All Questions
Tagged with convolution harmonic-analysis
15 questions
3
votes
1
answer
307
views
Approximate square root of Dirac delta function on $\mathrm{SL}_2(\mathbb{R})$
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\AdS{AdS}$I hope to find a sequence of complex-valued functions $\{f_i(g)\}$ on the group element $g$ of a locally compact group $\SL(2,\mathbb{R})$ so ...
- 175
0
votes
0
answers
96
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Derivative bounds for self convolution of the spherical measure in $R^d$
While reading this article on near $L^1$ estimates for the spherical lacunary maximal function, I came across the estimate
$$ |\partial^{\gamma} (\widetilde{\sigma} \ast \sigma)(x)| \lesssim |x|^{-(1 +...
- 1
6
votes
1
answer
343
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Integral convolution equation $\int_{B_n(R) } e^{- \| x - t\|} d\nu(t) = e^{- \|x \|^2/2}$ on $x \in B_n(R)$. Find measure $\nu$
Let $B_n(R)$ denote the $n$ ball centered at zero with radius $R$. We are interested in the following integral equation: given $R>0$ and $\lambda>0$, let
\begin{align}
\int_{B_n(R)} e^{- \...
- 671
5
votes
1
answer
510
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Recent progress restriction conjecture - Problem 2.7 (Terence Tao lecture notes)
I've been tackling the following problem for some time,
Problem 2.7. (a) Let $S:=\left\{(x, y) \in \mathbf{R}_{+} \times \mathbf{R}_{+}: x^2+y^2=1\right\}$ be a quarter-circle. Let $R \geq 1$, and ...
0
votes
1
answer
161
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Does convolution with $(1+|x|)^{-n}$ define an operator $L^p(\mathbb R^n) \to L^p(\mathbb R^n)$
Suppose that $f : \mathbb R^n \to \mathbb R^n$ is a locally integrable function. I am interested in the integral
$$ x \to \int_{\mathbb R^n} ( 1 + |y| )^{-n} f(x-y) \;dy $$
If the decay of the ...
- 641
0
votes
0
answers
23
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Is there a classification of 2D projective convolution kernels?
Is there any classification of all distributions on $\mathbb{R}^2$ such that they are equal to the convolution with themselves? i.e. given a distribution $\gamma$ under which conditions
$$ \gamma\star\...
3
votes
1
answer
667
views
Equivalent action of convolution of directional derivative
I have asked this question a while back on StackExchange but have not received any answer/comment. I received a suggestion to post the same question in here which is more research oriented.
Let $k*f(x)...
- 350
2
votes
0
answers
112
views
Anticommutation of convolution products on trace class operators of quantum groups
This question was originally posted to MathStackExchange.
Let $\mathbb{G}$ be a locally compact quantum group and let $W$ and $V$ be the left and right fundamental unitaries, i.e., they implement the ...
- 59
4
votes
1
answer
2k
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Norm of convolution operator
By Young's inequality for any $f\in L^p(\mathbf{R})$ the map $T_f:g\mapsto f\star g$ is a continuous operator from $L^q(\mathbf{R})$ to $L^r(\mathbf{R})$ where $1\leq p,q,r\leq \infty$ satisfy $1+\...
- 3,425
3
votes
1
answer
264
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Convolution of ball measures
It is well known that convolution of two ball measures (i.e. a uniform measure over a ball) in $\mathbb{R}^{n}$ is absolutely continuous with respect to the Lebesgue measure.
My question is - how to ...
- 39
3
votes
1
answer
167
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When are all the convolution roots of an infinitely divisible probability measure infinitely divisible?
Let $G$ be a topological group.
Let us say that a probability measure $\mu$ on $G$ is strongly infinitely divisible (SID) if $\mu$ is infinitely divisible and any probability measure $\nu$ on $G$ ...
- 128k
2
votes
0
answers
200
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Some detail in Fefferman's thesis
Recently, I am reading "Inequalities for strongly singular convolution operator" written by Fefferman. I have some question on the detail of proof of Theorem 2'.
Let $\theta \in (0,1)$.
Let $f \in ...
- 121
3
votes
0
answers
267
views
Link between standard convolution and Day convolution
There is a notion of convolution product between two functors called "Day convolution". (See here nlab for instance) I know that the definition of this notion is inspired by the discrete convolution $$...
- 1,017
1
vote
0
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154
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Convolution in Hardy spaces
Question Are there non-trivial restrictions on the coefficients of functions in Hardy spaces ($H_p(\mathbb{D})$, $p<1$) that make a subspace that is closed under convolution?
Definition The Hardy ...
- 323
3
votes
0
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741
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Justification of the convolution operation of $L^1(G)$ functions where $G$ is a LCA group (measurability)
Suppose $G$ is a locally compact abelian Hausdorff group (LCA), and $\lambda$ is the Haar measure on it. We all know the convolution of two $L^1(\lambda)$ functions $f$ and $g$ on $G$ is defined as
$$...
- 960