Distributions, Schwartz Space and Fractional Sobolev Spaces

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Distributions, Schwartz Space and Fractional Sobolev Spaces

Fredrik Joachim Gjestland

Master of Science in Physics and Mathematics Supervisor: Mats Ehrnstrøm, MATH

Department of Mathematical Sciences Submission date: July 2013

Norwegian University of Science and Technology

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TMA4900

Industrial Mathematics, Master’s Thesis:

Distributions, Schwartz Space and Fractional Sobolev Spaces

Fredrik Joachim Gjestland

July 8, 2013

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Abstract

This thesis derives the theory of distributions, starting with test functions as a basis. Distributions and their derivatives will be analysed and exemplified. Schwartz functions are introduced, and the Fourier transform of Schwartz functions is analysed, creating the basis for Tempered distributions on which we also analyse the Fourier transform. Weak derivatives and Sobolev spaces are defined, and from the Fourier transform we define Sobolev spaces of non-integer order. The theory pre- sented is applied to an initial value problem with a derivative of order one in time and an arbitrary differentiation operator in space, and we take a look at conditions for well-posedness under different differnetiation operators and present some minor results. The Riesz representation theorem and the Lax–Milgram theorem are pre- sented in order to offer a different perspective on the results from the initial value problem.

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Sammendrag

Denne oppgaven utleder teori om distribusjoner ved å definere testfunksjoner og analysere hvordan kontinuerlige lineære funksjonaler påvirker disse. Schwartz- funksjoner og Fouriertransformasjonen blir så introdusert, og egenskapene til Fouri- ertransformasjonen anvendt på Schwartz-funksjoner blir gjennomgått. Temperære distribusjoner blir introdusert som elementer i dual-rommet til Schwartz-rommet, og vi ser hvordan Fouriertransformasjonen påvirker disse. Svake deriverte og Sobolev- rom blir definert, og gjennom Fouriertransformasjonen finner vi en alternativ måte å definere svake deriverte, som lar oss introdusere Sobolevrom av fraksjonell orden.

Teorien blir så anvendt på et initialverdiproblem med førsteordens tidsderivert og en vilkårlig derivasjonsoperator i rom. Det undersøkes under hvilke tilefeller problemer er velstilt og noen mindre resultater presenteres. Så presenteres Riesz representasjonsteorem og Lax–Milgram teoremet som i noen grad oppsumerer re- sultatene fra initialverdiproblemet i form av at de begge påviser unik løsning under gitte vilkår.

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Preface

This master’s thesis is written during the spring semester of 2013 to complete the degree Master of Science through the course TMA4900, "Matematikk, Masteropp- gave" at the Norwegian University of Science and Technology (NTNU), where I have studied "Industriell Matematikk". The course has a value of 30 units and is the culmination of five years of studying mathematics, physics and engineering.

When embarking on this thesis, I knew surprisingly little about the theory I were to write about, and as such it has been a tremendous learning process in the subject itself, mathematical rigour, discipline and structure. In trying to convey the theory to the reader I had to spend at least twice as much time and energy accepting and understanding the theory myself, making the work process challenging, but rewarding. The work started in February 2013, and has, apart from the month of May 2013 when it was put on hold in order to prepare for two exams, been an ongoing process until July eight 2013.

I would like to express my gratefulness to my supervisor, førsteamanuensis/Associate Professor Mats Harald Andreas Ehrnström for his inputs, ideas and helpful com- ments. I would particularly like to thank him for having the time to meet me on a daily basis during the final week before submission of the thesis.

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Distributions

Distributions, or "generalized functions", is a concept which, as implied, allows us to expand the notion of a function. Take for instance the impulse created by a baseball bat hitting a baseball, which, when we consider the force of impact to occur at a singular instance in time, can not be described by a function, as it is a multiple of Dirac’s delta,δ, which, as we shall see, is a distribution.

One of the main motivations for distributions is the way an integrable function acts on bounded functions, when their product is integrated over Euclidean space

— such integrable functions define regular distributions. However, when one wants to generalise this, allowing for derivatives of, say, non-integrable functions, one gets inspiration from the chain rule, which for smooth and compactly supported functions yields the equality

Z

(D^αφ)ψ dx= (−1)^|α|

Z

φ(D^αψ)dx, (1.1)

where support, as mentioned in the previous paragraph, is taken in the usual sense:

Definition LetΩbe a domain¹. Forf ∈C(Ω), suppf ={x∈Ω :f(x)6= 0}

is called the support of f.

The "exchange of derivatives" in (1.1) can be used to define derivatives of functions that are not differentiable, but only if we require that the test functions (ψin (1.1)) have a corresponding degree of regularity and decay at infinity. This is the reason for the definition of test functions in section 1.1, in which we will use a multi-index notation:

1Throughout this paper, a domain will mean an open set inRⁿ.

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12 CHAPTER 1. DISTRIBUTIONS Definition When writing α∈Nⁿ0, we use a multi-index notation whereα is the n-tuple(α1, α2, α3,· · ·, αn). The notationD^αis defined as

D^α= ∂^α¹ (∂x1)^α¹

∂^α² (∂x2)^α²

∂^α³

(∂x3)^α³ · · · ∂^αⁿ (∂xn)^αⁿ, and the size ofαis defined as

|α|=α₁+α₂+α₃+· · ·+α_n.

Remark As we shall see, it is possible to extend this notion from integrable functions to general objects (distributions) which act in essentially the same way on test functions as integrable functions do. The space of test functions will therefore always determine the corresponding space of distributions.

1.1 Test Functions

Definition LetΩbe a domain inRⁿ. Then the functions contained in D(Ω) ={φ∈C^∞(Ω) :suppφcompact inΩ}

are called test functions. A sequence {φj}^∞_j=1 ⊂D(Ω) is said to be convergent in D(Ω)to φ∈D(Ω)if there is a compact setK⊂Ωwith

suppφj ⊂K, j∈N and

sup

K

|D^αφj−D^αφ| →0 for allαinNⁿ0. The notation φj−→

D φmeans "convergent inD(Ω)".

