Pierson Guthrey
pguthrey@iastate.edu
Norms k·k
0.1
Norms k·k
0.1
A norm on a linear space X is a function k·k : X → R with the properties:
• Positive Definite:
– kxk ≥ 0 ∀ x ∈ X (nonnegative)
– kxk = 0 ⇐⇒ x = 0 (strictly positive)
• kλxk = |λ| kxk ∀ x ∈ X, λ ∈ C(homogeneous)
• kx + yk ≤ kxk + kyk ∀ x, y ∈ X (triangle inequality)
Continuity Classes C m (Ω)
1
• C m (Ω) = {f : Dα is continuous on Ω ∀ |α| ≤ m}
• C ∞ (Ω) = {f : Dα is continuous on Ω ∀ |α|} (Smooth Functions)
• Analytic Functions: C ω (Ω) = {f : f ∈ C ∞ } and f equals the Taylor series expansion in a neighborhood
around every point in Ω
• Continuous and Bounded: CB(Ω) = {f : f is continuous and bounded on Ω}
Ry
• Absolutely Continuous: AC [a, b] = f (x) : f (y) − f (x) = x g(s)ds for some g ∈ L1 (a, b) such that
f ∈ AC [a, b] implies f 0 = g a.e. where f 0 is the pointwise derivative.
Facts.
• C m (Ω) ⊂ C ω (Ω) ⊂ C ∞ (Ω)
2
Vector Spaces
2.1
Norms
A norm on a linear space X is a function k·k : X → R with the properties:
• kxk ≥ 0 ∀ x ∈ X (nonnegative)
• kλxk = |λ| kxk ∀ x ∈ X, λ ∈ C(homogeneous)
• kx + yk ≤ kxk + kyk ∀ x, y ∈ X (triangle inequality)
• kxk = 0 ⇐⇒ x = 0 (stricly positive)
2.2
Vector Norms k·kp
kxkp =
X
p
|xi |
p1
Sum Norm, p = 1
kxksum = kxk1 =
1
X
|xi |
Pierson Guthrey
pguthrey@iastate.edu
Euclidean Norm, p = 2
kxkEuclidean = kxk2 =
qX
|x2i |
Maximum Norm, p = ∞
kxkmax = kxk∞ = max {|xi |}
N
Given a vector x in C ,
kxkp+a ≤ kxkp ∀ p ≥ 1 ∀ a ≥ 0
kxkp ≤ kxkr ≤ n( r − p ) kxkp ∀ p ≥ 1 ∀ a ≥ 0
√
Examples: kxk2 ≤ kxk1 and kxk1 ≤ nkxk2 .
1
1
Lp Spaces
3
The space of Lebesgue Integrable functions is defined for a given p:
n
o
Lp (Ω) = f : kf kLp (Ω) < ∞
Where the norm on the space is
Z
kf kLp (Ω) =
p1
|f | dx
p
Ω
• L∞ (Ω) is the space of essentially bounded functions. That is, f is bounded on a subset of Ω whose
complement has measure zero. The norm on L∞ (Ω) is the essential suprenum
kf k∞ = inf {M s.t. |f (x)| ≤ M a.e. ∈ Ω}
• L2 (Ω) is the space of square-integrable functions. This is a Hilbert Space.
• L1 (Ω) is the space of Lebesgue integrable functions
• L0 (Ω) is the space of measurable functions
• L1loc (Ω) is the space of locally integrable functions. That is, functions integrable on any compact subset
of an open set Ω ⊂ RN
L1loc (Ω) = {f : f ∈ L1 (K) ∀ K ⊂⊂ Ω, K compact}
Alternative definition
n
o
L1loc (Ω) = f : kf φkL1 (Ω) < ∞ ∀ φ ∈ C0∞ (Ω)
This definition generalizes to Lploc (Ω).
