All Questions
Tagged with convolution functional-analysis
302
questions
9
votes
1
answer
275
views
How to prove $(F\ast\sin)(x)=-\sin(x)$, where $F(x)=\frac{1}{2}|x|$?
Wikipedia states in this article about fundamental solutions that if $F\left( x \right) = \tfrac{1}{2} \left| x \right|$, then
$$\left( F \ast \sin \right)\left( x \right) := \int\limits_{-\infty}^{\...
- 139
1
vote
0
answers
134
views
convolution of the fundamental solution with the homogeneous solution
I have a question about the convolution of the fundamental solution with the homogeneous solution. Namely if the 2 are convoluble then the homogeneous solution is necessarily zero?
Let $U$ and $E$ ...
- 139
4
votes
1
answer
94
views
Convolution preserve the boundary condition
Here, I want to know if convolution will preserve the Neumann condition or not. Suppose $K$ is a continuous function on some interval $[-L,L]$, and $u$ is some 'good enouth' function on $[0,L]$ that ...
- 133
1
vote
1
answer
35
views
The operator norm regarding to the difference between a mollified function and the function itself
Let $\rho_\epsilon$ be a mollifier that has support in $B(0,\epsilon)$, define the operator
$$T_\epsilon (f)=\rho_\epsilon*f-f,$$
for every $f\in L^2(\mathbb R^d)$, can we prove or disprove that the ...
1
vote
0
answers
25
views
Classifying bounded linear functions satisfying convolution identity
Find all bounded linear functionals $T$ on $Y= \{f \in W^{1,1}[0,1]: f(0)=0\}$ such that there exists $K>0$ such that $$\int_0^1|Tf(\cdot-t)|\ dt\le K \|f\|_1\qquad \forall\ f\in Y\tag{1}$$
My ...
- 373
2
votes
2
answers
88
views
Does convolution preserve local integrability in the sense that if $f \in L^p_{\text{loc}}$ and $g \in L^1$, then $f \ast g \in L^p_{\text{loc}}$?
Consider the Euclidian space $\mathbb R^n$, where $n \in \mathbb N$ is a fixed integer, equipped with the Lebesgue measure. Let $L^p_{\text{loc}}(\mathbb R^n)$($1 \leqslant p < \infty$) denote the ...
- 1,141
2
votes
1
answer
50
views
Density of similiar Sobolev space
Consider the space of functions defined as,
$$
D = \{f \in L^p(0,\infty): f \in AC_{loc}(0,\infty) \text{ and } xf'(x) \in L^p(0,\infty)\},
$$
where $AC_{loc}(0,\infty)$ is the set of locally ...
4
votes
1
answer
74
views
Convolution between $f$ and $g$, with $g$ being in the Schwartz class. Does it follow that $f \ast g \in C^\infty$?
Usually, the convolution between two functions $f,g$ defined on $\mathbb R^n$ is given by
$$ (f \ast g)(x) = \int_{\mathbb R^n} f(x-y)g(y) \, dy. $$
Right now I am wondering about a specific property ...
0
votes
0
answers
35
views
How to show the limit of a series of convolutions exist?
Suppose $\rho:\mathbb{R}^d\to\mathbb{R}$ is an even, smooth test function, compactly supported in the ball of radius $1$. Define $\rho^{(n)}(x):=2^{nd}\rho(2^nx),$ and
$$\rho^{(n,m)}:=\rho^{(n)}\ast \...
- 329
1
vote
1
answer
71
views
Convergence of gaussian filtered function
Consider the function $f(x) = \frac{1}{\sqrt{x}}$ and the standard Gaussian filter $K_{\delta}(x) = \frac{1}{\delta \sqrt{2\pi}} e^{\frac{x^2}{2 \delta^2}}$. I am interested in finding out how fast ...
- 1,813
0
votes
0
answers
15
views
Does there exist a convolution semigroup with compactly supported density?
I am looking for a convolution semigroup $(P_t)_{t \geq 0}$ of operators acting on $C_0^\infty(\mathbb{R^d})$, of the form
$$P_tf(x) = (p_t * f)(x) = \int_{\mathbb{R}^d} p_t(x-y) f(y) dy,$$
where $p_t$...
- 1,633
5
votes
1
answer
110
views
Bound on convolution: $ | (h * f^2) (x)| \leq \| f\|^2_2 g(h)$
I am trying to find bounds for the following quantity. Take two functions $f,h \in L^{1} \cap L^2$ but $\|h \|_{\infty} = \infty$. Is there a way to obtain a bound of the following type:
$$ | (h * f^2)...
- 1,813
0
votes
1
answer
53
views
Rate of convergence of mollified exponential on $[0, \infty)$
Consider the function
$$
f(t)=e^{-at}
$$
with $a>0$, $t \in [0, \infty)$ and let $\rho$ a smooth function on $[0, \infty)$ such that
$$
\int_0^\infty \rho(t)=1
$$
and let $\rho_\delta(t):= \frac{1}{...
- 2,675
2
votes
0
answers
38
views
What extra condition on convolution can guarantee the associativity
Let $f,g:\mathbb R^n\to\mathbb C$ be two Lebesgue measurable functions. Thus $(x,y)\mapsto f(x-y)g(y)$ is also a mesaurable function. To be rigorous, we define let
$$f\ast g(x)=\int_{\mathbb R^n}f(x-y)...
- 2,269
1
vote
0
answers
29
views
Convolution of Schwartz and $C^\infty$ function with bounded increasing [duplicate]
I have the following problem: Let $f\in\mathcal{S}(\mathbb{R}^n)$ and $g\in C^\infty(\mathbb{R}^n)$ such that exists $\alpha>0$
$$
\left|g(x)\right|\leq \frac{1}{1+|x|^\alpha} \quad \forall x\in\...
- 377