1.2 Distributions

As stated in the introduction to the chapter, we can define distributions by how they interact with test functions:

Definition LetΩbe a domain inRⁿ and letD(Ω)be defined as above. D⁰(Ω)is the collection of all complex-valued linear continuous functionalsT overD(Ω):

T :D(Ω)→C, T :φ7→T(φ), φ∈D(Ω)

T(λ1φ1+λ2φ2) =λ1T(φ1) +λ2T(φ2), λ1, λ2∈C, φ1, φ2∈D(Ω) and

T(φ_j)→T(φ)wheneverφ_j −→

D φ.

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1.2. DISTRIBUTIONS 13 AnyT ∈D⁰(Ω) is called a distribution. By

Tj→T in D⁰(Ω), Tj, T ∈D⁰(Ω), j∈N we mean that

T_j(φ)→T(φ)in C, ifj→ ∞for anyφ∈D(Ω).

Remark Instead of writing T(φ), where φ ∈ D(Ω), it is conventional to write hT, φi. A simple example is Dirac’s delta:

δ(φ) =hδ, φi=φ(0).

Remark If f is locally integrable in a domainΩ, then the functional Tf defined by

T_f(φ) = Z

Ω

f(x)φ(x)dx

is said to be a regular distribution. A distribution that is not regular is said to be singular.

Example Dirac’s delta,δ(φ), is singular

Proof. We consider the point distribution atx= 0: δ(φ) =φ(0). Let us first check that δ really is a distribution. Is is clear that δ : D(Ω) → C, as for linearity, we can check

δ(λ₁φ₁+λ₂φ₂) =λ₁φ₁(0) +λ₂φ₂(0)

=λ1δ(φ1) +λ2δ(φ2).

Lastly,

δ(φ_j) =φ_j(0)

−−−→j→∞ φ(0)

=δ(φ) whenever

sup

K

|φj−φ| →0.

We need to show that there does not exist a locally integrable function, f, such

that Z

R

f φ=φ(0) (1.2)

for allφ∈D(Ω). Consider the test functionρ(x)=

(e⁻^1−|x|¹ ² if|x|<1 0 if|x| ≥1.

It is clear thatρ(x)has compact support, and through some calculation it can be

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14 CHAPTER 1. DISTRIBUTIONS seen that it is infinitely differentiable. Thus, ρ(x)∈D(Ω). If we assume that the locally integrable function in (1.2) exists, we would have

Z

R

f(x)ρ(nx)dx=ρ(0), ∀ n∈N. However,

1

e =|ρ(0)|

=| Z

R

f(x)ρ(nx)dx|

≤ Z

R

|f(x)||ρ(nx)|dx

= Z _n¹

−_n¹

|f(x)||ρ(nx)|dx

≤ Z _n¹

−_n¹

|f(x)|dx

−−−−→n→∞ 0,

where the second inequality follows from |ρ(x)| ≤1 for allx. So, by contradiction such an f can not exist.

Definition We defineη∈D(Ω) to be η(x)=

( cηe⁻

1

1−|x|2 if|x|<1 0 if|x| ≥1, where the positive constantcη is chosen such thatR

Rⁿη(x)dx= 1. For every >0, let

η(x) = 1 ⁿη(x

).

We callη the standard mollifier. The functionsη are infinitely differentiable, and Z

Rⁿ

ηdx= 1, supp(η) =B(0, ),

where B(0, ) denotes the n-dimentional sphere centred at the origin and with radius. Thus,η∈D(Rⁿ).

Definition Ifuis a locally integrable function inRⁿ, we define its mollification u(x) =

Z

Rⁿ

η(y)u(x−y)dy= Z

Rⁿ

η(x−y)u(y)dy.

Lemma 1.2.1. For any open setΩ∈Rⁿ,D(Ω)is dense inL^p(Ω)for1≤p <∞.

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1.3. DISTRIBUTIONAL DERIVATIVE 15 Proof. As any function inL^p(Ω) can be approximated by a step function,

g=

m

X

j=1

ajχQ_j, aj∈C,

where χQ_j are the characteristic functions of open cubes Qj with Qj ∈ Ω, it is enough to show thatχQ, the characteristic function of an arbitrary cube inΩcan be approximated by functions inD(Ω).

Forh <∞,(χQ)h(x)∈D(Ω), and k(χQ)h−χQk_Lp(Ω)=k

Z

Rⁿ

ηh(x−y)(χQ(y)−χQ(x))dyk_Lp(Ω)

=kcη

hⁿ Z

B(x,h)

η(x−y

h )(χQ(y)−χQ(x))dyk_Lp(Ω)

≤ kc_η hⁿ

Z

B(x,h)

|χQ(y)−χ_Q(x)|dykL^p(Ω)

≤ k 1

|B(x, h)|

Z

B(x,h)

c|χQ(y)−χQ(x)|dyk_Lp(Ω), (1.3) where |B(x, h)|denotes the size of the ball B. By Lebesgue’s differentiation theorem, (1.3) tends to the value of its integrand for everyxashgoes to zero, and we obtain

k(χQ)_h−χ_QkL^p(Ω)→0, where (χQ)h∈D(Ω), proving the lemma.

1.3 Distributional Derivative

While functions are limited in the sense that they do not necessarily have derivatives which are functions (i.e. the derivative of the Heavyside function is not a function), all distributions have derivatives which are in turn distributions. In- spired by integration by parts we make the following definition for the derivative of a distribution:

Definition Letα∈Nⁿ0 andT ∈D⁰(Ω). Then the derivativeD^αT is given by D^αT(φ) = (−1)^|α|T(D^αφ)

or, by inner-product notation:

hD^αT, φi= (−1)^|α|hT, D^αφi.

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16 CHAPTER 1. DISTRIBUTIONS Example of a distributional derivative

Letg(x) =|x|. We will look at ^dg_dx. hg⁰, φi=−hg, φ⁰i

=− Z

R

g(x)φ⁰(x)dx

=− Z

R

|x|φ⁰(x)dx

= Z 0

−∞

xφ⁰(x)dx− Z ∞

0

xφ⁰(x)dx

=xφ(x)

0

−∞− Z 0

−∞

φ(x)dx−xφ(x)

∞ 0 +

Z ∞ 0

φ(x)dx

=− Z 0

−∞

φ(x)dx+ Z ∞

0

φ(x)dx

=hsgn, φi.

