All Questions
Tagged with convolution distribution-theory
116
questions
1
vote
1
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76
views
Proof review Tempered Distributions: $(T*\varphi)^\wedge = (2\pi)^{n/2} \hat{\varphi}\hat{T}$ and computing $(x^\alpha)^\wedge$
I am learning about tempered distributions on schwartz space. But there are 2 questions in this text where Im unsatisfied with my solution... I ask kindly for your review of my solutions below, and if ...
- 3,266
1
vote
1
answer
66
views
Positive integral appears negative after applying the convolution theorem
I am Fourier transforming momentum representation of quantum mechanical wave functions to the position representation. In the weak sense (tempered distribution) we have
$$\int_{\mathbb{R}^3} d^3k \ e^{...
- 514
2
votes
0
answers
64
views
Problem with Mellin transform of convolution production formula
I have came across a case where the standard formula for the Mellin transform of a convolution of two functions is not the product of the Mellin transforms and I don't understand exactly where the ...
- 21
0
votes
1
answer
53
views
Rate of convergence of mollified exponential on $[0, \infty)$
Consider the function
$$
f(t)=e^{-at}
$$
with $a>0$, $t \in [0, \infty)$ and let $\rho$ a smooth function on $[0, \infty)$ such that
$$
\int_0^\infty \rho(t)=1
$$
and let $\rho_\delta(t):= \frac{1}{...
- 2,675
0
votes
0
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43
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Can the convolution of distributions be defined for distributions "whose Fourier transform is non-smooth on null set"
In this https://web.abo.fi/fak/mnf/mate/kurser/fourieranalys/chap3.pdf Section 3.7: Convolutions (Page 79) lecture notes, it is said that the fourier transform of a convolution of distributions $\...
- 165
0
votes
2
answers
46
views
How to linearize time-variant ordinary differential equation (ODE), in particular, if there is a convolution term?
Letting $(\ * \ )$ denote convolution, I have an ODE of the form
\begin{align}
\dot{r} &= -7.4r -1.6f - 8.8 (f(t)*e^{-t}) - 10.4(f(t)*(te^{-t})) \\
\dot{f} &= 0.25r
\end{align}
with initial ...
- 286
0
votes
0
answers
26
views
Proving that $\rho_\varepsilon\ast U\to U$ as $\varepsilon\to 0^+$
I'm trying to understand the proof of the following fact about tempered distributions. Let $U\in \mathscr{S}'$ be a tempered distribution, and let $\{\rho_\varepsilon\}$ be the family of standard ...
- 3,536
0
votes
0
answers
54
views
Convolution of two characteristic functions
Given two intervals on XY-plane: $I_1$
lying on OX-axis, $I_2$
lying on OY-axis. Compute convolution of two characteristic functions of these intervals.
$$\chi_{I_1} \ast \chi_{I_2}.$$
MY ATTEMPT:
$(\...
- 39
1
vote
0
answers
73
views
Is the convolution of a tempered distribution and a Schwartz function also a function?
Let $T \in \mathscr{S}'(\mathbb{R}^n)$ and $f \in \mathscr{S}(\mathbb{R}^n)$. We may define their convolution as
$$(T * f)(\varphi) = T(\tilde{f}*\varphi)$$
where $\tilde{f}(x) = f(-x)$. The above ...
- 6,275
0
votes
1
answer
40
views
Calculate the limit of $\langle \partial_x f,\rho_\varepsilon \ast \chi_B \rangle $
Let $f(x_1,x_2)=x_2^2\chi_E$ where $E=\{(x_1,x_2)\in \mathbb{R}^2:x_1\geq 0 \}$ and let $B$ be the unit ball in $\mathbb{R}^2$. Since $f$ is locally integrable, it can be interpreted as a distribution ...
- 3,536
1
vote
1
answer
131
views
Understanding of Fourier transform of the convolution of two distribution
In Yosida's Functional Analysis, he covers a topic on the Fourier transform of the convolution given by distributions (at the end of Section 3, Chapter 6). He begins by pointing out the Fourier ...
- 440
2
votes
0
answers
56
views
Inverse Fourier Transform of product of two functions in $L^p$
If $f \in L^{q}$ for all $1 \leq q \leq \infty$ and $g \in L^{p}$ for some $1<p \leq 2$, how is it possible to get that
\begin{equation}
\mathcal{F}^{-1}(f \mathcal{F}(g))(x)=\mathcal{F}^{-1}f*g(x)?...
- 119
4
votes
1
answer
106
views
Is the convolution of the Heaviside distribution with itself well defined?
During a lecture, we were told that for two distributions $S,T \in \mathcal{D'}(\mathbb{R})$ the convolution $S* T$ is defined if either $S$ or $T$ had a compact support, or, if both of them were ...
- 197
3
votes
0
answers
105
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Are there functions with a Fourier transform but no Laplace transform?
Let
$$A = \text{the collection of functions with a Fourier transform}$$
and
$$B = \text{the collection of functions with a Laplace transform.}$$
What is the relationship between $A$ and $B$? Based on ...
- 75
0
votes
1
answer
163
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Tempered distribution by convolution with singular kernel
I am trying to learn more about convolutions with singularities and was hoping that someone could point me to a reference and perhaps check the following (i have it in my notes that this is from a MSE ...
- 505