All Questions
Tagged with convolution measure-theory
169
questions
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
0
votes
0
answers
43
views
Solution to convolution integral equation
I have two positive definite kernels $k_1,k_2 : \mathbb R^d \to \mathbb R$ and a probability measure $\rho$ on $\mathbb R^d$. The kernels are of the form $k_i(x,y)=K_i(x-y)$ for some function $K_i:\...
- 182
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:
$$ | (h * f^2)...
- 1,813
0
votes
0
answers
43
views
Can the convolution of distributions be defined for distributions "whose Fourier transform is non-smooth on null set"
In this https://web.abo.fi/fak/mnf/mate/kurser/fourieranalys/chap3.pdf Section 3.7: Convolutions (Page 79) lecture notes, it is said that the fourier transform of a convolution of distributions $\...
- 165
1
vote
0
answers
43
views
Strict positivity of convolution integral
Let $K : \mathbb R^n \to (0,\infty)$ be radially symmetric function such that $\|K\|_{L^1(\mathbb R^n)}=1$ and let $T: L^2(\mathbb R^n)\to L^2(\mathbb R^n)$ be defined as $u\mapsto T \star u$.
...
- 373
1
vote
1
answer
78
views
Convolution of random variables - confusion about change of variables
Consider
I understand everything except of the first line of the note. In particular
$$
\int_\Omega 1_{X(\omega ) + Y (\omega )\in A} \;d \mathbb P(\omega ) =\iint \mathbf{1}_A (x+y) \; \;d(\mu, \nu)...
- 2,302
1
vote
0
answers
28
views
Question about step in proof of Young's convolution Inequality
This is the inequality we are attempting to prove: Let $f \in \mathcal{L}^p(\mathbb{R}^n), \ g \in \mathcal{L}^1(\mathbb{R}^n)$ and $1 \leq p < \infty$. Then $f * g$ is defined almost everywhere ...
- 11
0
votes
1
answer
39
views
Inequalities with a normal convolution
Let $\mathbb{P}$ be any probability measure concentrated on an interval $[-1, 1]$. Why is it that for $\varphi$, the standard normal density, the following holds?
$$ (\varphi * \mathbb{P}) (x) = \int_{...
- 177
1
vote
1
answer
190
views
Fourier transform of a convolution of a measure and mollifier converges uniformly to the fourier transform of the measure
Let $\psi$ be a mollifier with $spt(\psi) \subset B_1(0)$. Define $\psi_\epsilon$ as usual. Now let $\mu$ be a finite borel measure. Define $\mu_\epsilon = \psi_\epsilon * \mu$. Then $\hat{\mu_\...
- 438
1
vote
1
answer
107
views
Show the renewal function is differentiable
Definition.
For a sequence of $\text{i.i.d.}$ random variables $ \left\{X_{k},k\ge 1 \right \},$called
inter-renewal times, let $S_{n}=\sum_{k=1}^{n}X_{k},n\ge 0,$ be the time instant of the $n$th ...
- 204
2
votes
1
answer
158
views
If $f*g(x)=0$ then is $f$ identically zero?
Let $f$ be a function in $L^1(\mathbb R).$ The convolution of $f$ and $g$ is defined by $f * g(x) = \int_{\mathbb R} f(x-t)g(t)dt.$
Let $g(x) = e^{-x^2}.$
Suppose it is given that $f*g(x)=0$ for all $...
- 751
4
votes
1
answer
99
views
Prove that either $m(E)=0$ or $m(\mathbb{R}\setminus E)=0$.
$\textbf{Problem :}$
Let $E\subset\mathbb{R}$ be a Lebesgue measurable set such that $m(E\setminus(E+x))=0$ for all $x\in\mathbb{R}$ ; where $E+x:=\{e+x:e\in E\}$.
Prove that either $m(E)=0$ or $m(\...
- 850
1
vote
0
answers
38
views
How do you express convolution of measures in term of functionals?
This may be a silly question and I am certainly missing something obvious, but I can't figure it out at the moment. Following Rudin's RCA, Theorem 6.19, the space $M(\mathbb{R})$ of regular Borel ...
- 176
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
1
vote
0
answers
104
views
Showing that the convolution of two Borel measures is a Borel measure
A book I am reading (Mattila's Fourier Analysis and Hausdorff Dimension) defines the convolution of two Borel measures $\mu, \nu$ over $\mathbb{R}^n$ sort of implicitly as $\int_{\mathbb{R}^n}\varphi(...
- 1,221