Where

L¹_loc3sgn(x) =

(−1 ifx <0 1 ifx >0.

Thus, the derivative of the absolute value ofxis the signum function, as one would expect.

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Chapter 2

Schwartz Space

When working with the Fourier transformation we require Ω = Rⁿ. However, D(Rⁿ) is, in a sense, too small for for the Fourier transform, which also makes D⁰(Rⁿ) too large. For the purpose of the Fourier transformation, the Schwartz spaces, S(Rⁿ)(named in honour of Laurent Schwartz) andS⁰(Rⁿ), as introduced below, are optimally adapted in the sense that they are both closed under the Fourier transform.

Definition Forn∈N, S(Rⁿ) ={φ∈C^∞(Rⁿ) : sup

x∈Rⁿ

(1 +|x|²)^k² X

|α|≤l

|D^αφ(x)|<∞, for allk, l∈N0} (2.1) is called the Schwartz space of all rapidly decreasing infinitely differentiable functions, or Schwartz space for short.

A sequence{φj}^∞_j=1⊂S(Rⁿ)is said to converge inS(Rⁿ)toφ∈S(Rⁿ)if kφj−φkk,l→0 forj→ ∞and allk, l∈N0.

Where

kφkk,l= sup

x∈Rⁿ

(1 +|x|²)^k² X

|α|≤l

|D^αφ(x)|.

Proposition 2.0.1. The Schwartz space,S(Rⁿ)is a subspace ofL^p(Rⁿ)for every pinN.

Proof. For every functionφinS(Rⁿ), there exists a constantK such that

|φ(x)| ≤ K 1 +|x|² for every xin Rⁿ. Thus,

Z

Rⁿ

|φ(x)|^pdx≤K^p Z

Rⁿ

1

(1 +|x|²)^pdx <∞ 17

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18 CHAPTER 2. SCHWARTZ SPACE for every p <∞. Forp=∞, it follows from the definition, (2.1), that

sup

x∈Rⁿ

|φ(x)|<∞,

and we conclude thatS(Rⁿ)⊂L^p(Rⁿ)for every pin N.

Proposition 2.0.2. Ifφ∈S(Rⁿ), so are bothx^αφ andD^αφforα∈Nⁿ0.

Proof. This follows directly from the definition ofS(Rⁿ), (2.1). All functionsφ∈ S(Rⁿ) are rapidly decreasing (i.e. go to zero when multiplied with an arbitrary polynomial), and so do all of their derivatives.

2.1 The Fourier Transformation on S( R

ⁿ

)

The Fourier transform, named after Joseph Fourier is an important tool, and it has several applications in physics and engineering. As we shall see, it allows us to, amongst others, transform differentiation operators in our regular dimension (usually time in the applied sense) into polynomials in another dimension (usually frequency in the applied sense), a property of great use in solving differential equations.

Definition For φ ∈ S(Rⁿ), the Fourier transform, F, and the inverse Fourier, F⁻¹, are given by

(Fφ)(ξ) = (2π)⁻ⁿ² Z

Rⁿ

e^−ixξφ(x)dx,

(F⁻¹φ)(ξ) = (2π)⁻ⁿ² Z

Rⁿ

e^ixξφ(x)dx,

forξ∈Rⁿ.

Remark With the purpose of shortening notation, we will sometimes write F(φ(x))(ξ) = ˆφ(ξ).

Proposition 2.1.1. The Fourier transformation ofD^αφ(x)is given by

F(D^α_xφ(x))(ξ) =i^|α|ξ^αφ(ξ).ˆ (2.2)

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2.1. THE FOURIER TRANSFORMATION ONS(R^N) 19 Proof. The proof forn= 1is straightforward by calculation:

F(D_x^αφ(x))(ξ) = (2π)⁻ⁿ² Z

R

e^−ixξD_x^αφ(x)dx

=D_x^α−1φ(x)e^−ixξ

∂R+ (2π)⁻ⁿ² Z

R

iξe^−ixξD^α−1_x φ(x)dx

= (2π)⁻ⁿ² Z

R

iξe^−ixξD^α−1_x φ(x)dx

= (2π)⁻ⁿ² Z

R

(iξ)²e^−ixξD_x^α−2φ(x)dx ...

= (2π)⁻ⁿ² Z

R

(iξ)^αe^−ixξφ(x)dx

= (iξ)^α(2π)⁻ⁿ² Z

R

e^−ixξφ(x)dx

= (iξ)^αφ(ξ).ˆ

To extend ton >1we note that

F(D^α_xφ(x))(ξ) = (2π)⁻ⁿ² Z

Rⁿ

e^−ixξD^α_x(x)dx

= (2π)⁻ⁿ² Z

R

Z

R

· · · Z

R

e^−ixξD^α_x(x)dx

= (iξ₁)^α¹(iξ₂)^α²· · ·(iξ_n)^αⁿφ(ξ)ˆ

=i^|α|ξ^αφ(ξ)ˆ

Proposition 2.1.2. The Fourier transform ofx^αφ(x)is given by

F(x^αφ(x)) =i^|α|D_ξ^αφ(ξ).ˆ (2.3)

Proof. Again, the proof forn= 1straightforward by calculation, but we start with

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20 CHAPTER 2. SCHWARTZ SPACE the right hand side of (2.3).

(iD_ξ)^αφ(ξ) =ˆ i^αD^α_ξ(2π)⁻ⁿ² Z

R

e^−ixξφ(x)dx

=i^α(2π)⁻ⁿ² Z

R

D^α_ξe^−ixξφ(x)dx

=i^α(2π)⁻ⁿ² Z

R

−ixD_ξ^α−1e^−ixξφ(x)dx

=i^α(2π)⁻ⁿ² Z

R

(−ix)²D^α−2_ξ e^−ixξφ(x)dx ...

=i^α(2π)⁻ⁿ² Z

R

(−ix)^αe^−ixξφ(x)dx

=i^α(−i)^α(2π)⁻ⁿ² Z

R

e^−ixξx^kφ(x)dx

=F(x^kφ(x)).