3.1
Useful Facts
• Embeddings:
kukLp (Ω) ≤ meas(Ω)( p − q ) kukLq (Ω)
1
1
Thus if Ω is bounded, then Lq (Ω) ⊂ Lp (Ω) ⊂ Lploc (Ω) for all q > p
L∞ (Ω) ⊂ L2 (Ω) ⊂ L1 (Ω) ⊂ L1loc (Ω)
2
Pierson Guthrey
pguthrey@iastate.edu
• Continuous functions of compact support are dense in Lp (Ω)
• One can discuss weighted Lp spaces with
Z
kf kLp (Ω,wdµ) =
4
p1
p
w(x) |f | dµ(x)
Ω
`p (N) Space
∞
This is the space of bi-infinite sequences {xn }−∞ such that
∞
P
p
|xn | < ∞ for 0 < p ≤ ∞. `p (N) is
n=−∞
complete with respect to the norm
∞
X
kxkp =
! p1
p
|xn |
,
p≥1
n=−∞
and as well for in the case of p = ∞
kxk∞ = sup |xn |
n
There is no norm for 0 < p < 1, but there is a metric
∞
X
d ({xn } , {yn }) =
p
|xn − yn |
n=−∞
• `∞ is the space of bounded sequences
• `2 is the space of square-summable sequences. This is a Hilbert Space.
• `1 is the space of summable sequences
5
Banach Spaces
6
Hilbert Spaces
A Hilbert Space is a vector space with a defined inner product hu, vi which has the following properties
• Conjugate symmetry: hu, vi = hv, ui
• Linear in the first argument: hau + bw, vi = a hu, vi + b hw, vi
• Antilinear in the second argument: hu, av + bwi = ā hu, vi + b̄ hw, vi
• Positive definite when u = v:
– hu, ui ≥ 0
– hu, ui = 0 if and only if u = 0
A Hilbert Space is complete with respect to its norm:
kxkH =
q
hx, xiH
3
6.1
Pierson Guthrey
pguthrey@iastate.edu
Convergence
6.1
Convergence
• xn ∈ H converges strongly to x ∈ H if kxn − xkH → 0.
• xn ∈ H converges weakly to x ∈ H if
hxn , yi → hx, yi ∀ y ∈ H
Thus kxk ≤ lim inf kxn k which is strict inequality when convergence is not strong.
n→∞
6.2
Important Hilbert Spaces
• `2 is the space of square-summable sequences with
hx, yi =
∞
X
xi yi
i=−∞
• L2 (Ω) is the space of square integrable fuctions with
Z
hu, vi =
uvdx
Ω
• W s,2 = Hs are the Sobolev Spaces
Z
Z
Z
hu, vi =
ugdx +
Du · Dgdx + ... +
Ds u · Ds gdx
Ω
• C
n×m
Ω
Ω
is a Hilbert space which is the space of all m × n matrices with the inner product
hA, Bi = tr(A∗ B) =
m X
n
X
āij bij
i=1 j=1
with the Hilbert-Schmidt norm (aka Frobenius norm)
v
uX
n
um X
2
kAk n×m = t
|aij |
C
i=1 j=1
• H∗ , the dual of H, is the Hilbert space of continuous linear functionals from H into the base field. For
every element u ∈ H, there exists a unique element φu ∈ H∗ defined by
φu (x) = hx, ui
The norm is
kφkH∗ =
6.3
sup
|φ(x)|
kxk =1,x∈H
Hilbert Bases
To check if a given sequence If {xn } is a basis for H, we check the following
1. {xn } is closed if the set of all finite linear combinations of them is dense in H
2. {xn } is complete if and only if there is no nonzero vector orthogonal to all xn , that is
hx, xn i = 0 ∀ n ⇐⇒ x = 0
3. If {xn } are orthonormal they are maximal orthonormal if {xn } is not contianed in any strictly larger
orthonormal set.
4
6.3
6.3.1
Pierson Guthrey
pguthrey@iastate.edu
Hilbert Bases
Orthonormal Basis
∞
Let {en }n=1 be orthonormal, the following are equivalent
1. {en } is maximal orthonormal
2. {en } is closed
3. {en } is complete
4. {en } is a basis of H
5. Bessel’s Inequality holds for all x ∈ H
6.
