1
vote
0answers
41 views

Absolute continuity and convolution

Suppose that $\mu$ is a finite Borel measure on the real line, $f, g\in L^1(\mu)$. Define $\nu=\mu\ast\mu$. Do I understand correctly that the convolution $f\mu\ast g\mu$ is absolutely continuous wrt ...
0
votes
0answers
23 views

Proof commutativity of (differential) convolution operater

I tried to proof a claim and I'm not sure if I did it right. It would be great if someone could have a look at it! First I give a definiton: Let $h : [0, \infty ) \rightarrow \mathbb{R}$. We define ...
0
votes
1answer
37 views

Proof of a corollary about associativity of (differential) convolution operater

I m working on the proof the following corollary for ages... I would appreciate any help!! Cor: Let $h : [0, \infty ) \rightarrow \mathbb{R}$. We define the convolution operator $*$ for the ...
1
vote
1answer
192 views

The issue of treating an inverse Fourier transform in terms of a tempered distribution.

Consider the wave equation $$ u_{tt}=\Delta{u} \quad u(x,0)=f(x) \quad u_t(x,0)=g(x) \tag{*} $$ A solution to this equation is given by $$ u(.,t)=f*\partial_t\Phi_t+g*\Phi_t \tag{**} $$ where ...
4
votes
1answer
96 views

Convolution of a probability measure with a smooth function

If $f\in L^1(\mathbb{R}^n)$ and $g\in L^p(\mathbb{R}^n)$ then by Young's convolution inequality we have the estimate: $$ \|f*g\|_{L^p}\leq \|f\|_{L^1}\|g\|_{L^p}.$$ Question: Let $\mu$ be a ...
1
vote
1answer
125 views

$f$ is bounded and continious $\Rightarrow$ the convolution integral $\int f(\tau)g(x-\tau)\text{ d}\tau$ is bounded and continuous

Let $g\in L^1(\mathbb{R}^n)$ and $f:\mathbb{R}^n\to\mathbb{R}$ be bounded and continuous. Why is the convolution integral $$f*g:\mathbb{R}^n\to\mathbb{R}\;,\;\;\;\int f(\tau)g(x-\tau)\text{ d}\tau$$ ...
3
votes
1answer
96 views

Show that for any $f\in L^1$ and $g \in L^p(\mathbb R)$, $\lVert f ∗ g\rVert_p \leqslant \lVert f\rVert_1\lVert g\rVert_p$.

I write the exact statement of the problem: Show that for any $g \in L^1$ and $f ∈ L^p(\mathbb{R})$, p $\in (1, \infty)$, the integral for f ∗g converges absolutely almost everywhere and that $∥f ∗ ...
2
votes
1answer
112 views

Convolution of integrable function with bounded function

Let $H$ Lebesgue integrable. Let $f$ be measurable and bounded on $\mathbb{R}$ with $\lim_{\left|x\right|\rightarrow \pm \infty}f(x)=0$. Let $F(x)=K\ast f=\int_{\mathbb{R}} K(x-s)f(s) \, ds$ be the ...
1
vote
1answer
69 views

convergence of convolutions and approximation of unity

Let $\phi : \mathbb{R}\rightarrow \mathbb{R}$ be an integrable function with $\int \phi(x)dx = 1$. Let us define $\phi_\delta = \delta^{−1}\phi(\delta^{-1}x)$. Show that for every continuous ...
2
votes
1answer
81 views

Haar measure, convolution and involutions

I have some problems to follow the proof of the anti commutativity property of the convolution and involution operations defined using a Haar measure as presented in Pedersen's book "analysis Now", ...
4
votes
2answers
134 views

Let $S$ be the Schwartz class. Show that if $f,g\in S$, then $fg\in S$ and $f*g\in S$, where $*$ denotes convolution.

Let $S$ be the Schwartz class. Show that if $f,g\in S$, then $fg\in S$ and $f*g\in S$, where $*$ denotes convolution. To differentiate $fg$, we may apply Leibniz's rule ( ...
1
vote
0answers
70 views

Is the Convolution of a Schwartz Function with an $ L^{p} $-Function a Smooth $ L^{p} $-Function?

Let $ n \in \mathbb{N} $ and $ p \in \mathbb{R}_{\geq 1} $. If $ f \in \mathscr{S}(\mathbb{R}^{n}) $ and $ g \in {L^{p}}(\mathbb{R}^{n}) $, then it is a well-known fact from real analysis that the ...
3
votes
0answers
54 views

Convolution-like operator on (probability) measures on $[0,1]$ yielding measures on $[0,1]$.