We extend ton >1in the same manner as the previous preposition:

F(x^kφ(x)) = (iDξ₁)^α¹(iDξ₂)^α²· · ·(iDξ_n)^αⁿφ(ξ)ˆ

=i^|α|D^α_ξφ(ξ).ˆ

Theorem 2.1.3. If φ∈S(Rⁿ), so isFφandF⁻¹φ.

Proof. We need to show that ifφ∈S(Rⁿ), then sup

ξ∈Rⁿ

(1 +|ξ|²)^k² X

α≤l

|D^αφ(ξ)|ˆ <∞ for every k, l.

We look at each term in the series individually, so for everyk, l inNⁿ0: sup

ξ∈Rⁿ

ξ^k|D_ξ^lφ(ξ)|ˆ = sup

ξ∈Rⁿ

|ξ^k

i^lF(x^lφ(x))(ξ)| by (2.3)

= sup

ξ∈Rⁿ

| 1

i^l+kF(D_x^kx^αφ(x))| by (2.1.1)

= sup

ξ∈Rⁿ

| 1

i^l+k(2π)⁻ⁿ² Z

Rⁿ

e^−ixξD_x^k(x^lφ(x))dx|

≤ sup

ξ∈Rⁿ

| 1

i^l+k(2π)⁻ⁿ²| Z

Rⁿ

|e^−ixξD^l_x(x^lφ(x))dx|

≤ sup

ξ∈Rⁿ

(2π)⁻ⁿ² Z

|D^k_x(x^αφ(x))|dx.

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2.1. THE FOURIER TRANSFORMATION ONS(R^N) 21 Now, by proposition 2.0.2, sinceφ(x)∈S(Rⁿ), so isx^αφ(x), and thusD_x^k(x^αφ(x)), and sinceS(Rⁿ)⊂L¹(Rⁿ)by proposition 2.0.1, we conclude thatφˆ∈S(Rⁿ).

Proposition 2.1.4. Let φ(x) =e⁻

2|x|2

2 where is a positive, real constant. The Fourier transform of this Gaussian is given by

φ(ξ) =ˆ ⁻ⁿe⁻^|ξ|

2 22.

Proof. Let us first consider the casen= 1, by propositions 2.1.1 and 2.3 we have Dξφ(ξ) =ˆ F(x

ie⁻

2ξ2 2 )(ξ)

=1

iF(−⁻² d dxe⁻

2x2 2 )(ξ)

=− 1 i²iξφ(ξ)ˆ

=−1 ²ξφ(ξ).ˆ We can rewrite this to

d

dξ lnφ(ξ) =−1 ²ξ.

By integrating both sides with respect toξ and taking exponents, we obtain φ(ξ) =ˆ Ce⁻^ξ

2 22.

The constantC is obviously equal toφ(0), which we can find:ˆ φ(0) = (2π)ˆ ⁻¹²

Z

R

e⁻

2|x|2

2 e^−ix0dx

= (2π)⁻¹² Z

R

e⁻

2|x|2

2 dx

= (2π)⁻¹² r2π

² (2.4)

= 1

where (2.4) follows from taking a contour integral and calculating residues. Thus, we have for n= 1:

φ(ξ) =ˆ 1 e⁻^ξ

2 22. For n >1 we use the fact that |x|² =Pn

j=1(xj)² together with the properties of

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22 CHAPTER 2. SCHWARTZ SPACE the exponential function:

φ(ξ) =ˆ Z

Rⁿ

e⁻

2|x|2

2 e^−ixξdx

= Z

Rⁿ

e⁻

2 2

Pn

j=1(xj)²e⁻ⁱ^Pⁿ^k=1^ξ^j^k^x^k^jdx

= Z

Rⁿ n

Y

j=1

e⁻

2

2(x_j)²e^−iξ^j^x^j dx

=

n

Y

j=1

Z

R

e⁻

2

2(x_j)²e^−iξ^j^x^jdx

=

n

Y

j=1

⁻¹e⁻

ξ2 j 22

=⁻ⁿe⁻²¹²^Pⁿ^j=1ξ²_j

=⁻ⁿe^|ξ|

2 22.

Proposition 2.1.5. BothF F⁻¹ andF⁻¹F are identity operators onS(Rⁿ),

φ=F⁻¹Fφ=F F⁻¹φ. (2.5)

Proof. Assumingφ, ψ∈S(Rⁿ), Fubini’s theorem gives Z

Rⁿ

(Fφ)(ξ)e^ixξψ(ξ)dξ= (2π)⁻ⁿ² Z

Rⁿ

φ(y) Z

Rⁿ

e^{−i(y−x)ξ}ψ(ξ)dξdy

= Z

Rⁿ

φ(y)(Fψ)(y−x)dy

= Z

Rⁿ

φ(x+y)(Fψ)(y)dy. (2.6)

By letting

ψ(x) =e⁻

2|x|2

2 , >0, x∈Rⁿ we obtain

(Fψ)(y) =⁻ⁿ(Fe⁻^|x|

2 2 )(y

)

=⁻ⁿe⁻^|y|

2

22 (2.7)

by proposition 2.1.4. We then insert (2.7) into (2.6) and substitutey=z:

Z

Rⁿ

(Fψ)(ξ)e^ixξe⁻

2|ξ|2 2 dξ=

Z

Rⁿ

φ(x+z)e⁻^|z|

2 2 dz.

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2.2. CONVOLUTIONS OF CONTINUOUS FUNCTIONS 23 Finally we let →0and obtain

(F⁻¹Fφ)(x) = (2π)⁻ⁿ² Z

Rⁿ

(Fψ)(ξ)e^ixξdξ

= (2π)⁻ⁿ²φ(x) Z

Rⁿ

e⁻^|z|

2 2 dz

= (2π)⁻ⁿ²(2π)ⁿ²φ(x)

=φ(x).

Thus,F⁻¹F is an identity operator onS(Rⁿ), the same property can be shown for F F⁻¹ in a similar fashion.

Theorem 2.1.6. BothF and F⁻¹ map S(Rⁿ)one-to-one onto itself,

FS(Rⁿ) =S(Rⁿ), F⁻¹S(Rⁿ) =S(Rⁿ). (2.8) Proof. By applying (2.5) toφ=F⁻¹ψ,

φ=F⁻¹Fφ

=Fψ,

thus,FS(Rⁿ) =S(Rⁿ), and similarly one obtainsF⁻¹S(Rⁿ) =S(Rⁿ). In addition, ifFφ1=Fφ2, (2.5) yieldsφ1=φ2, henceF andF⁻¹ are one-to-one mappings of S(Rⁿ)onto itself.