∞
P
hx, en i hen , yi = hx, yi ∀ x, y ∈ H
n=1
So we know a complete orthonormal set forms a basis, with hx, en i known as the nth Generalized Fourier
Coefficient of x with respect to {en }
6.3.2
Gram Schmidt
If M = L {e1 , e2 , ..., en } is linearly independent but not orthonormal, then create a new orthonormal basis
M = L {f1 , f2 , ..., fn } with
n
X
f0
fn = 0n
fn0 = en −
hen , fi i fi
kfn k
i=1
6.3.3
Reisz Fischer Critertion
For an infinite dimensional basis,
∞
∞
P
P
2
cn en converges in H if and only if
|cn | < ∞
n=1
This implies
∞
P
n=1
6.3.4
n=1
∞
cn en converges if and only if {cn }i=1 ∈ `2
Riemann Lebesgue Lemma
Version 1
∞
P
∞
2
|hx, en i| < ∞ implies lim hx, en i = 0
If given {en }n=1 is ON and x ∈ H then
n→∞
n=1
6.3.5
Bessel’s Inequality
For an infinite dimensional basis, SN =
P
hx, en i en = PMN x where MN = L {e1 , e2 , ..., en }, which implies
∞
X
2
2
|hx, en i| ≤ kxk
n=1
5
Pierson Guthrey
pguthrey@iastate.edu
7
Sobolev Spaces
Intuitively, a Sobolev space is a space of functions with sufficiently many derivatives for some application
domain, such as partial differential equations, and equipped with a norm that measures both the size and
regularity of a function. It is a vector space of functions equipped with a norm that is a combination of Lp
norms and the function itself as well as derivatives up to a given order. The derivatives are understood in
a suitable weak sense to make the space complete, thus a Banach space. Their importance comes from
the fact that solutions of partial differential equations are naturally found in Sobolev spaces, rather than in
spaces of continuous functions and with the derivatives understood in the classical sense.
7.1
Weak Derivatives
Given f ∈ Lp (Ω), Dα f is defined in the D0 sense for any α if ∃ g ∈ Lq (Ω) s.t. Dα f = g in the D0 sense then
we say g is the weak α−derivative of f in Lq (Ω). That is,
Z
Z
f (x)Dα φ(x)dx = (−1)|α| g(x)φ(x)dx
∀ φ ∈ C0∞ (Ω)
Ω
• Weak derivatives need not be uniformly convergent
• Weak derivatives are defined almost everywhere
• Weak and classical derivatives coincide when u ∈ C k (Ω) and |α| = k
Example: If f (x) = |x| on (−1, 1), then f 0 = [f 0 ] + ∆f0 δ = sign (x). So since f ; ∈ Lq (Ω) for 1 ≤ q ≤ ∞, f
has a weak first derivative in Lq (Ω)
Example: If f (x) = H(x) then f 0 = δ ∈
/ Lq (Ω) for any q. So f has does not have a weak first derivative in
q
L (Ω) for any q
7.2
Definition
The Sobolev space W k,p (Ω) of order k is defined to be the set of all functions u ∈ Lp (Ω) such that for every
multi-index α with |α| ≤ k, the weak partial derivative Dα u belongs to Lp (Ω), i.e. for an open set Ω and for
1≤p≤∞
W k,p (Ω) = {u ∈ Lp (Ω) : Dα u ∈ Lp (Ω) ∀ |α| ≤ k}
• The first (default) of norm on W k,p (Ω) is

! p1


P

p

kDα ukLp (Ω)
kukW k,p (Ω) =
|α|≤k



