1
vote
1answer
22 views

Spectral theory - continuous spectrum

imagine that I have some differential operator $D$ that is defined on an interval $[a,b]$. Now, assume that we take the boundary conditions in such a way that this operator is self-adjoint. Then, I ...
2
votes
1answer
51 views

Sequence of orthogonal vectors in a Hilbert space

Let $\{x_n\}_{n\in\mathbb{N}}$ be a sequance of pairwise orthogonal vectors in a Hilbert space $H$. Show that the following are equavalent: (a) $\sum_{n=0}^\infty x_n$ converges in the norm topology ...
0
votes
2answers
41 views

Show Uniqueness of Solution for Boundary Value Problem

Let $G \subseteq R^n$ be a simple, connected and bounded region with smooth boundary and let $f : \overline G \to \mathbb R$, $g : \partial G \to \mathbb R$ be continuous. Show that the following ...
1
vote
1answer
41 views

Eigenvalues and Eigenfunctions of Integral Equation

Compute the eigenvalues and eigenfunctions of $$ \varphi(x) - \lambda \int_0^1 e^{x+s} \varphi(s) ds = f(x) $$ Are there functions $f$ such that the inhomogenous equation has for every real $\lambda$ ...
0
votes
0answers
25 views

Show that different eigenfunctions of integral kernel are orthogonal

Consider the integral operator $K \varphi := \int_0^1 k(x,s) \varphi(s) ds$ with a continuous and symmetric kernel $k : [0,1]^2 \to \mathbb R$ which has at least two different eigenvalues $\lambda_1$ ...
0
votes
1answer
30 views

Bounded from below or not?

if i have that a functional $J$ defined on a Hilbert space is weakly lower semi continuous and coercive is it bounded from below ??? Please help me Thank you
2
votes
0answers
56 views

Show a set is dense in $C(X)$

Let $X$ be a totally discontinuous compact space. Show that the algebra generated by $$\{f_F; ~f_F=\chi_F-\chi_{X/F},F \text{ is a clopen subset of }X\}$$ is dense in $C(X)$. My attempt: Suppose ...
1
vote
0answers
26 views

can we expect a bounded approximate identity in $L^{1}(\mathbb R)$, whose Fourier transform at particular point is zero?

Let $\phi:\mathbb R \to \mathbb C$ be a function; and define $$\phi_{t}(x):=\frac{1}{t}\phi (x/t), (t>0).$$ We let, $\phi\in L^{1}(\mathbb R),$ then $\phi_{t}\in L^{1}(\mathbb R);$ in fact, we ...
1
vote
1answer
41 views

Fredholm index for 1-d Schroedinger operator

if I look at a Schroedinger-operator $-\frac{d^2}{dx^2} +V$ on a compact intervall $[a,b] \subset \mathbb{R}$ and I take boundary conditions that this operator is self-adjoint (for example periodic ...
2
votes
1answer
45 views

Properties of the Double Layer Potential

Consider the double layer potential $$ W_{\nu}(x) = \int_{\partial\Omega} \nu(y) \frac{\partial}{\partial n_y}\left( \frac{1}{|x - y|} \right) d \sigma_y $$ for a bounded region $\Omega \in \mathbb ...
0
votes
1answer
31 views

If $S$ is dense in $L^{2}$. Is it true that $pS=\{pf| f\in S, pf\in C^{\infty}\} $ is dense?

Let $S=\{f\}$ be a set of function defined in a compact subset $\Omega\subset \mathbb{R}^{n}$ such that $S$ is dense in $L^{2}(\Omega)$. Is it true that for $p\neq 0$ a rational function $pS=\{pf| ...
2
votes
1answer
86 views

Does a function that is twice weakly differentiable have a version that is classically differentiable?

