1
vote
1answer
28 views

Lipschitz constants of projections

Consider two compact sets $A, B \subset \mathbb{R}^n$. Assume that the projection mappings $P_A: \mathbb{R}^n \rightarrow A$, $P_B : \mathbb{R}^n \rightarrow B$ have Lipschitz constant $1$ and $L$, ...
6
votes
1answer
46 views

Which normed space have Fatou's property?

There is tool in mathematics, more specifically in Analysis, which has immense applications in handling delicates estimates, limiting arguments, etc... ; namely Fatou property. Let $E$ be a normed ...
0
votes
0answers
24 views

Fixed point of projected operator

Let $X \subset \mathbb{R}^n$ be a compact convex set and let $f: X \rightarrow X$ be Lipschitz continuous and such that $$ \left( f(x) - f(y) \right)^\top \left( x-y\right) \leq 0 $$ for all $x, y ...
2
votes
3answers
57 views

The “Circle” is a Vector Space?

Consider the set of angles $C = [0, \ 2\pi)$ and, for all $x,y \in C$, define the $sum$ operation as the sum modulo $[0, \ 2\pi)$. The identity element of the addition is the angle $0$. The inverse ...
1
vote
2answers
56 views

$2\pi$ in the Definition of Fourier Transform

Most textbooks I read define Fourier transform of a function $f \in L^2(\mathbb R)$ as $$ \hat f (\xi) := \int_\mathbb R f(x) e^{-2\pi i x \xi} dx. $$ However, in class my teacher defines it without ...
0
votes
1answer
32 views

Dense subset of $L^{2}$ such that $x^{-1/2}f \in L^{1}$ and $\int_{[0, 1]}x^{-1/2}f\, dx = 0$

Does there exist a dense set of functions $f \in L^{2}([0, 1])$ such that $x^{-1/2}f(x) \in L^{1}([0, 1])$ and $\int_{0}^{1}x^{-1/2}f(x)\, dx = 0$? I've noticed that $\int_{0}^{1}x^{-1/2}f(x)\, dx = ...
0
votes
0answers
38 views

Question about a theorem from Chang's book: Methods in Nonlinear analysis

I have this this theorem from Chang's book: Methods in Nonlinear analysis, with it's proof, but i don't understand it, for example what it means $K(f_{\sigma_i})$ ? Please help me thank you
1
vote
0answers
33 views

A proof for Jensen’s inequality

I’m trying to prove a version of Jensen’s inequality, but I end up with the wrong result. I’d appreciate any help or comments. The theorem states: let $\varphi :{{R}^{k}}\to R$ be convex. Then for ...
1
vote
1answer
33 views

Norm of the multiplication operator $f\mapsto (x\mapsto xf(x))$ on $L^2[a,b]$ [duplicate]

We have a linear operator $T : L^2[a,b] \rightarrow L^2[a,b]$ (with $|a| \le |b|$), $f \mapsto (x \mapsto xf(x))$ Now I shall determine what $\Vert T\Vert$ is. We clearly have $\Vert x \mapsto ...
3
votes
0answers
27 views

Every Cauchy net is convergent [duplicate]

Prove that in a Banach space every Cauchy net is convergent. I have trouble to prove this, please help.Thanks Edit:Let $A$ be a directed set and $\{f_{\alpha}\}_{\alpha\in A}$ is a net in $X$ ...
2
votes
1answer
45 views

Definition of exponential for operators

if I have a self-adjoint operator $T:D(T) \rightarrow L^2$, then I define its unitary exponential operator by $$e^{iT}(f) := \lim_{k \rightarrow \infty} e^{iT_{k}}(f),$$ where $T_k(f):=\frac{1}{2} ...
0
votes
1answer
28 views

Domain Schrödinger equation

If I have a self-adjoint (TIME INDEPENDENT) operator $H: D(H) \subset L^2([0,x]) \rightarrow L^2([0,x])$ with and I want to define the appropriate domain for $$ (Hf)(x,t) = i \partial_t f(x,t),$$ ...
1
vote
2answers
57 views

Proper domain for Laplacian

it is well known that the spherical harmonics are eigenfunctions to the 3D Laplacian(angular part). But my question is: What is the right domain for this operator so that we actually get these ...
2
votes
1answer
38 views

Limit of the p-norm of a function on subdomains equals the p-norm of the function on the union domain

Let $\Omega$ be an open subset in $\mathbb{R}^n$. Given a measurable function $f$, define $$ ||f||_{p,\Omega}=\inf_{a\in\mathbb{R}}||f-a||_{L^p{(\Omega})}. $$ Let $\{\Omega_n\}$ be a sequence of open ...
1
vote
0answers
38 views

amazing boundedness problem from maximal function

Let $n\geq 2$. For any $M>1$, prove that there exists a constant $C_M>1$ such that for any ball $B$ in $\mathbb{R}^n$, if we denote $MB$ as the concentric ball of $B$ with $M$ times radius of ...
0
votes
2answers
24 views

Space of bounded functions vs. bounded space of functions.

Suppose I have a bounded set of functions, say $B\subset C[0,1]$. What exactly does this mean? I.e. is a bounded set of continuous functions equivalent to a set of continuous bounded functions? For ...
1
vote
0answers
30 views

Different Formulations of Riesz' lemma

Version I: Let $U$ be a closed subspace of the normed space $X$ with $U \ne X$. Also let $0 < \delta < 1$, then there exists $x_{\delta} \in X$ with $||x_{\delta}|| = 1$ and $$ || x_{\delta} ...
1
vote
1answer
47 views

If holomorphic $\{f_n\}\to f$ uniformly on compact subsets of $U$, then do $f_n$ and $f$ eventually have the same number of zeros?

Let $U$ be an open subset of $\mathbb{C}$. Let $\{f_n\}$ be a sequence of holomorphic functions on $U$ such that $f_n\to f$ uniformly on any compact subset $K$ in $U$. Suppose $f$ is not constant, ...
2
votes
2answers
50 views

For holomorphic functions, if $\{f_n\}\to f$ uniformly on compact sets, then the same is true for the derivatives.

Let $\Omega$ be an open subset in $\mathbb{C}$. Let $\{f_n\}$ be a sequence of holomorphic functions on $\Omega$ such that $f_n\to f$ pointwise and converges uniformly on any compact subset ...
3
votes
1answer
46 views

Is $AA^*$ and $A^*A$ self-adjoint?

if I have a densely defined closed linear operator $A$ and $A^* = -A$(same domain also closed). Is this sufficient that $AA^*$ and $A^*A$ are proper self-adjoint operators, assuming that we can also ...
0
votes
1answer
24 views

Determining the semigroup and relating it to the generator via the functional calculus

In the theory of Hille Yoshida, we have a (definitely real) banach space $L$ and on it a contractive, strongly continuous semigroup of operators. ($T_tT_s=T_{t+s}$ and $T_0=I$.) We then define the ...
0
votes
0answers
35 views

Proving the existence of limit of an integral

Let $g:\mathbb{R}^d\to\mathbb{R}$ a smooth function and $B:\mathbb{R}^d\to\mathbb{R}^d$ a Lipschitz continuous vector field. I have to study the limit of the following integral ...
4
votes
2answers
101 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
60 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
43 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
26 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
59 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
29 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
42 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
46 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
35 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
92 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
66 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 ...
8
votes
0answers
275 views

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
93 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
157 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
46 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
33 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
24 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
20 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
3answers
118 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
24 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
50 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
31 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
36 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
52 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
29 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} ...