 max kDα ukL∞ (Ω)
|α|≤k
1≤p<∞
p=∞
• An equivalent norm is
kukW k,p (Ω) =


 kukp p
L (Ω)
α
p1
p
+ Dk uLp (Ω)

 max kD ukL∞ (Ω)
|α|≤k
• Another equivalent norm on W k,p (Ω) is
0
kukW k,p (Ω) =
X
|α|≤k
6
kDα ukLp (Ω)
1≤p<∞
p=∞
7.3
Pierson Guthrey
pguthrey@iastate.edu
Embeddings
• W k,p (Ω) is a Banach space
• Alternative definiton: We say f has a strong α−derivative in Lp (Ω) if ∃ fn , g ∈ C ∞ (Ω) s.t. fn → f in
Lp (Ω) and Dα fn → g in Lp (Ω)
It turns out that the space of f ∈ Lp (Ω) that have a strong α−derivative for |α| ≤ k is the same space
as W k,p (Ω). That is for |α| ≤ k f has a strong α−derivative if and only if it has a weak α−derivative
7.3
Embeddings
• W k,2 (Ω) is the Hilbert space Hk (Ω) with norm k·kW k,2 (Ω) that can be equivalently defined as
(
k
H (Ω) =
2
f ∈ L (Ω) :
∞
X
2
1 + n + n + ... + n
fˆ(n) < ∞
2
4
)
2k
n=−∞
with norm
2
kf kHk (Ω) =
∞
X
2
2
(1 + |n| )k fˆ(n)
n=−∞
and inner product
hf, giHk (Ω) =
k
X
i
D f, Di g L2 (Ω)
i=0
– H01 (Ω) is subspace of H1 (Ω) where functions vanish on the bondary ∂Ω , complete with respect
to its norm
Z 21
2
2
|f | + |∇f |
kf kH1 (Ω) =
Ω
– f ∈ H1 (T) if cn =
1
2π
R 2π
0
∞
P
f (x)e−inx dx satisfies
2
|ncn | < ∞
n=−∞
– H1 (T) ⊂ L2 (T)
7.3.1
Sobolev Embedding Theorem
– If k > l, 1 ≤ p, q ≤ ∞, (k − l)p < n, and
1
p
−
1
q
=
k−l
n
then
W k,p (RN ) ⊂ W l,q (RN )
When k = 1 and l = 0, let p∗ be the Sobolev conjugate of p:
1
p∗
=
1
p
−
1
n
∗
W 1,p (RN ) ⊂ Lp (RN )
– Existence of sufficiently many weak derivatives implies some continuity of the classical derivatives.
Hs (RN ) ⊂ C0k (RN ) ∀ s > k +
Notably,
H1 (R) ⊂ C0 (R)
7
N
2
7.4
Pierson Guthrey
pguthrey@iastate.edu
Facts
7.4
Facts
– For finite p, W k,p (Ω) is a separable space (contains a countable dense subset).
If E ⊂ W k,p (Ω), E 6= ∅, {un }n∈N ∈ W k,p (Ω), then ∃ n s.t. un ∈ E
– W 0,p (Ω) is simply Lp (Ω)
– W 1,1 (Ω) is the space of absolutely continuous functions AC(Ω)
– W 1,∞ (Ω) is the space of Lipschitz continous functions on Ω
∞
P
• Formally if f (x) =
n=−∞
cn ineinx
n=−∞
Bessel’s equality says kf 0 kL2 (RN ) =
8
∞
P
cn einx then the weak derivative of f is f 0 (x) =
∞
P
ncn 2 < ∞
n=−∞
Linear Operators
The space of Linear Operators LX, Y The norm on this space is the Operator norm
kT k =
kT xk
=
sup
x∈D(T ),kxk
x∈D(T ),x6=0 kxk
kT xk =
sup
=1
sup
kT xk
x∈D(T ),0<kxk ≤1
which has the following properties
• kT uk ≤ kT k kuk
• kT + Sk ≤ kT k + kSk
• kT Sk ≤ kT k kSk
• kT k = 0 if and only if it is the Zero map 0u = 0
8.1
Convergence
• The space of Bounded linear operators B(H) is a Banach space complete with respect to the operator
norm with the property that kT k < ∞
• If kTn − T k → 0, we say Tn → T in B(H) uniformly
• If Tn → T pointwise or strongly if Tn x → T x ∀ x ∈ H Uniform convergence implies strong convergence. Converse is false.