I have been wondering about the idea of functions that are weakly differentiable. My intuition tells me that the weak derivative allows one to differentiate functions that either have a removable ...
2
votes
1answer
54 views

Differentiation under the integral if and only if we have an $L^1$ dominator

Let $f(x)\in L^2(\mathbb{R})$ and define $$g(t) = \int_\mathbb{R} f^2(x)\exp(-tx^2)dx$$ for $t\geq0$. We want to show that $g(t)$ is continuously differentiable if and only if $xf(x)\in ...
5
votes
0answers
193 views
+100

norm of a singular integral operator

My question is from Harmonic Analysis, about the study of singular kernels (in the Calderon Zygmund sense.) Suppose that a kernel $K$ is a singular kernel, extending to a bounded operator on $L^p$, ...
1
vote
1answer
89 views

Spectral theory

I have absolutely no idea about Spectral theory and want to ask some fundamental questions. 1.) What does it mean that the resolvent of an operator is Hilbert-Schmidt? (Cause I saw a theorem that was ...
0
votes
0answers
19 views

Minimization of an evaluation under the weak* topology

I'm self studying (for fun) the book "Functional Analysis, Calculus of Variations and Optimal Control" (by Clarke), and I'd appreciate some feedback for my proposed solution for an exercise from the ...
3
votes
1answer
135 views

Monotone Convergence Theorem for Riemann Integrable functions

I'm having a really hard time proving this statement (this is not homework): If $f_{n} : [0,1] \rightarrow \mathbb{R}$ is a Riemann integrable function for all $n \in \mathbb{N}$, and $0 \leq f_{n + ...
2
votes
1answer
45 views

Are these two expression square integrable?

I have two expressions (let's call them functions $f,g$) on $[0,1]$, where I want to find out whether they are square-integrable or better: for which $m \in \mathbb{Z}$ they are square-integrable ( ...
0
votes
0answers
32 views

Convergence of norms

I have this space $H_{0,p}^1=\lbrace u\in AC([0,+\infty),\mathbb{R}),u(0)=u(+\infty)=0, \sqrt{p} u'\in L^2(0,+\infty)\rbrace $ endowed with the norm $||u||^2=\int_0^{+\infty} p(t) u'^2(t) dt$ ...
1
vote
1answer
21 views

How to choose, $\phi \in C^{\infty}_{c}(\mathbb R)$ such that its Fourier transform $\hat{\phi}$ is 1 in some neighbourhood of the given point?

Put $C_{c}^{\infty}(\mathbb R)=$ The space of $C^{\infty}$ functions on $\mathbb R$ whose support is compact. Fix $x_{0}\in \mathbb R.$ My Question is : Can we expect to choose, $\phi \in ...
0
votes
1answer
19 views

$f\in L^{p}(\mathbb R)\cap C_{0}(\mathbb R); (1<p<\infty), g\in C^{\infty}_{c}(\mathbb R) \implies f\ast g \in C^{k}(\mathbb R)$?

We put, $C_{0}(\mathbb R)=$ The space of continuous functions on $\mathbb R$ vanishing at $\infty$; $C^{k}(\mathbb R)=$ The space of all functions $\mathbb R$ whose derivative of order $\leq k$ exist ...
2
votes
2answers
104 views

How to reduce this to Sturm-Liouville form?

I have the ODE $$-(1-x^2) \frac{d^2 f(x)}{dx^2} + x \frac{df(x)}{dx}+g(x)f(x)=\lambda f(x)$$ and I want to reduce it to Sturm-liouville form. The problem is that we don't have $2x$ but just $x$. ...
1
vote
1answer
22 views

Weighted $L_2$ Hilbert space

this is a question where I am trying to find a reference for a result but I haven't been able to find one at all. Define $L_2(\mathbb R,d\mu) = \{g\in \mathbb R: \int g^2d\mu <\infty\}$. I am ...
0
votes
0answers
47 views

How to find an example

I want to find a function $f\in C^1([0,+\infty)\times\mathbb{R},\mathbb{R})$ such that $f(t,0)=0$ $f(t,u)\leq \alpha u+\beta$, $\alpha<\lambda_1,\beta\geq 0$ $f(t,u)\geq C_1 |u|^{\sigma}$ where ...
3
votes
1answer
30 views

Classify the continuous bilinear functional on $L^p \times L^q$.

