3
votes
1answer
32 views

Norm of Hardy-Littlewood maximal operator

We define Hardy-Littlewood maximal operator $M$ by \begin{equation} Mf(x)=\sup_{r>0} \frac{1}{|B(x,r)|} \int_{B(x,r)} |f(y)| dy \end{equation} where $B(x,r)$ denotes the ball centered at $x \in ...
2
votes
1answer
62 views

Exercise 31 from chapter 4 (“Hilbert Spaces: An Introduction”) of Stein & Shakarchi's “Real Analysis”

The following problem shows that $\{e^{inx}\}_{n \in \mathbb{Z}}$ is an orthonormal basis of $L^2([-\pi, \pi])$. It is taken from [1] (exercise 31 of chapter 4: "Hilbert Spaces: An Introduction", pp. ...
0
votes
0answers
28 views

Solution of nonhomogeneous problem using semigroup of linear operators

Let $X$ be a complex Hilbert space; and let $A:D(A)\subset X \to X$ be a $\mathbb C-$linear operator. Assume that $A$ is self-adjoint(so that $D(A)$ is a dense subset of $X$) and that $A\leq 0$ (i.e., ...
1
vote
2answers
84 views

Fourier transform as diagonalization of convolution

I've read this in a lot of places but never quite got how this is true or meant. Let's say we have a convolution Operator $$ A_f(g) = \int f(\tau)g(t-\tau)d\tau $$ and apply it to $g(t)=e^{ikt}$. ...
1
vote
1answer
39 views

Convolution operator positive definite?

Let $\mu$ be a compactly supported Borel probability measure on $\mathbb{R}^n$. Consider the convolution operator $T: L^2(\mathbb{R}^n) \rightarrow L^2(\mathbb{R}^n)$ defined by $$ Tf = f \ast \mu $$ ...
0
votes
1answer
23 views

characterization of unital Fourier multipliers on $L^\infty(\mathbb{R})$?

Does there exist a characterization of Fourier multipliers $T \colon L^\infty(\mathbb{R}) \to L^\infty(\mathbb{R})$ which are unital, i.e. $T(1)=1$? In the case of the torus $\mathbb{T}$, it is easy ...
3
votes
2answers
111 views

Two questions in spectral theory: the spectrum of the Fourier transform and the Hamiltonian of the hydrogen atom.

I have the following two questions: The Fourier transform defines a unitary (provided that it is normalized properly) map $\hat{\cdot}:L^2(\mathbf{R})\rightarrow L^2(\mathbf{R})$. I figured out its ...
1
vote
1answer
53 views

Fourier-Transformation of Operator

I have an operator $\hat{L}$ which gives $$\hat{L} f(x) = \lambda \cdot f(x)$$ where $\lambda$ is the eigenvalue. Now I Fourier-Transform my function $f(x)$: $$\mathcal{F}(f)(p) = g(p)$$ Question: ...
0
votes
1answer
42 views

Spectral norm of a Hadamard product

Let $F$ be the $n\times n$ DFT matrix, i.e., $F_{i,j} = \exp\left(-\tfrac{2\pi(i-1)(j-1)}{n}\imath\right)$, for $1\le i,j\le n$ and $\imath =\sqrt{-1}$. Futhermore, let $\Vert \cdot\Vert$ and $\odot$ ...
0
votes
0answers
30 views

Proving something is a convolution operator…

If we define the operator $K(a)=F^{−1}aF$ where $ F:L^2({\mathbb R})\to L^2({\mathbb R})$, is the fourier transform given by $$\left(Ff\right)\left(x\right)=\int_{{\mathbb ...
0
votes
2answers
24 views

How to show something is a convolution operator?