• The Identity Map I X → X is bounded on any normed space and has kIk = 1.
P∞
A series n=1 Tn converges uniformly or strongly if the corresponding sequence of partial sums does. We
∞
∞
P
P
know
Tn converges uniformly if
kTn k < ∞
n=0
n=0
8
8.2
Pierson Guthrey
pguthrey@iastate.edu
Compact Operators
Example T ∈ B, S =
∞
P
n=0
Tn
n!
converges uniformly since
∞
∞ n
n
X
X
T kT k
≤
<∞
n! n!
n=0
n=0
Limit is deifned to be eT
∞
P
Let E ∈ B(H), kEk < 1. Then S =
E n is convergent since
n=0
∞
∞
X
X
j
j
E ≤
kEk =
j=0
If SN =
n
P
j=0
1
<∞
1 − kEk
En,
n=0
SN (I − E) = (I + E + E 2 + ... + E N )(I − E) = I − E N +1
N +1
as N → ∞, E N +1 ≤ kEk
→ 0 so
S(I − E) = I =⇒ S = (I − E)−1
or
(I − E)−1 =
∞
X
En
n=0
∞
P
j
In particular, (I − E)−1 ≤
kEk =
j=0
8.2
1
1−kEk
This implies 1 ∈ ρ(E)
Compact Operators
The space of compact operators is the linear operators L(X, Y ) such that Y is precompact (the closure of
Y is compact).
• Compact operators are bounded and contiuous.
• Any operator with finite rank is a compact operator.
• Any compact operator is a limit of finite rank operators.
• If Tn ∈ K(H) and kTn − T kL(X,Y ) → 0, then T is compact
9
Pierson Guthrey
pguthrey@iastate.edu
9
Distribution Theory
9.1
Test Functions
For open Ω ⊂ Rn , C0∞ (Ω) = D(Ω) is the vector space of test functions
D(Ω) = {f ∈ C ∞ (Ω) : supp f ⊂⊂ R}
(f with compact support)
• C0∞ is not a metric space, but rather a Topological Vector Space (TVS)
9.1.1
Convergence in C0∞ (RN )
Let φn ∈ C0∞
We say φn → 0 in C0∞ if
1. ∃ K ⊂⊂ Ω s.t. supp φn ⊂ K ∀ n
2. kDα φn kC(Ω) → 0 as n → ∞ ∀ α
And so we say φn → φ in C0∞ if
1. kφn − φk∞ → 0 (uniformly) in C0∞
9.1.2
Properties of C0∞
C0∞ (Ω) is closed under the following operations:
Given φ1 (x) ∈ C0∞ (Ω), C0∞ is closed under the following operations:
• Scaling φ(x) = αφ1 (x)
• Translation φ(x) = φ1 (x − c)
• Dilation φ(x) = φ1 (αx) for any α 6= 0
• Differentiation φ(x) = Dα φ1 (x) for any multiindex α
• Sums of such terms
• Products of such terms
• If φ1 (x) ∈ C0∞ (Ω) then φ1 (|x|) ∈ C0∞ (Ω)
Example: The Common Mollifier
(
φ(x) =
9.2
1
e x2 −1
0
|x| < 1
|x| > 1
Distributions T ∈ D0 (Ω)
Distributions T are continuous linear functionals on C0∞ (Ω) that satisfy
1. T : C0∞ (Ω) → C
2. T (c1 φ1 + c2 φ2 ) = c1 T (φ1 ) + c2 T (φ2 ) ∀ c1 , c2 ∈ C and ∀ φ1 , φ2 ∈ C0∞ (Ω)
3. If φn → φ in C0∞ (Ω) then T (φn ) → T (φ)
That is, D0 (Ω) is the set of distributions over Ω , and it is the dual space of C0∞
10
Pierson Guthrey
pguthrey@iastate.edu
Distributions T ∈ D0 (Ω)
9.2
9.2.1
Convergence in D0 (Ω)
Tn → T in D0 (Ω) if Tn (φ) → T (φ) for all φ ∈ C0∞ (Ω)
hTn , φi → hT, φi ∀ φ =⇒ Tn → T
9.2.2
Regular Distributions
T is a regular distribution if T = Tf : φ → Tf (φ) for some f ∈ L1loc
• The set of Tf s.t. f ∈ L1loc (Ω) may be regarded as a subset of D0 (Ω) and is defined
Z
Tf (φ) =
f (x)φ(x)dx
Ω
• Tf1 = Tf2 if and only if f1 = f2 a.e.