Let $1<p<\infty$, $1/p+1/q=1$ and let $L(\cdot,\cdot)$ be continuous bilinear functional on $L^p(\mathbb{R}) \times L^q(\mathbb{R})$. The continuity means that if $f_{n} \rightarrow f$ in $L^p$ ...
2
votes
1answer
34 views

Spherical coordinates and Lebesgue integral

I found a textbook that says that $r \in [0,\infty), \theta \in [0,\pi], \phi \in [-\pi,\pi]$ are the spherical coordinates. Then he goes on with: This corresponds to a unitary map $L^2(\mathbb{R}^3) ...
2
votes
1answer
50 views

Series in a space which is not complete

Let $X$ be a normed vector space and $\left\lbrace x_n \right\rbrace_{n \in \mathbb{N}} \in X^{\mathbb{N}}$ with $$\sum_{n=1}^{\infty} \|x_n\| < \infty \wedge \sum_{n=1}^{\infty} x_n \notin X,$$ ...
1
vote
1answer
26 views

Norms and convergence in $\mathcal{C}^{\infty}(O)$

Let $O \in \mathbb{R}^d$ be open, $K \subseteq O$ compact and $n \in \mathbb{N}$. For $f \in \mathcal{C}^{\infty}(O)=\mathcal{E}(O)$ we define $$\|f\|_{n,K}^{(1)}:=\sup_{|\alpha| \leq n} \|D^{\alpha} ...
3
votes
1answer
40 views

The Dirac Delta Distribution $\delta_0 : D \to \mathbb R$ is not regular

I have a question on the following proof, that $\delta_0$ is not a regular distribution. We define $\delta_0$ as the linear function on test function with $$ \delta_0(\varphi) = \varphi(0) $$ for ...
1
vote
1answer
41 views

The existence of $f \in C^\infty(R^n)$ with $ f=0$ on closed $E$, otherwise $f>0$

This is problem 6.3 in 'Rudin's Functional analysis If $E$ is an arbitrary closed subeset of $R^n$, show that there is an $f \in C^\infty(R^n)$ such that $f(x)=0$ for every $x \in E$ and ...
4
votes
2answers
44 views

Can we expect, $S(\mathbb R)\ast L^{p}(\mathbb R) \subset L^{p}(\mathbb R), (1<p<\infty) $?

It is well-known that $L^{1}(\mathbb R)$ is a closed with respect to convolution(product), that is, $L^{1}(\mathbb R)\ast L^{1}(\mathbb R)\subset L^{1}(\mathbb R),$ more specifically, if $f, g\in ...
2
votes
1answer
46 views

The dual space of locally integrable function space

I'm strongly interested in dual spaces. I learned the dual spaces of $L^p$, $L^{\infty}$, $C(X)$ and $M(X)$. I wonder the dual space of locally integrable function space in $\mathbb{R}^n$. The ...
2
votes
3answers
102 views

Prove that $f_n(x) → f(x)$ uniformly on $E$ as $n → ∞.$

Let $E ⊂ \mathbb{R}$ be a compact (i.e., closed bounded) set of real numbers. Suppose $\{f_n\}$ is a sequence of real-valued continuous functions which converges pointwise on $E$ to a function $f$ ...
2
votes
1answer
58 views

Show solvability of ODE without explicitly calculating solution

Show that $$ u + u^{(4)} - u^{(2)} = f $$ has a solution $u \in H^4(\mathbb R)$ (without explicitly calculuting it) for every $f \in L^2(\mathbb R)$! What criteria for solvability for such ODE's ...
5
votes
1answer
71 views

$\widehat{f\ast g}= \hat{f} \cdot \hat{g}$ for $f, \hat{f} \in L^{p}(\mathbb R)\cap C(\mathbb R) (1<p<\infty, p\neq 2), g\in \mathcal{S}(\mathbb R)$?

It is well-known that, for $f,g \in L^{1}(\mathbb R).$ Then, by Fubini's theorem, one can derive, $\widehat{f\ast g} = \hat{f} \cdot \hat{g},$ (that is, Fourier transform takes, convolution to point ...
1
vote
1answer
32 views

Weakly * continuous definition

What does it mean that $$t\to u(t,\cdot)$$ is waakly* continuous from $[0,T]$ to $L^\infty(\mathbb{R}^d)$? I guess that I have to see $L^\infty(\mathbb{R}^d)$ as the dual of $L^1(\mathbb{R}^d)$ and ...
0
votes
1answer
26 views

Formula for $L^{q}$ norm using $C_{c}^{\infty}$ functions

We put, $L^{p}=L^{p}(\mathbb R), L^{q}=L^{q}(\mathbb R);$ $\frac{1}{p}+\frac{1}{q}=1;$ ($p$ and $q$ are conjugate exponents); and $<f,g> =\int_{\mathbb R} f(x)g(x) dx.$ Fix $g\in L^{q}, ...
0
votes
0answers
50 views

Proof that a set is open.

