All Questions
Tagged with convolution lebesgue-integral
59
questions
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
0
votes
0
answers
47
views
$g: R \to R$ is integrable and has compact support. $f(y) =sin^2(y) ln|y|$. Show that $(f ∗ g)(x)$ is well defined for $x \in R$ and of class $C^1$?
$g : \mathbb{R} \to \mathbb{R}$ is integrable over $\mathbb{R}$ and has a compact support.
$f(y) = \sin^2(y) \ln|y|$
Show that the convolution $(f ∗ g)(x)$ is well defined for $x \in \mathbb{R}$ and ...
- 1,096
4
votes
1
answer
81
views
Convergence of the difference of the convolution of sequence of functions and a function
Let $(g_j)_{j \in \mathbb{N}} \subseteq \mathscr{L}_1(\mathbb{R}^n)$ with $\|g_j\| = 1$ and $g_j \geq 0$ for all $j \in \mathbb{N}$. Suppose that $\lim_{j \to \infty} d_j = 0$ where $d_j := \sup\{\|x\|...
- 115
1
vote
0
answers
119
views
Define multiplication on $L^1(\mathbb{R})$
I want to define multiplication on $L^1(\mathbb{R})$, of course, convolution is an allowed multiplication which makes $(L^1(\mathbb{R}),*)$ is a Banach algebra, I want to know if there are other ...
- 11
2
votes
1
answer
75
views
Differentiability of convolution without compact support
Let $g \in C^1(\mathbb R), f, g, g' \in L^1(\mathbb R)$.
How do I then show $f * g$ is differentiable? Variations of this question have been asked where $g$ or $g'$ have compact support, but these ...
- 878
0
votes
0
answers
82
views
About the Convex Hull of a the image of a function
Let $ f \in L^1_{loc}(\mathbb R^n)$ and $ \varphi $ a mollifier, consider $\varphi_{\varepsilon}:=\frac{1}{\varepsilon ^n} \varphi \left( \frac{x}{\varepsilon} \right)$ show that $f * \varphi_{\...
- 311
2
votes
0
answers
37
views
Prove that if either $f \in (L^p \cap C^m)(\mathbb{R}^n)$, or $g \in (L^q \cap C^m)(\mathbb{R}^n)$, then $f \ast g \in C^m$
Let $1 \leq p \leq \infty$, with $\frac{1}{p} +\frac{1}{q}=1$. Show that if $f \in L^p(\mathbb{R}^n)$ and $g \in L^q(\mathbb{R}^n)$, then
a) The convolution $f\ast g$ is bounded and continuous on $\...
- 1,575
1
vote
1
answer
124
views
Compound Poisson Distribution and its Expected Value
I am trying to understand the proof of Theorem 16.14 of Probability Theory by A. Klenke (3rd version) about the Levy-Khinchin formula.
I would like to know how to prove this:
$$E[X]=\int x e^{-v(\...
- 563
3
votes
0
answers
79
views
Differentiating under the integral sign in $n$ variables to show $\partial^{\alpha}(f*g)=f*\partial^{\alpha}g$
As an easy corollary of Dominated Convergence theorem, Folland gives a criterion for differentiating under the integral sign and uses it to prove Proposition $8.10$:
Proposition $8.10:$ If $f \in L^1$...
- 1,777
0
votes
1
answer
470
views
Uniform Convergence of Convolution of $f$ and Good Kernel
My task is to prove that if $f$ is continuous on $\mathbb{R}^d$ with compact support and $\{K_\delta\}_{\delta>0}$ is a family of good kernels then $f \ast K_\delta \xrightarrow{\text{unif.}} f$ as ...
3
votes
0
answers
92
views
Convolution derivative, $(f*g)'=f*g'$
Let $f,g \in L^1$ and $g$ is differentiable and $g' \in L^1$, then $(f*g)'=f*g'$
PROOF: For this proof I am asked to use the dominated convergence theorem
It can be quickly arrived at that
$$\begin{...
- 178
1
vote
0
answers
28
views
A Convolution Type Integral is Convergent in Limit
I am having trouble with solving the following problem:
Let $\alpha \in (0, 1)$ and $T: [0, \infty) \to [0, \infty)$ be a continuous function with $\lim_{x \to \infty} x^\alpha T(x) = a_0 \in (0, \...
- 593
1
vote
0
answers
44
views
$\|\chi_A*\chi_B\|_{\mathbb L^\infty}\leq \min\{\lambda(A),\lambda(B)\}$
Let $A,B \subseteq \mathbb R^d$ lebesgue measurable with finite Lebesgue measure and $f=\chi_A*\chi_B$
Show that $\|f\|_{L^\infty}\leq \min\{\lambda(A),\lambda(B)\}$
I know that $\|f\|_\infty\leq \...
- 242
1
vote
1
answer
161
views
prove that $L^1$ is not unital under the convolution product
Prove that $L^1([-\pi, \pi))$ (the space of integrable $2\pi$ periodic functions using the norm $\lVert f\rVert_1 = \frac{1}{2\pi}\int_{-\pi}^\pi |f(x)|dx$) is not unital under the convolution product....
- 1,166
0
votes
2
answers
111
views
convolution of integrable functions
Prove that the convolution product $\star$ on $L^1 ([-\pi,\pi])$ ($2\pi$ periodic Lebesgue integrable functions on $\mathbb{R}$ with respect to the $L^1$ norm satisfies $f\star g(\theta) = \frac{1}{2\...
- 1,166