2.2 Convolutions of Continuous Functions

Definition For two functions φ and ψ, both in C(Rⁿ) and at least one of them having compact support, the convolution (φ, ψ) 7→ φ∗ψ is defined through the continuous function

Lemma 2.2.1. If eitherφ∗ψ orψ∗φ exist,φ∗ψ=ψ∗φ.

(φ∗ψ)(x) = Z

Rⁿ

ψ(x−y)φ(y)dy.

Proof. The proof follows from a simple substitution:

(φ∗ψ)(x) = Z

Rⁿ

φ(x−y)ψ(y)dy, u=x−y, du=−dy

=− Z

Rⁿ

φ(u)ψ(x−u)du

= Z

Rⁿ

ψ(x−u)φ(u)du

= (ψ∗φ)(x).

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24 CHAPTER 2. SCHWARTZ SPACE Theorem 2.2.2. Wheneverf, g∈L¹(Rⁿ), and their convolution is defined,f[∗g= (2π)ⁿ²fˆg.ˆ

Proof. Becausef∗g∈L¹(R)and|e^−ixξ|= 1, Fubini’s theorem implies that the Fourier transformation is well-defined, and

F(f∗g)(ξ) = (2π)⁻ⁿ² Z

Rⁿ

Z

Rⁿ

f(x−y)g(y)dy

e^−ixξdx

= (2π)⁻ⁿ² Z

Rⁿ

Z

Rⁿ

f(x−y)e^−ixξdx

g(y)dy

= (2π)⁻ⁿ² Z

Rⁿ

Z

Rⁿ

f(z)e^−i(z+y)ξdz

g(y)dy

= (2π)ⁿ²

(2π)⁻ⁿ² Z

Rⁿ

f(z)e^−izξdz (2π)⁻ⁿ² Z

Rⁿ

g(y)e^−iyξdy

= (2π)ⁿ²Ff(ξ)Fg(ξ)

= (2π)ⁿ²fˆˆg.

Lemma 2.2.3. Withφ∈C^jandψ∈C^k, we can, in the derivative of a convolution as defined above, and with |α| ≤ j, |β| ≤ k interchange the derivative and the convolution in the following way:

D^α+β(φ∗ψ) = (D^αφ∗D^βψ).

Proof. From theorem 2.2.2, we have

F(φ∗ψ) = (2π)ⁿ²F(φ)F(ψ), and further,

F(D^α+β(φ∗ψ)) =i^|α+β|x^α+βF(φ∗ψ)(x)

=i^|α+β|x^α+β(2π)ⁿ²F(φ)F(ψ)

= (2π)ⁿ²i^|α|x^αF(φ)i^|β|x^βF(ψ)

= (2π)ⁿ²F(D^αφ)F(D^βψ)

=F(D^αφ∗D^βψ).

The proof is completed by the uniqueness of the Fourier-transform.

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2.2. CONVOLUTIONS OF CONTINUOUS FUNCTIONS 25 Theorem 2.2.4. Let j, k≥0. Ifφ∈C₀^j andψ∈L¹_loc,φ∗ψ∈C^j+k ifψ∈C^k. Proof. Firstly, let f, g ∈ C⁰(Rⁿ), g ∈ L¹_loc(Rⁿ = and either f or g have compact support. We defineh(x) =f∗g(x). We obtain

|h(x)−h(x+δ)|=| Z

Rⁿ

(f(x−y)−f(x−y+δ))g(y)dy| (2.9)

≤ Z

Rⁿ

(|f(x−y)−f(x−y+δ))||g(y)|dy (2.10)

≤ Z

Rⁿ

|g(y)|dy (2.11)

≤k, (2.12)

for some k, thus the convolution of two continuous functions is a continuous function.

Next, let f ∈ C¹(Rⁿ), g ∈ L¹_loc(Rⁿ)∩C⁰(Rⁿ) and either f or g have compact support. Now,

h(x)−h(x0) x−x₀ =

Z

Rⁿ

f(x−y)−f(x0−y) x−x₀ g(x)dx.

We need to show that ^{f(x−y)−f(x}_x−x ⁰^−y)

0 converges uniformly to f⁰(x0−y). By the mean value theorem, we have

f(x−y)−f(x0−y) = Z 1

0

df(x0−y+t(x−x0))

dt dt

= Z 1

0

f⁰(x0−y+t(x−x0))dt

(x−x0).

Thus,

f(x−y)−f(x0)−y) x−x0

−f⁰(x0−y) = Z 1

0

(f⁰(x0−y+t(x−x0))−f⁰(x0−y))dt.

(2.13) Because f⁰ is continuous by assumption and has compact support, it is uniformly continuous. Thus, for any >0 there exists aδ >0such that if|x−x0|< δ, then

|f⁰(x₀−y+t(x−x₀))−f⁰(x₀−y)|< . (2.14) Thus, (2.13) tends to zero uniformly for allx0andyandh∈C¹(Rⁿ). The assertion then follows from induction onj orkfixed.

Theorem 2.2.5. If 0 5 φ ∈C₀^∞, R

Rⁿφ(x)dx = 1, and u ∈C₀^j(Rⁿ), then u_φ = u∗φ∈C₀^∞(Rⁿ)by theorem 2.2.4. Further,

sup|D^αu−D^αuφ| →0 (2.15)

whenever suppφ→ {0} and|α| ≤j.

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26 CHAPTER 2. SCHWARTZ SPACE Proof. By using theorem 2.2.4 and lemma 2.2.3 and, it is sufficient to prove (2.15) forα= 0. Let|y|< δ in suppφ, and we obtain

|u(x)−uφ(x)|=| Z

(u(x)−u(x−y))φ(y)dy|

≤ sup

|y|<δ

|u(x)−u(x−y)|

−−−→δ→0 0.

