 0
Z ∞
df
dϕ
=
ϕ(x)dx − f (0− ) ϕ(0) − lim
f (x) dx ,
r→0 r
dx
−∞ dx
Z 0
Z ∞
df
df
=
ϕ(x)dx − f (0− ) ϕ(0) +
ϕ(x)dx + f (0+ ) ϕ(0) ,
dx
dx
0
Z−∞
df
ϕ(x)dx + J(f, 0) ϕ(0) ,
=
R dx
df
=h
+ J(f, 0)δ, ϕi ,
dx
which shows that f 0 =
df
dx
+ J(f, 0)δ . 2
References
[GeSh64] Gelfand, I.M. and Shilov, G.E.: Generalized functions.
Academic Press, 1964.
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