All Questions
Tagged with functional-analysis or fa.functional-analysis
9,772 questions
-2
votes
2
answers
324
views
$f\in (W^{1,p}(\Omega)\cap C(\Omega) \cap L^{\infty}(\Omega))\setminus C(\bar{\Omega})$, $f=0$ on $\partial \Omega$ imply $f\in W^{1,p}_{0}(\Omega)$?
Q1:
Let $p\geq 1$, and let $f\in W^{1,p}(\Omega)\cap C(\Omega)$. Assume also
$f\in L^{\infty}(\Omega)$ and $f=0$ on $\partial \Omega$. Is it true that
$f\in W^{1,p}_{0}(\Omega)$ even if $f\notin C(\...
- 852
-2
votes
1
answer
146
views
a measure convolution equation
My question is:
Given a function $f$ in the Schwartz class, we are looking for a measure $\mu$ which is a solution of the convolution equation: $f = e^{-|.|^2/2} \ast \mu$, where $e^{-|.|^2/2}$ is ...
- 367
-2
votes
1
answer
80
views
Density property for Sobolev spaces
My question is as follows: is the space $ C_c^{\infty}(\mathbb{R}^3 \setminus \mathcal{C}) $ dense in $ H^1( \mathbb{R}^3) $ where $ \mathcal{C} $ is the circle $ \{(x,y,z) \in \mathbb{R}^3 \mid x^2 +...
- 1
-2
votes
1
answer
314
views
Series representation for $\log(|\zeta(\frac{1}{2}+it)|)$
(Question is short and straight-forward. )
What is/are "nice and non-trivial" series representation/s of $\log(|\zeta(\frac{1}{2}+it)|)$ ??
By "nice and non-trivial" I mean contains no ...
- 375
-2
votes
1
answer
158
views
About local maxima of multivariable polynomials
Lets say I have a real valued function which is writable as a polynomial in terms of Frobenius norms of a pair of matrices as in it is of the form, $f_B(A) = f(||A||_F^2, ||AB||_F^2, ||A^TAB||_F^2)$ ...
- 2,246
-2
votes
1
answer
193
views
Analysis of Sobolev spaces [closed]
I just wanted to know wthether the following is OK or not.
Let $X$ be $H_0^1(\Omega)\bigcap L^{\infty}(\Omega)$, thought of as a subspace of $H^1_0(\Omega)$ and endowed solely with the usual $H^1$ ...
- 157
-2
votes
1
answer
295
views
When does the adjoint operator map closed convex subsets to closed convex subset?
Let $T:X\rightarrow Y$ be a linear continuous map between Banach spaces $X$ and $Y$ and denote by $T':Y'\rightarrow X'$ the norm adjoint of $T$. Let $M\subseteq U'$ be a subset of
the unit sphere $U'$ ...
- 101
-3
votes
1
answer
315
views
Are the injective functions dense in $C([0,1]^n,\mathbb R^n) $?
Let $n\geq 2$. Are injective functions dense in $C([0,1]^n,\mathbb R^n) $ with the uniform norm?
- 4,074
-3
votes
1
answer
232
views
Function satisfying $f(x)^{f^{-1}(x)}=x^2$ with $ f^{-1}$ is a compositional inverse of $f$ and $f:\mathbb{R+}\to \mathbb{R+}$?
Let $f$ be a function such that :$f:\mathbb{R+}\to \mathbb{R+}$ and $f^{-1}$ is a compositional inverse of $f$ , I have tried to find solution of the following functional $f(x)^{f^{-1}(x)}=x^2$, I ...
-3
votes
2
answers
768
views
Question on Linear Operators
Let $V$ be a normed infinite dimensional vector space. Let $L: V \longrightarrow V$ be a bounded linear operator. Moreover assume that $L$ is 'locally nilpotent' that is:
$$ \forall v \in V \quad \...
- 741
-3
votes
1
answer
634
views
compactly supported harmonic functions [closed]
Do a significant class of compactly supported smooth functions u on Ω⊂Rn such that Δu≥0 exist?
