Tagged Questions

1
vote
1answer
51 views

Convolution of distributions.

We are given with distributions $f,g \in D'(\Bbb R)$. If $suppf\subset (-\infty,a)$ and $supp(g)\subset(b,\infty)$ then prove that $f*g$ is well defined distribution. where $a$ and $b$ are real ...
1
vote
2answers
103 views

How to prove that operator is not compact in $L_2 (\mathbb{R})$

I have the operator $(Af)(x) = \int _{\mathbb{R}} e^{{-(x-t)^2}/2} f(t) dt$. It seems to me that it isn't compact and I'm looking for some general <=> criterion for integral operators to be ...
2
votes
1answer
58 views

The differentiability of convolutions

Yes, again, this type of question. Similar ones this and this. I come with another variant. Let $f\in\mathcal{S}$, i.e. Schwartz function, and $g\in L^{p}(\mathbb{R}^d),p\in[1,\infty]$. The following ...
3
votes
2answers
56 views

Problem of convolution.

If we are given with a polynomial $\mathcal P$ and a compactly supported distribution $g$. Can we prove that their convolution will be a polynomial again?
1
vote
0answers
35 views

General approach to prove the smoothness of convolution

Consider $\mathbb{R}^d$ and $D_i = \partial /\partial x_i$, in many cases $$D_i(f*g) = D_if*g,$$ given one of $f,g$ is smooth and the other is $L^p$ integrable. I am wondering if there is a general ...
0
votes
1answer
110 views

What will be the support of the convolution of two test functions.

If $g\in C^{\infty}_c$ defined on $\Bbb R^n$ and K is the support of function $g$. I want to find the support of $g_\epsilon$. Where $g_\epsilon$ is regularization of $g$. Regularization of $g$ is ...
4
votes
2answers
35 views

partially reconstruct information of function convoluted with boxcar kernel

the function (f) I want to reconstruct partially could look like this: The following properties are known: It consists only of alternating plateau (high/low). So the first derivation is zero ...
2
votes
1answer
50 views

convolution-distributions

We denote by $E'(\mathbb{R})$ the set of distribution with compact support , and $\mathcal{D}(\mathbb{R})$ is the set of function $\mathcal{C}^{\infty}$ with a compact support. 1) I want to compute ...
1
vote
1answer
87 views

$\mathcal{L}^p$ spaces and convolution

Suppose that $f \in \mathcal{L}^p$ and $g \in \mathcal{L}^q$, and $p,q$ are conjugate exponents. Then prove that (a) $h(x) = \int_{-\infty}^{\infty} f(t) g(x+t) \, dt$ defines a bounded continuous ...
3
votes
0answers
83 views

The norm of an operator

Let $\rho(x)$ be a weight function in a unit sphere, such that \begin{equation} \begin{array}{l} \displaystyle 1. \rho(x)\ge 0,\int_{\mathbb{R}^n}\rho(x)=1\\ \displaystyle 2. \rho(x)\in ...
1
vote
1answer
148 views

Double tophat convolved with a gaussian

I need some help calculating the analytically expression of this convolution. The functions in question are: 1) a gaussian ($g(x)$) 2) a "double tophat function" (in lack of a better name). i.e. ...
1
vote
0answers
91 views

convolution of L1 function with a harmonic oscillation

I have to show that the convolution of a function $f \in L^1(\mathbf{R})$ with the harmonic oscillation $\phi_\omega (t) = \exp(2 \pi i t \omega)$ is equal to the Fourier Transform of $f$, ...
2
votes
0answers
46 views

bound on Hilbert transform

Consider $\widehat{Tf(\xi)}=m(\xi)\hat{f}(\xi)$, where $m(\xi)=(1-\vert\xi\vert)1_{[-1,1]}$, i.e. $T$ is the operation of taking Fourier transform and multiplying with the function $m(\xi)$. I am ...
7
votes
2answers
699 views

convolution of a function with itself equals itself

In a homework question, I was asked to show: (1) in $L^1(R)$, if $f*f = f$, then $f$ must be a zero function. (2) In $L^2(R)$, find a function $f*f=f$. I don't know how to proceed. for (1), $f*f=f$ ...
2
votes
2answers
314 views

Is convolution operator compact?

I know convolution is not a Hilbert–Schmidt integral operator, but it needs more to tell if convolution is compact or not.
5
votes
2answers
404 views

Convolution of measures

We say that a family of measures $\mu_{t}\to \mu$ weakly if for any $g\in C_{0}$, $\int g d\mu_{t} \to \int g d\mu$. Show that if $\mu_{t}\to \mu$ weakly, then $\nu*\mu_{t}\to \nu*\mu$ weakly, where ...
2
votes
1answer
76 views

Convolution on noncommutative group algebras

If $G$ is a non-Abelian locally compact group, and $f$ is in $L^1{(G)}$ and $u$ is in $L^{\infty}(G)$, and $f\ast u=0$ can it be concluded that $u\ast f=0$?
5
votes
3answers
781 views

Convolution of compactly supported function with a locally integrable function is continuous?

Can someone show me the proof that the convolution of a compactly supported real valued function on $\mathbb{R}$ with a locally integrable function is also continuous? I feel that this is a standard ...
2
votes
1answer
123 views

Preservation of Lipschitz Constant by Convolutions

The following is a step in a proof: $f$ is a Lipschitz function from $E$ to $F$ where $E$ is a finite-dimensional Banach space and $F$ an arbitrary Banach space. $\phi\geq 0$ is a $C^\infty$ function ...
5
votes
3answers
531 views

Is $L^2(\mathbb{R})$ with convolution a Banach Algebra?

Is $L^2(\mathbb{R})$ a Banach algebra, with convolution? I am pretty sure the answer is no, because I think that $f,g \in L^2(\mathbb{R})$ does not imply that $f*g \in L^2(\mathbb{R})$. However, I ...