Is there a "correct" or "best" way to define convolution of two (Borel) probability measures on $[0,1]$ to yield another probability measure on $[0,1]$? Recall that the convolution, $\mu * \nu$, of ...
2
votes
1answer
133 views

Convolution and uniform continuity

If $f\in L^{\infty}(\mathbb{R}^n)$ and $f$ is continuous at $x$, then $$\lim_{k\to\infty}(f*\phi_k)(x)=cf(x)$$ If $f\in L^{\infty}(\mathbb{R}^n)$ and is uniformly continuous, then $f*\phi_k\to cf$ ...
3
votes
1answer
72 views

Interesting inequality $\|F\|_p\le \frac{\pi}{\sin(\pi/p)}\|f\|_p$ over $L^p$

Consider the function $$F(x)=\int_0^\infty \frac{f(y)}{x+y} \, dy, \quad0<x<\infty$$ Prove that if $1<p<\infty$, $$\|F\|_p\le \frac{\pi}{\sin(\pi/p)}\|f\|_p$$ and show that the constant ...
0
votes
1answer
61 views

Give an example of $f\in L^1$, $g\in L^{\infty}$, such that $f*g\notin C_0$ (meaning $\lim_{|x|\to\infty}(f*g)(x)\neq0$)

Give an example of $f\in L^1$, $g\in L^{\infty}$, such that $f*g\notin C_0$ (meaning $\lim_{|x|\to\infty}(f*g)(x)\neq0$) Here's a theorem from my real analysis book: Assume $1\le p\le \infty$ and ...
0
votes
1answer
239 views

Convolution of functions with compact support

I have a question regarding convolution with compact support: Suppose $f \in L^1(\mathbb{R})$ and $g \in L^p(\mathbb{R})$, and both of them have compact support. Show that $f*g$ (convolution ...
4
votes
1answer
116 views

Convolution of an $L^1$ function and a function that tends to $0$ results in a function that tends to $0$

I'm trying to solve the following problem in review for a test, but have only partly succeeded: Let $K \in L^1(\mathbb{R})$ and $f$ be a bounded, measurable function on $\mathbb{R}$, with ...
2
votes
0answers
47 views

Weakest Conditions for Convolution to be Differentiable

I was going through various posts about differrentiability of convolutions. What I would like to ask is: Suppose $f \in C^{1}(\mathbb{R})$. Then what conditions on the function $g$ would ensure that ...
1
vote
2answers
278 views

Differentiability of convolution

First let me say that I have used the search bar and looked through all the "differentiability of convolution" questions that I saw, but none of them seem to cover this case. (If one of them did and I ...
1
vote
1answer
133 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
209 views

Convolution Exercise Homework

Put $\varphi(t)= 1- \cos \;t\;\;\;$ if $\;\;\;0 \leq t \leq 2 \pi$, $\varphi(t) = 0$ for all other real $t$. For $-\infty < x < \infty $, define $$ f(x)= 1,\;\;\;\;\;\;\;\;\;\;g(x) = ...
2
votes
1answer
65 views

Convolution of $L_1$ functions is $L_1$

Let $f,g\in L_1(\mathbb R, m)$ where m is the Lebesgue measure. Prove that: a) $f(x-t)g(t) \in L_1(\mathbb R, m)$ as a fuction of $t$ almost all $x$ b) $h\in L_1(\mathbb R, m)$ where $h=(f \ast ...
5
votes
1answer
108 views

Compactness of Convolutions of Compact Measures

This regards measures on $d$-dimensional Euclidean space $\mathbb R^d$ and their associated densities. A super-level set of a density $f : \mathbb R^d \to \mathbb R^+$ at level $t$ is the set $\{x \in ...
2
votes
3answers
426 views

Convolution of an integrable function of compact support with a bump function.

Let $f\in L^1(\mathbb{R})$ be of compact support and $\psi(x)=C \exp(-(1-x^2)^{-1})$ where $C$ is chosen so that $\int_{\mathbb{R}} \psi =1$. Show that the convolution $f*\psi(x)=\int_{\mathbb{R}} ...
8
votes
1answer
500 views

How to show convolution of an $L^p$ function and a Schwartz function is a Schwartz function

We have the Schwartz space $\mathcal{S}$ of $C^\infty(\mathbb{R^n})$ functions $h$ such that $(1+|x|^m)|\partial^\alpha h(x)|$ is bounded for all $m \in \mathbb{N_0}$ and all multi-indices $\alpha$. ...
1
vote
1answer
106 views

Show the convolution of a $C_c^\infty (\Bbb R^n)$ function with a $L^p(\Bbb R^n)$ function is in $C^\infty(\Bbb R^n)$, $1\le p\le\infty$

Let $f \in L^p\left(\Bbb R^n\right)$ and $g \in C_c^\infty \left(\Bbb R^n\right)$. Show $f \ast g \in C^\infty\left(\Bbb R^n\right)$ for $1 \le p \le \infty$. Let $x=(x_1,x_2,\ldots,x_n)$ and ...
0
votes
2answers
118 views

Show $\exists \{g_n\} \subset C_c^\infty$ such that $(f \ast g_n)(x) \to f(x)$ a.e., when $f \in L^1_\text{loc}$

Let $C_c^\infty$ denotes the set of real valued function with compact support. Show $\exists \{g_n\} \subset C_c^\infty$ such that $(f \ast g_n)(x) \to f(x)$ a.e., when $f \in L^1_\text{loc}$. If ...
5
votes
3answers
287 views

if convolution of $f$ with itself remains same, then $f=0$ a.e?

I'm trying to answer the question above.. But I'm not certain in either way. I tried to prove it by giving counter examples.. But it always failed.. Then i also tried to draw contradictions But ...
6
votes
2answers
773 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
137 views

Integrability of the function $f_1(y)f_2(x-y)$ for almost all $x$. (convolution)

What I'm trying to prove is that for $f_1,f_2 \in \mathcal{L}_1$ the function $y \mapsto f_1(y)f_2(x-y)$ is integrable for almost all $x$, or: $$ F(x) = \int f_1(y)f_2(x-y)\,dy < \infty \text{ ...