Thanks!
- 25
-3
votes
1
answer
392
views
A generalization of Chebyshev's sum inequality
From some my previous questions here and here and well-known rearrangement inequality. I pose an inequality as follows and I am looking for the proof or a reference.
Inequality: Let $y=f(x,y)$ is ...
- 1,776
-3
votes
1
answer
76
views
Minimal norm problem with linear combination of translation operator to be estimated
Follow up question from this one
Suppose $X = L^2(G)$, where $G$ is some locally compact group. Let $x, y \in G$ I for fixed $n$ I am seeking for an operator $H \in B(X)$ of the form
$$
H = H(\alpha_1,...
- 115
-3
votes
1
answer
63
views
How to show $\lambda_i \in \sigma_A(x)$?
Let $\sigma_A(x)$ be the spectrum of $x$ in $A$, and linear functional $\phi$ satisfying $\phi(x)\in \sigma_A(x)$ for every $x \in A$, consider $p(\lambda)=\phi((\lambda e-x)^n)$, and denote its roots ...
- 33
-3
votes
0
answers
66
views
Exercise generalizing (related to) Hölder's inequality
I came across this exercise and feel absolutely stuck:
Let $p, q, r \in (1, \infty]$ be such that $1/p + 1/q = 1 + 1/r$. Suppose that $F : \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ satisfies ...
- 1
-3
votes
1
answer
451
views
Exponential decay of kernel
Let $A: \ell^2 \rightarrow \ell^2$ be a bounded operator given by
\begin{equation}
(Au)(\alpha) = \sum_{\beta}A(\alpha,\beta)u(\beta)
\end{equation}
where $\left|A(\alpha,\beta) \right|\le Ce^{-|\...
- 11
-4
votes
2
answers
530
views
Inverse square-law as a positive definite kernel?
Newtons law for gravity states that:
$$F_{12} = \frac{G m_1 m_2} {|x_1-x_2|^2}$$
The function :
$$k(x,y):=\exp(-| x-y|^2)$$
is known to be a positive definite function, called the RBF-kernel.
It ...
- 3,064
-4
votes
2
answers
286
views
Does the Laplacian commutes with the indicator function [closed]
We define the laplacian operator $\Delta$ with the Neumann boundary conditions on the space $H^2(\Omega)$, where $\Omega$ is an open set of $\mathbb{R}^n$ with a smooth boundary $\partial\Omega$, and ...
-4
votes
1
answer
200
views
How I can choose $(t_1,t_2,...,t_{r}) \in (0,1)^{r}$ such that $f^{(k)}\left(1-2\prod_{j=1}^{k}t_{j}\right)=0$?
Let $f:\mathbb{R} \to \mathbb{R}$ be a real analytic function. Assume that $f$ has simple trivial zeros at each nonpositive integer. Then, all the $k$-th derivatives $f^{(k)}$ of $f$ have necessarily ...
- 1,197
-4
votes
1
answer
370
views
Is delta function symmetric against real axis? [closed]
Is $\delta\left(a+bi\right)=\delta\left(a-bi\right)$?
I wonder whether Dirac Delta (as defined via Fourier transform) is symmetric against the real axis.
We can write Delta function as
$$\delta(z) = \...
- 10.1k
-4
votes
1
answer
144
views
Coordinate free computation of the second derivative of a functional [closed]
Let $F(g(f))$
be a functional that sends functions of a vector variable (from $n$-dimensional vector space) to $\mathbb R$.
$g$
is some function of scalar valued functions $f$.
I'm interested in a ...
- 1
-6
votes
1
answer
180
views
An analog of Anderson's result in C* algebra setting [closed]
Let $\mathcal{A}$ be a unital $C^{*}$-algebra and $S(\mathcal{A})$ denote the states space of $\mathcal{A}$.
For $a\in \mathcal{A}$ , define $W(a) =\{\phi(a):\phi\in S(\mathcal{A})\}$
It's known that $...
- 307