Let $(\Lambda_i)_{i\in I}$ a collection of linear operators from $X$ (Banach space) to $Y$ (Normed space). Let $\alpha : X \rightarrow [0,\infty]$ be the function $\alpha(x):=\sup_{i \in I} ...
0
votes
0answers
24 views

Question about convergence

If i have that $$\int_0^{+\infty} a(t)|u_n(t)-u(t)|^2 dt \rightarrow 0 $$ how we can deduce that $$\int_0^{+\infty} a(t)|~|u_n(t)|-|u(t)|~|^2 dt \rightarrow 0 $$ where $a>0, a\in ...
0
votes
1answer
48 views

Definition of weak divergence [closed]

Can anyone give me the definition of the divergence of a vector field in the distributional sense?
1
vote
0answers
37 views

Real Hardy space question

Please see this related question for the definition of the grand maximal function and the class of normalised test functions $\mathcal{T}$. I will refer to them in this question. Let $f\in ...
2
votes
1answer
54 views

If $\phi\in \mathcal{S}(\mathbb R) $ then $\phi_{t}(x)=\frac{1}{t} \phi(x/t)\in\mathcal{S}(\mathbb R)$?

Let $\phi:\mathbb R \to \mathbb C$ be a function; and define $$\phi_{t}(x):=\frac{1}{t}\phi (x/t), (t>0).$$ We note that, if $\phi\in L^{1}(\mathbb R),$ then $\phi_{t}\in L^{1}(\mathbb R);$ in ...
0
votes
0answers
76 views

Is Fourier transform density preserving?

I know my question is not well-defined since Fourier domain and codomain are not the same, but one knows that they are actually homomorphic. Now what I mean by density preserving is as follows: ...
1
vote
1answer
39 views

Constructing a specific normalised test function.

Given the following class of normalised test functions: \begin{equation} \mathcal{T}=\{\phi\in C_c^{\infty}(\mathbb{R}^n)\ |\ \text{supp }\phi \subseteq B(0, 1),\ \|D\phi\|_{\infty}\leq 1\} ...
0
votes
1answer
52 views

Space of continuous functions vanishing at infinity

Let's denote with $C_0(X)$ the space of continuous functions $f$ on $X$ such that for every $\epsilon>0$ there exists a compact set $K_\epsilon\subset X$ satisfying $sup_{x\notin K_\epsilon ...
3
votes
1answer
30 views

Every unitary representation of a compact group is a direct sum of irreducible representations.

I've read nice proofs of a few different variants of the Peter-Weyl theorem and its corollaries. For instance I know that for $G$ a compact group, $L^2(G)$ is a Hilbert space direct sum of the matrix ...
1
vote
1answer
75 views

Any idea with this problem of distances???

Let $E$ a normed linear space and $H$ the closed hyperplane $H=\ker f$, where $f\in L(E,\mathbb{R})$, $f\not\equiv 0$. Show that if $a\in E$ then $$d(a,H)=\frac{|f(a)|}{||f||}$$ And the problem have a ...
2
votes
1answer
60 views

How can I prove that $f$ is continuous at $0$?

Let $E$ and $F$ linear normed spaces, and consider a linear function $f:E\to F$ which satisfies for all $(x_n)\in E^{\mathbb{N}}$ such that $x_n\to 0$ then $f(x_n)$ is bounded in $F$. Then I have to ...
0
votes
1answer
24 views

a non-decreasing sequence of functions with bounded L^p-norm is a Cauchy sequence in L^p space

Let $\{f_k\}_{k=1}^\infty$ be a sequence in $L^p(\mathbb{R})$ for $1\leq p<\infty$. Suppose $f_1\leq f_2\leq\cdots$ and $\sup ||f_k||_p<\infty$. Prove that $\{f_k\}_{k=1}^\infty$ converges in ...
1
vote
1answer
35 views

unbounded self-adjoint operator

Given an operator $T:D_1(T)\subset L^2 \rightarrow L^2$ and the same operator $T:D_2(T) \subset L^2 \rightarrow L^2$, such that the operator is both times self-adjoint and closed, with $D_1(T) \subset ...