All Questions
Tagged with integration functional-analysis
1,150
questions
-2
votes
1
answer
61
views
If the integral of a monotonic f converges, does it mean f approaches 0? [closed]
I have come across this question:
Say $f(x): [0,\infty) \rightarrow \mathbb{R}$ is monotonic non-increasing, and $\int_{0}^{\infty} f(x)dx$ converges.
Does it mean that $\lim_{x\to\infty}{f(x)}=0$? If ...
0
votes
1
answer
53
views
$ L^p ( X ) \cap L^{\infty}( X) $ is a Banach space with respect only to the $p$-norm $\| \cdot \|_p$, $p<\infty$?
The space $πΏ^π(π) \cap πΏ^\infty(π)$, $p<\infty$, with the norm $||π||_{πΏ^π \cap πΏ^\infty}=||π||_π+||π||_\infty$ is a Banach space. I imagine that if we remove the norm $||π||_\infty$ ...
- 3,004
4
votes
1
answer
94
views
Convolution preserve the boundary condition
Here, I want to know if convolution will preserve the Neumann condition or not. Suppose $K$ is a continuous function on some interval $[-L,L]$, and $u$ is some 'good enouth' function on $[0,L]$ that ...
- 133
0
votes
1
answer
59
views
Estimating integrals and measures over Hilbert space using finite dimensional projections
Let $H$ be a separable Hilbert space with orthonormal basis $\{e_k\}_{k \in \mathbb{N}}$. Let $P_n$ be the projection onto the $n$-dimensional subspace of $H$:
$$P_n x = \sum_{i=1}^n \langle x, e_i\...
- 6,275
0
votes
1
answer
47
views
Defining and computing the mean of a Gaussian measure in an infinite dimensional separable Hilbert space
Let $H$ be an infinite dimensional separable Hilbert space and $\mathscr{B}(H)$ the Borel $\sigma$-algebra on $H$. In Da Prato's book An Introduction to Infinite-Dimensional Analysis he defines the ...
- 6,275
0
votes
0
answers
27
views
Extension of a $W^{1,p}_0(\Omega)$ function
Let $u \in W^{1,p}_0(\Omega)$ and let $\tilde{u}$ the function defined extending $u$ to zero on $\mathbb{R}^N \setminus \Omega$. I should prove that $\tilde{u}\in W^{1,p}(\mathbb{R}^N)$.
The exercise ...
- 133
4
votes
1
answer
55
views
If $g_1, g_2\in\mathscr{L}^{\infty}(X,\mathscr{A},\mu)$ are equal locally $\mu$-almost everywhere, then $T_{g_1}=T_{g_2}$.
Background
Suppose that $(X,\mathscr{A},\mu)$ is an arbitrary measure space, that $p$ satisfies $1\leq p<+\infty$, and that $q$ is defined by $\frac{1}{p}+\frac{1}{q}=1$. Let $g$ belong to $\...
- 2,493
0
votes
0
answers
40
views
Question regarding integral mean value theorem
I am sorry for cross-posting a question that I asked on mathoverflow. It is about the integral form of the mean value theorem which states that for the range $[a,b]$ for two continuous functions $f$ ...
- 198
3
votes
1
answer
67
views
Prove that $\int _Xf_ngd\mu \overset{n\to\infty}{\to}\int _Xfgd\mu$ ,$\forall g\in \mathcal{L}^\infty (\mu )$ if it's true $\forall g\in C_b(X)$
Let $X$ be a Polish space and $\mu :\mathfrak{B}_X\to\overline{\mathbb{R}}$ a finite measure on the Borel subsets of $X$. Suppose $(f_n)_{n\in\mathbb{N}}$ is a sequence of $\mathcal{L}^1(\mu )$ and $f\...
- 1,209
2
votes
1
answer
82
views
Singular integral kernel
Define
$$ I : L^2(\mathbb R) \to L^2(\mathbb R), (If)(y) = \int_0^\infty \frac{f(x+y)}{\sqrt{x}} dx. $$
Why is this well-defined and a bounded operator?
Since $\int_1^\infty \frac 1 {\sqrt{x}} dx$ ...
- 878
1
vote
0
answers
39
views
Continuous function supported in $[2-\delta,2]$, satisfying integral of $f(x)\sqrt{4-x^2}$ equals $1$.
Fix $\delta >0$.
I want to show the existence of continuous function $f$, satisfying $supp(f) \subset [2-\delta, 2]$ and $$\frac1{2\pi}\int_{2-\delta}^{2} f(x)\sqrt{4-x^2}\,\mathrm dx=1\;.$$
How do ...
- 125
2
votes
0
answers
65
views
A generalization of integration by parts
Many years ago, I came up with a short generalization of integration by parts that was definitely known, but I could never find a reference for it. I was considering throwing it on arxiv, but before I ...
- 332
0
votes
0
answers
37
views
Is this calculus derivation process correct?
This is part of the economics romer model, finding the Lagrangian value
$$\mathcal{L}=\int_{i=0}^Ap(i)L(i)di-\lambda([\int_{i=0}^AL(i)^\phi di]^{\frac{1}{\phi}}-1)\\
s.t.\int_{i=0}^{A}L(i)^{\phi}di=1
$...
- 101
0
votes
0
answers
26
views
Bounding $L^2$ norm independently of $\hbar$ constant
Let the dimension be $d$ and $k\in\mathbb R^d$ be a constant I am trying to bound the $L^2_p$ norm of $$\hbar^{d-1}\Big(g(\hbar(p-k))-g(\hbar(p+k))\Big)e^{-i\hbar k\cdot p}$$ in a way that is ...
- 461
1
vote
1
answer
45
views
Weak convergence in $L^2$ equivalence
Problem statement: Denoting by $B_r$ the open ball of $\mathbb{R}^N$ centered at the origin with radius $r$, consider a sequence $f_n \in L^2(B_1)$ which is bounded in the $L^2$ norm. Prove that $f_n$ ...
- 608