Theorem 2.2.6. Ifφ(x, y)∈C^∞(X, Y)whereY is an open set inRⁿ, and if there is a compact setK⊂X such thatφ(x, y) = 0 whenx6∈K, then

y7→u(φ(·, y)) is aC^∞ function ofy ifu∈D⁰(X), and

D_y^αu(φ(·, y)) =u(D_y^α(·, y)).

Proof. We fixy∈Y and use Taylor’s formula to obtain φ(x, y+h) =φ(x) +X

hj

∂

∂yj

φ(x, y) +ψ(x, y, h),

where

sup

x

|D_x^αψ(x, y, h)|=O(|h|²), ash→0.

Hence,

u(φ(·, y+h)) =u(φ(·, y)) +X hju( ∂

∂yj

φ(·, y)) +O(|h|²).

Thus,y7→u(φ(·, y))is differentiable and

∂

∂yj

u(φ(·, y)) =u( ∂

∂yj

φ(·, y))

and the theorem is proved by iterating this argument.

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2.3. TEMPERED DISTRIBUTIONS 27

2.3 Tempered Distributions

In the same fashion as defining Distributions to be functionals acting on test functions, we define a smaller set of functionals acting on Schwartz functions:

Definition Let S(Rⁿ) be as previously defined. S⁰(Rⁿ) is the collection of all complex-valued linear continuous functionalsT overS(Rⁿ):

T :S(Rⁿ)→C, T :φ7→T(φ), φ∈S(Rⁿ).

T(λ1φ1+λ2φ2) =λ1T(φ1) +λ2T(φ2), λ1, λ2∈C, φ1, φ2∈S(Rⁿ).

We furnishS⁰(Rⁿ)with the the simple convergence topology

T_j →T inS⁰(Rⁿ), T_j ∈S⁰(Rⁿ), j∈N, T ∈S⁰(Rⁿ), means that

T_j(φ)→T(φ)inCifj→ ∞for anyφ∈S(Rⁿ).

AnyT ∈S⁰(Rⁿ)is called a tempered distribution.

Remark All test functions are elements in the Schwartz space: D(Rⁿ)⊂S(Rⁿ).

It follows from this that S⁰(Rⁿ)⊂D⁰(Rⁿ). Thus, as implied by the name, every tempered distribution is a distribution.

Definition LetT ∈S⁰(Rⁿ). Then the Fourier transformFT and its inverseF⁻¹T are given by

(FT)(φ) =T(Fφ), and(F⁻¹T)(φ) =T(F⁻¹φ), φ∈S(Rⁿ). (2.16) Theorem 2.3.1. The Fourier transform is a continuous linear one-to-one and onto mapping on S⁰(R)ⁿ, and for allT inS⁰(Rⁿ),

T =F⁻¹FT =F F⁻¹T (2.17)

Proof. The mapping T 7→ FT from S⁰(Rⁿ) is clearly linear. And if Tn → 0 in S⁰(Rⁿ), then

hFTn, φi=hTn,Fφi →0 asn→ ∞.

The same proof works forF⁻¹. For allφ∈S(Rⁿ).

hF F⁻¹T, φi=hT,F⁻¹Fφi

=hT,F F⁻¹φi

=hF⁻¹FT, φi

=hT, φi,

which proves (2.17). Thus,F andF⁻¹ being one-to-one ontoS⁰(Rⁿ)follows from F andF⁻¹ being one-to-one onS(Rⁿ).

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28 CHAPTER 2. SCHWARTZ SPACE

2.3.1 Example of Fourier Transformation of Distributions

Example 1

Let us first use our standard example: Dirac’s delta function. If we insert the

"function" into the classical definition of the Fourier transformation we obtain

F(δ(x))(ξ) = (2π)⁻ⁿ² Z

Rⁿ

e^−ixξδ(x)dx

= (2π)⁻ⁿ²e^−i0ξ

= (2π)⁻ⁿ².

If we on the other hand use (2.16) we obtain

(Fδ)(φ) =δ(F(φ))

=F(φ)(0)

= (2π)⁻ⁿ² Z

Rⁿ

φ(x)e^−ix0dx

= (2π)⁻ⁿ² Z

Rⁿ

φ(x)dx

=h(2π)⁻ⁿ², φi

Thus, we obtain

hFδ, φi=h(2π)⁻ⁿ², φi,

obtaining the same result as in our more classical approach.

Example 2

Consider the constant function f(x) = 1. In the sense of functions, the Fourier transform

F(1)(ξ) = (2π)⁻ⁿ² Z

Rⁿ

e^−ixξ1dx

does not make sense, ase^−ixξ∈/ L¹(Rⁿ). In the sense of distributions, however, we have

h1, φi= Z

Rⁿ

φ(x)dx,

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2.3. TEMPERED DISTRIBUTIONS 29 giving us

hF1, φi=h1,Fφi

= Z

Rⁿ

φ(ξ)dξˆ

= (2π)⁻ⁿ² Z

Rⁿ

Z

Rⁿ

e^−ixξφ(x)dxdξ

= (2π)⁻ⁿ² Z

Rⁿ

e^i0ξ Z

Rⁿ

e^−ixξφ(x)dxdξ

= (2π)ⁿ²F⁻¹(F(φ))(0)

= (2π)ⁿ²φ(0)

=h(2π)ⁿ²δ, φi.

Thus,

F(1)(ξ) = (2π)ⁿ²δ(ξ) (2.18) Example 3

In this example we will consider the distributione^iαx. F(e^iαx)(ξ) = (2π)⁻ⁿ²

Z

Rⁿ

e^−ixξe^iαxdx

= (2π)⁻ⁿ² Z

Rⁿ

e^ix(α−ξ)dx

=F(1)(ξ−α)

= (2π)ⁿ²δ(ξ−α), where we used (2.18).

Example 4

In this example we will use the result from example 3 to find the Fourier transform of the trigonometric functionssin(αx)andcos(αx):

F(sin(αx)) =F

e^iαx−e^−iαx 2i

= 1

2i F(e^iαx)− F(e^−iαx)

=(2π)ⁿ²

2i (δ(ξ−α)−δ(ξ+α)). And in similar fashion one finds

F(cos(αx)) = (2π)ⁿ²

2 (δ(ξ−α) +δ(ξ+α)).