• If fn → f in the L1loc sense, then Tfn → Tf since |Tfn (φ) − Tf (φ)| ≤ kfn − f kL1 (K) kφkL∞ (K) → 0
We abuse notation and say fn → f in the D0 sense
• Every distribution is either regular or singular: δ is singular
9.2.3
Properties of D0
• D0 is closed under linear combinations.
• Multiplication for distributions is not defined in a general way.
• For a(x) ∈ C ∞ , we define (aT )(φ) = T (aφ) since aφ ∈ C0∞ and linearity is preserved. Similiarly,
aTf = Tf (aφ).
Example: T = δ, (aδ)(φ) = δ(aφ) = a(0)φ(0) = a(0)δ(φ) so aδ = a(0)δ
• Distributional Derivatives are defined as hT 0 , φi = − hT, φ0 i.
For T = Tf for f ∈ C 1 (R) we define Tf0 (φ) = −Tf (φ0 )
• In RN , for a multiindex α, (Dα T )(φ) = (−1)α T (Dα φ). Rather,
hDα T, φi = (−1)α hT, Dα φi
• Any locally integrable function (f ∈ L1loc (Ω)) has a distributional derivative
• T 0 (φ) ∈ D0 since T 0 : φ → C and is linear and if φn → 0 then −T (φ0n ) → 0
• All distributions are infinitely differentiable in the sense of distribution, and you may suffle mixed partial
derivatives.
∂
∂
∂a
• Product Rule For a ∈ C ∞ , ( ∂x
(aT
))(φ)
=
a
T
(φ) + ∂x
T (φ), or
∂xi
i
i
h(aT )0 , φi = ha0 T, φi + haT 0 , φi
• Dα : D0 (Ω) → D0 (Ω). That is, if Tn → T in D0 (Ω) then Dα Tn → Dα T in D0 (Ω) for all α
Example: If fn → δ then fn0 → δ 0
Classically function convergence does not imply derivative convergence
11
Pierson Guthrey
pguthrey@iastate.edu
Distributions T ∈ D0 (Ω)
9.2
9.2.4
Properties of Distributions
• T (φ) = φ(x0 ) for some x0 ∈ Ω and φ ∈ C0∞ is a distribution but not a function
• The Dirac Delta Function is defined as such: pick x0 ∈ Ω, let T (φ) = φ(x0 ), denoted
δx0 : δx0 (φ) = φ(x0 )
δ : δ(φ) = φ(0)
1. δx0 (x − x0 ) = 0 if x 6= 0
R∞
2. −∞ δ(x)dx = 1
• The Dipole Distributionis defined
T (φ) = δ 0 = −φ0 (x0 ) ∈ D0
• H 0 (φ) = φ(0) = δ(φ) and H 00 (φ) = δ 0 (φ) = −δ(φ0 ) = −φ0 (0) (the dipole distributon)
• Convolutions in D0 is defined for f ∈ C0∞ : f ∗ T
hf ∗ T, φi = T, fˇφ ,
(f ∗ T )(x) = T (τx fˇ)
and for S ∗ T where T is a distribution of compact support:
S ∗ (T ∗ φ) = (S ∗ T ) ∗ φ
9.2.5
Aproximations to the Identity
f is said to be an approximate identity if f ∈ L1 (Rn ) such that
R
1. Rn fn (x)dx = 1 ∀ n
2. ∃ c s.t. kfn kL1 (Rn ) ≤ c
R
3. lim |x|> |fn (x)| dx = 0 ∀ > 0
n→∞
Then fn → δ in D0 sense
9.2.6
Principle Value Distributions
We see that
Z
∞
T (φ) =
−∞
looks like Tf for f (x) =
1
x
but f (x) =
1
x
∈
/
L1loc ,
Z
T (φ) = lim
→0
φ(x)
dx
x
|x|>
so instead we consider the Principle Value of the integral
Z ∞
φ(x)
φ(x)
1
dx = pv
dx = pv
x
x
−∞ x
Note that the principle value distribution is not the same as the improper integral.
