Tagged Questions
3
votes
1answer
43 views
Geometric intuition behind the Uniform Boundedness Principle
Is there a way to visualize why the Uniform Boundedness Principle should be true? I understand the statement of the theorem but I'm having a hard time seeing a picture of it in my head.
-1
votes
0answers
17 views
Help with Toeplitz operators applications. [closed]
I am trying to find a physics problem which solution involves Toeplitz operators.
0
votes
0answers
16 views
Question on a third-order boundary value problems
This is the corollary $2.1$, from the article "Positive solutions of third order semipositone boundary value problems"
if $$u'''=\lambda \left(\sum_{i=1}^m c_i(t)u^{\mu_i}-d(t)\right)+e(t), t\in ...
2
votes
3answers
79 views
Show that $c$ is closed in $l^{\infty}$
Let $$c=\{ (a_i)_{i \in \mathbb{N}} \ ; \ \ a_i \in \mathbb{R}\ ,\ \forall i \in \mathbb{N} \ , \ \mbox{exist} \ \displaystyle \lim_{i \to \infty}(a_i)\}$$
$$l^{\infty}=\{ (a_i)_{i \in \mathbb{N}} \ ...
3
votes
1answer
64 views
About Lusin's condition (N)
We say that $f:[0,1]\to \mathbb{R}$ satisfies Lusin's condition (N) provided
$$m(f(B))=0 \quad\mbox{whenever}\quad B\subseteq [0,1] \mbox{ with }m(B)=0$$
where $m$ stands for the Lebesgue measure on ...
0
votes
1answer
33 views
A particular weak subadditivity
Given a function $f: \mathbb{R}^n \rightarrow \mathbb{R}$, consider the following property.
For all $(x^1, ..., x^n) \in \left(\mathbb{R}^n \right)^n$ such that $f(x^i) \geq 0$ $\forall i \in [1,n]$, ...
3
votes
0answers
30 views
SVD, infinite matrices and normal operators from a function
I'm trying to understand the behavior the Singular Value Decomposition on a deeper level, and why it might give a particular result. Take the function
$$
f(x,y) = \frac{1}{(1+2x+y)^2}
$$
and ...
1
vote
3answers
90 views
$\|f*g\|_q\leq \|g\|_q \|f\|_1$ and $\|f*g\|_\infty\leq \|g\|_q \|f\|_{q^{'}}$, $(1/q+1/q^{'}=1)$?
Now I'm reading the Young inequality. It says that if $f \in L^p(R)$, $g \in L^q(R)$, $1\leq p,q\leq \infty$, $1/p+1/q\geq 1$. Then how could we have the following inequalities:
$$\|f*g\|_q\leq ...
1
vote
1answer
25 views
Difference between Rician distribution and Gaussian distribution
could any one please tell me the difference between Rician and Gaussian Distribution and the advantages of using one over other please.With some mathematical proof would be truly appreciated
Thank ...
1
vote
2answers
67 views
Sequence of Functions Converging to 0
I encountered this question in a textbook. While I understand the intuition behind it I am not sure how to formally prove it.
Define the sequence of functions $(g_n)$ on $[0,1]$ to be $$g_{k,n}(x) = ...
1
vote
2answers
35 views
Function Spaces
What is exactly the difference between $L^2$ space and ${\ell}^1$ space? I believe that one of them is the space of square of square integrable functions.
Does it have to do with one is for series ...
0
votes
0answers
35 views
Green's function
Please can someone told me how to find the Green's function $G(t,x)$ of BVP : $$u'''(t)=0 , \quad t\in (0,1)$$ and BC : $$u(0)=u'(p)=\int_q^1 w(s)u''(s) ds =0 $$ where $\frac12 < p<q<1$ are ...
1
vote
1answer
71 views
Newton-Raphson in $\mathbb{R}^n$
Let $U\in\mathbb{R}^n$ be open and $f:U\to\mathbb{R}^n$ be a $\mathcal{C}^1$ map. $\exists p\in U$ such that $\;f(p) = 0$ and $Df_{|p}$ is invertible. Define $\phi(x)= x-(Df_{|x})^{-1}f(x)$, show ...
1
vote
1answer
50 views
What are some non-constant functions in the following sets? Are these sets a subspace of $Y = C^1[a,b]$?
Let $Y = C^1[a,b]$. What are some non-constant functions in the following sets? Are these sets a subspace of $Y = C^1[a,b]$.
a) $D = \{ y \in Y: y'(a) = 0,\, y(b) = 1\} $
b) $D = \{y \in Y : ...
2
votes
0answers
52 views
(localized) L^2 norm of quasimode for Laplacian
Lately I've been thinking about the following distribution on the flat torus $\mathbb{T}^2$:
$u_k=\frac{1}{\sqrt{2\lfloor k^{0.99}\rfloor+1}}\sum_{|l|\leq ...