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30 CHAPTER 2. SCHWARTZ SPACE Example 5

LetT ∈S⁰(Rⁿ), thenD^αT ∈S⁰(Rⁿ). Let us findF(D^αT)andD^αFT, analogous to (2.1.1) and (2.3), and properties we will need in chapter 4.

h[D^αT , φi=hD^αT,φiˆ

= (−1)^|α|hT, D^αφiˆ

= (−1)^|α|hT,F

(−i)^|α|ξ^αφ i

= (−1)^|α|hT ,ˆ (−i)^|α|ξ^αφi

= (−1)^|α|h(−i)^|α|ξ^αT , φiˆ

=hi^|α|ξ^αT , φi.ˆ

hD^αT , φiˆ = (−1)^|α|hT , Dˆ ^αφi

= (−1)^|α|hT,D[^αφi

= (−1)^|α|hT, i^|α|ξ^αφiˆ

= (−1)^|α|hi^|α|ξ^αT,φiˆ

= (−1)^|α|hi^|α|\ξ^αT , φi

=h(−i)^|α|ξd^αT , φi Thus,

D[^αT =i^|α|ξ^αT ,ˆ D^αTˆ = (−i)^|α|ξd^αT . (2.19)

2.3.2 Convolutions with Tempered Distributions

In order to define the convolution for tempered distributions we start with a fixed ψ ∈S(Rⁿ), we do this because products are not defined for all distributions, but with one factor inS(Rⁿ)this will not be a problem. A convolution withψis then an operation which preservesS⁰(Rⁿ), so to defineψ∗f forf ∈S⁰(Rⁿ)we find

Z

ψ∗φ1(x)ψ2(x)dx= Z Z

ψ(x−y)φ1(y)φ2(x)dydx.

Note that Z

ψ(x−y)φ2(x)dx= ˜ψ∗φ2(y), ψ(x) =˜ ψ(−x).

Thus,

Z

ψ∗φ1(x)φ2(x)dx= Z

φ1(y) ˜ψ∗φ2(y)dy.

Definition We defineψ∗f, where ψ∈S(Rⁿ)andf ∈S⁰(Rⁿ)by hψ∗f, φi=hf,ψ˜∗φi.

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2.3. TEMPERED DISTRIBUTIONS 31 Lemma 2.3.2. The Fourier transform of the convolution between a tempered distribution and a Schwartz function is given by the product of their Fourier tranforms:

F(ψ∗f) = (2π)ⁿ²ψˆf .ˆ Proof.

hF(ψ∗f), φi=hψ∗f,φiˆ

=hf,ψ˜∗φiˆ

=hf ,ˆF⁻¹( ˜ψ∗φ)iˆ

=hf ,ˆ(2π)ⁿ²(F⁻¹ψ)φi˜

=h(2π)ⁿ²f ,ˆψφiˆ

=h(2π)ⁿ²ψˆf , φi.ˆ

Example of convolution with a tempered distribution Iff =δandψ∈S(Rⁿ),

hψ∗f, φi=hψ∗δ, φi

=hδ,ψ˜∗φi

= ( ˜ψ∗φ)(0)

= Z

Rⁿ

ψ(y)φ(y)dy

=hψ, φi Thus,

ψ∗δ=ψ.

We also note that

F(ψ∗δ) = (2π)ⁿ²ψˆˆδ= ˆψ, as ˆδ= (2π)⁻ⁿ².

Theorem 2.3.3. If u∈D⁰(Rⁿ)andφ∈D(Rⁿ), thenu∗φ∈C^∞(Rⁿ)and supp(u∗φ)⊂supp u+supp φ.

For any multi-index αwe have

D^α(u∗φ) = (D^αu)∗φ=u∗(D^αφ).

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32 CHAPTER 2. SCHWARTZ SPACE Proof. By theorem 2.2.6, u∗φ∈C^∞, and

D^α(u∗φ) =u∗D^αφ.

proving the second equality.

Note that u∗ φ(x) = 0 unless x−y ∈ suppφ for some y ∈ suppu. Thus, x∈suppy+suppφ. And this is a closed set because suppφis compact.

Theorem 2.3.4. Let 0 5 φ ∈ D(Rⁿ) and R

Rⁿφdx = 1. If u ∈ D⁰(Rⁿ), uφ = u∗φ∈C^∞(Rⁿ)anduφ→uinD⁰(Rⁿ)as supp φ→ {0}.

Proof. We haveu(ψ) =u∗ψ(0)˜ ifψ∈D(Rⁿ), where again,ψ(x) =˜ ψ(−x), giving uφ(ψ) =uφ∗ψ(0)˜

=u∗φ∗ψ(0)˜

=u( ˜φ∗ψ).

Theorem 2.2.5 gives us that φ˜∗ψ→ψ in C₀^∞ as suppφ→ {0}. Thus, u_φ(ψ)→ u(ψ).

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Chapter 3

Sobolev Spaces

With the foundation of distributions and the Fourier transform we are almost equipped with the tools necessary to define Sobolev spaces. A Sobolev space is a vector space which is a combination of L^p norms of the function itself and its derivatives up to a given order. However, for the obtained space to be a Banach space, we need to look at the norm of its so-called weak derivatives:

Definition A function f inL¹_loc(R)is called weakly differentiable if there exists a function∂xf in L¹_loc(R)such that

Z

Rⁿ

(∂xf)φdx=− Z

Rⁿ

f ∂xφ, for allφ∈D(R).

Furthermore, if for everyk= 0,1,· · ·, n, there exists∂_x^kf ∈L¹_loc(R)with Z

Rⁿ

(∂_x^kf)φdx= (−1)^k Z

Rⁿ

f ∂^k_xφ, for allφ∈D(R)

we say thatf isntimes weakly differentiable with corresponding weak derivatives

∂_x^kf.

Definition Fork∈Nⁿ0 and1≤p <∞,

W_p^k(Rⁿ) ={f ∈L^p(Rⁿ) :D^αf ∈L^p(Rⁿ)∀α∈Nⁿ0,|α| ≤k},

where the derivatives are taken in the weak sense. W_k^p is then called a Sobolev spaces. When furnished with the norm

kfk_Wk p(Rⁿ)=



 X

|α|k

kD^αfk^p_Lp(Rⁿ)





1 p

W_p^k(Rⁿ)becomes a Banach space. With the inner product hf, gi_Wk

2(Rⁿ)= X

|α|≤k

Z

Rⁿ

D^αf(x)D^αg(x)dx

33

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34 CHAPTER 3. SOBOLEV SPACES W₂^k(Rⁿ)become Hilbert spaces.