12
9.3
Pierson Guthrey
pguthrey@iastate.edu
Tempered Distributions
9.2.7
Pseudofunctions
For asymptotic singularities,
we cannot use ∆f (c)
(
log(x) x > 0
, note that f ∈
/ L1loc
Example: f (x) =
0
x<0
Tf0 (φ)
Z
= − lim+
→0
∞
Z
∞
log(x)φ (x)dx = lim+ − [log(x)φ(x)] +
0
→0
∞
φ(x)
dx
x
Neither limit exists but their sum does due to cancellation. Using φ() = φ(0) + φ0 (θ) by MVT,
Z ∞
H(x)
φ(x)
0
dx = pf
Tf (φ) = lim log()φ(0) +
x
x
→0+
Where pf means ’pseudofunction’ or ’finite part’, and is itself a distribution
9.3
Tempered Distributions
Norm on the Schwartz Space
pα,β (φ) = kφkα,β = sup xα ∂ β φ(x)
x∈RN
9.3.1
Definition of the Schwartz Space S
The Schwartz Space S(RN ) consists of all functions for which all derivatives are bounded and all derivatives are bonded by multiplication by xα for any α. That is,
S = φ ∈ C ∞ (RN ) : pα,β (φ) < ∞ ∀ α, β ∈ Zn+
If φ ∈ S, then there exists a constant Cd,α such that
|∂ α φ(x)| ≤
Cd,α
d
(1 + |x|) 2
∀ x ∈ RN
Thus, and element of S is a smooth function such that the function and all of its derivatives decay faster than
the reciprocal of any polynomial (that is it decays faster than any polynomial grows) as |x| → ∞. Elements
of S are called Schwartz functions and known as rapidly decreasing functions.
Schwartz Functions Facts
• C0∞ (RN ) ⊂ S
• Gaussians multiplied by any finite degree polynomial are Schwartz functions
X
2
q(x)e−c|x−x0 | ∈ S ∀ c > 0, x0 ∈ RN , q(x) =
cα xα
|α|≤d
• S ⊂ L1 (RN )
• S is dense in Lp (RN ) for 1 ≤ p < ∞ since C0∞ is dense in RN
13
9.4
Pierson Guthrey
pguthrey@iastate.edu
Distributions of Compact Support
9.4
Distributions of Compact Support
E is the subspace of distributions with compact support.
Define E = C ∞ (RN ) with φn → φ ∈ E if kDα (φn − φ)kL∞ (K) → 0 for all K ⊂⊂ RN
That is, T ∈ E if and only if T ∈ D0 and supp T ⊂⊂ RN
So C0∞ (RN ) ⊂ S ⊂ E are toplogical inclusions and so E 0 ⊂ S 0 ⊂ D0
• Support of Distributions: x ∈ supp T if and only if ∀ > 0 ∃ φ ∈ C0∞ (B (x)) s.t. T (φ) 6= 0
• T = δ If x 6= 0 let =
|x|
2
then φ ∈ C0∞ (B (x)) then δ(φ) = φ(0) = 0 and so supp δ = {0}
• T̂ is defined for all T ∈ E 0 since E 0 ⊂ D0
14