I have the operator $W(a)$ defined by $$W(a)=F^{-1}aF$$ where $F$ denotes the fourier transform and $a$ is a function on $L^{\infty}$. I need to prove that this is convolution operator, but I don't ...
2
votes
0answers
37 views

Formula for trace of particular operators

Let $\mathcal{H}$ be the Hilbert space $L^2(\mathbb{R})$. View the Fourier transform as a unitary operator $\mathcal{F} \in B(\mathcal{H})$. For each function $f \in C_0(\mathbb{R})$, let $T(f) \in ...
0
votes
0answers
32 views

Split Step Fourier Algorithm

Consider the NLSE (Nonlinear Schroedinger equation) that can be written as the following partial differential equation: $$ \frac{\partial{A}}{\partial{z}}=({\cal{L+N}})A\quad\quad(1) $$ where $A: ...
0
votes
1answer
33 views

composition and commutators of Fourier multiplier operators

I am working with some Fourier multiplier operators arising in study of a PDE. I have a general question: Suppose $S$ and $T$ be two Fourier multiplier operators (on some space) with multipliers $m_1$ ...
2
votes
1answer
67 views

Are these operators and the fourier transform compact?

I do not want a proof but rather an explanation. I just read that $T_k:L^2(\mathbb{R}) \rightarrow L^2(\mathbb{R})$ such that $(T_kf)(s) = \int_{\mathbb{R}} k(s,t)f(t) dt $ is compact. (in this ...
1
vote
1answer
300 views

Fourier transform of inverse of Laplace operator

On $L^2(\mathbb{R}^n)$ consider the operator $(-\Delta+z)^{-1}$, for $\Delta$ being the Laplacian and $z\in\mathbb{C}$, on $\mathcal{D}(\Delta^{-1})$. How can one show that for $\phi\in ...
3
votes
0answers
48 views

elliptic operator and wave front set

Let us $f(x) \in C^\infty $ on $\mathbb{R}^n$, and the pseudo-diff. operator $ Q$ is defined by: $(Qu)(x)=(2\pi)^{-n}\int_{\mathbb{R}^{n}}e^{ix\xi }f(x)\left | \xi \right |\hat{u}(\xi) d\xi$ Where ...
5
votes
1answer
339 views

Fourier transform as a Gelfand transform

One question came to my mind while looking at the proof of Gelfand-Naimark theorem. Is Fourier transform a kind of Gelfand transform? Are there any other well-known transforms which are so?
1
vote
2answers
220 views

operator norm of this multiplier operator

I am having some trouble with some basic properties of a given operator. Firstly, the operator T is defined as taking the fourier inverse transform of the function ...
1
vote
1answer
155 views

Boundedness of supremum of an Integral operator

I am trying to find an $L_2$ - bound on a certain class of operators, and on my way I produced an estimate for which I need to show that \begin{equation} \sup_{x \in \mathbb{R}^n} \, ...
1
vote
0answers
85 views

positively homogeneous asymptotic expansion associated to the symbol of a pseudodifferential operator

I am currently reading about pseudodifferential operators and their symbols, and I came across the notion of classical pseudodifferential operators. For these it is possible to find an asymptotic ...
1
vote
0answers
93 views

Introductory book on Distribution theory [duplicate]

Possible Duplicate: Distribution theory book Is there a good alternative to Friedlander Introduction to the Theory of Distributions ? Many thanks !
2
votes
0answers
168 views

What does it mean to carry out calculations modulo $S^{- \infty}$

I have trouble making sense of the following remark that is given in a set of lecture notes I am currently going through on my own: " Oscillatory integrals are used for the study of singularities of ...
3
votes
1answer
132 views

Help understanding a proof on Taylor's formula in Schwartz space $S(\mathbb{R}^n)$

I am having trouble understanding a proof to establish a specific version of Taylor's formula. I'll first give the statement and then below cite the part where I am stuck, so here is what I'd like to ...
14
votes
3answers
4k views

Differential equations and Fourier and Laplace transforms

Why do both the Fourier transform and the Laplace transform appear in the study of differential equations? I've never understood why there are some situations where the Fourier transform is used and ...