Remark The spaces mentioned thus far are connected in the following way:

D(Rⁿ)⊂S(Rⁿ)⊂W_p^k(Rⁿ)⊂L^p(Rⁿ)⊂S⁰(Rⁿ)⊂D⁰(Rⁿ). (3.1) Theorem 3.0.5. D(Rⁿ) is dense inD⁰(Rⁿ).

Proof. We begin by choosing a sequence χj ∈ D(Ω) such that on any compact subset ofΩwe haveχj= 1for all sufficiently largej. Then we chooseφj ∈D(Rⁿ) satisfying theorem 2.3.4 and with small enough support to satisfy

suppφ_j+suppχ_j ⊂Ω, |x|<1

j ifx∈suppφ_j. Sinceχjuis a compactly supported distribution we can form

u_j= (χ_ju)∗φ_j,

thus obtaining a function in D(Ω) by theorem 2.3.3 and theorem 2.3.4, and we have as in the proof of theorem 2.3.4

uj(ψ) = (χju)( ˜φj∗ψ)

=u(χj( ˜φj∗ψ)).

Now, because supp φ˜_j∗ψ belongs to any neighborhood of supp ψ wheneverj is large enough, and we obtain χj( ˜φjψ) = ˜φjψ for those same j. It follows that uj(ψ)→u(ψ)as required.

Remark Because D(Rⁿ)is dense in D⁰(Rⁿ), and all inclusions in (3.1) are con- tinuously embedded, every inclusion in (3.1) is dense.

Lemma 3.0.6. If

(1 +|ξ|²)ⁿ²g(ξ)∈L2(Rⁿ), n∈N0,

there exists ann times weakly differentiable function f in L2(Rⁿ) with fˆ=g and weak derivatives inL2(Rⁿ)such that

∂_x^kf −^F→(iξ)^kg(ξ)∈L2(Rⁿ), k= 0,1,· · ·, n.

Proof. It is clear thatgis inL2(Rⁿ), asξⁿgis inL2(Rⁿ), so there existsf ∈L2(Rⁿ) and a sequence {φj}_j∈Z_>0⊂S(Rⁿ)such that

j→∞lim kf−φjk²_L2(Rⁿ = 1 2π lim

j→∞kg−φˆjk²_L

2(Rⁿ)= 0.

Because (1 +|ξ|²)ⁿ²g(ξ) is in L₂(Rⁿ), so is |ξ|^kg(ξ) for k = 1,2,· · · , n. From Lebesgue’s dominated theorem we obtain

j→∞lim Z

|ξ|^2k|g(ξ)−φˆj(ξ)|dξ= 0

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35 sinceφˆj →g and the convergent series {φˆj}j is bounded. Thus(iξ)^kφˆj →(iξ)^kg in L2(Rⁿ)and

Z

Rⁿ

φ_j∂_x^kψdx= (−1)^k Z

Rⁿ

∂_x^kφ_j ψdx

= (−1)^k Z

Rⁿ

F⁻¹ (iξ)^kg ψdx,

which by definition make∂_x^kφj, and hence alsoF⁻¹((iξ)^kg), weak derivatives.

Now that we have introduced a notation for derivatives that do not require integer- order we can define Sobolev spaces of non-integer order.

Definition

H^s(Rⁿ) ={f ∈S⁰(Rⁿ) : (1 +|x|²)^s²Ff ∈L2(Rⁿ)}, s∈R With the scalar product

hf, gi_Hs(Rⁿ)= Z

Rⁿ

(1 +|x|²)^s²Ff(x)(1 +|x|²)^s²Fg(x)dx H^s(Rⁿ)are Hilbert spaces. The associated norm is given by

kfk_Hs(Rⁿ=q

hf, fi_Hs(Rⁿ)

= Z

Rⁿ

(1 +|x|²)^sfˆ(x) ˆf(x)dx ¹₂

= Z

Rⁿ

(1 +|x|²)^s|fˆ(x)|²dx ¹₂

=k(1 +|x|²)^s²fˆ(x)kL²(R, which is equivalent, in the sense of norms, to

kfk_Hs(Rⁿ)=k(1 +|x|^s) ˆfk_L2(Rⁿ, which is theH^s-norm we will use in the next chapter.

Proposition 3.0.7. For natural numbersk, we have:

H^k(Rⁿ) =W₂^k(Rⁿ), k∈N0

Proof. Iff is inH^s(Rⁿ), then by Lemma 3.0.6, it is also on inW₂^k(Rⁿ). Giving us H^s(Rⁿ)⊆W₂^k(Rⁿ).

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36 CHAPTER 3. SOBOLEV SPACES On the other hand, iff is inW₂^k(Rⁿ),f and all its derivatives up to orderkare in L²(Rⁿ). Thus,

Ff ∈L²(Rⁿ) F(Dxf) = (2πiξ) ˆf ∈L²(Rⁿ) F(D_x²f) = (2πiξ)²fˆ∈L²(Rⁿ), and so on. We conclude that

W₂^k(Rⁿ)⊆H^s(Rⁿ).

And the sets are equal.

Distributions, Schwartz Space and Fractional Sobolev Spaces

Distributions, Schwartz Space and Fractional Sobolev Spaces

Fredrik Joachim Gjestland

TMA4900

Industrial Mathematics, Master’s Thesis:

Distributions, Schwartz Space and Fractional Sobolev Spaces

Fredrik Joachim Gjestland

July 8, 2013

Abstract

Sammendrag

Preface

Contents

Chapter 1

Distributions

1.1 Test Functions

1.2 Distributions

1.3 Distributional Derivative

Chapter 2

Schwartz Space

2.1 The Fourier Transformation on S( R

)

2.2 Convolutions of Continuous Functions

2.3 Tempered Distributions

2.3.1 Example of Fourier Transformation of Distributions

2.3.2 Convolutions with Tempered Distributions

Chapter 3

Sobolev Spaces