All Questions
Tagged with convolution fourier-transform
240
questions
4
votes
1
answer
73
views
Evaluating the convolution of $e^{-at^2}$ and $e^{-bt^2}$ via Fourier transforms
Problem Statement: Use the convolution theorem on the function $ f(t) = e^{-at^2} $ and $ f(t) = e^{-bt^2} $, $ a, b \in \mathbb{R} $. Calculate $ (f \ast g)(t) $.
I got a hint that I should first ...
- 21
-1
votes
1
answer
65
views
Inverse Fourier Transform - convolution of exponential and rectangular window
I'm trying to get the response in the time domain of the convolution between the exponential $u(t)e^{-at}$ and the rectangular window ($u(t+1)-u(t-1)$). I had already obtained its result by ...
0
votes
0
answers
80
views
The Fourier transform of product of derivatives
I have the task to compute the Fourier transform of the product in matlab:
$$ \left( \frac{\partial u(t, x)}{\partial x} \right)^2 \left( \frac{\partial^2 u(t, x)}{\partial x^2 } \right)$$
I was ...
- 391
0
votes
0
answers
19
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Weighted inequality on torus
In the Torus (circle). Let $[0,2\pi]\to\mathbb ]0,\infty[\colon \theta\mapsto w(\theta)$ a weight function, i.e. nonnegative and integrable on $[0,2\pi]$. If $\mathbb{Z}\to\mathbb{R}\colon k\mapsto m(...
- 3,650
1
vote
3
answers
101
views
Try to give the solution of PDE with initial boundary
The equation is
\begin{align}
\partial_{t}\!\operatorname{u}\!\left(x,t\right) & =
x^{2}\,\partial_{x}^{2}\operatorname{u}(x,t) +
x\,\partial_{x}\operatorname{u}\left(x,t\right),\quad\quad(t,x)\in\...
- 67
0
votes
0
answers
34
views
Fourier transform with and without convolution theorem not equivalent
This is a problem involving Fourier transforming an integral relevant to the computation of Feynman diagrams, which is of the form:
$S(r_1,r_2)=\int d^3 r_3 \space v(r_1,r_3)f(r_3,r_2),$ where $v(r_1,...
- 13
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
$$\int_{\mathbb{R}^3} d^3k \ e^{...
- 514
1
vote
1
answer
45
views
Fourier Transform of a Triangular function (using the formula and the convolution theorem)
I am studying physics at a degree level, and I was given a seemingly simple question: FT of a triangular function. $\\f(x)=\frac{1}{a}(a-|x|)$ and I needed to show $\DeclareMathOperator{\sinc}{sinc} \...
- 21
4
votes
1
answer
202
views
Convolution theorem: proof via integral of Fourier transforms
Given two functions $f$ and $g$, their convolution is defined by:
$$
(f \ast g) (t) = \int_{-\infty}^{\infty} f(x) g(t-x) dx \tag{1}\label{conv}
$$
The convolution theorem states that:
$$
F(f \ast g) (...
- 233
0
votes
0
answers
40
views
Convolution of sinc and dirac comb
I just learned about the dirac comb
$$ Ш_T(t) = \sum_n \delta(t-nT) $$
and wanted to use it and the convolution theorem to understand the spectrum of a train of sinc pulses
$$ x(t) = \text{sinc}(t)* ...
- 1
0
votes
0
answers
71
views
Fourier transform and convolution of product of Heaviside and indicator of a ball
Let $r > 0$ and $\mathbf{w}\in\mathbb{R}^n$, and denote by $rB_n$ the ball of radius $r$ in $n$ dimensions. Consider the functions: $f_1(\mathbf{x}) = \mathbf{1}(\langle\mathbf{w},\mathbf{x}\rangle ...
- 993
0
votes
0
answers
78
views
Relationship between Fourier inversion theorem and convergence of "nested Fourier series representations" of $f(x)$
In this question I use the term "nested Fourier series representation" to refer to an infinite series of one or more Fourier series versus a single Fourier series. Whereas a single Fourier ...
- 7,631
0
votes
1
answer
27
views
Computing the 2D greyscale image produced from putting a row and column of zeros between every two rows and two columns in the Fourier transform $F$
I have the 2 dimensional image $f(x,y)$ of dimensions $K\cdot T$, and there is its Fourier transform $F(u,v)$.
Now, I want to compute the image $g(x,y)$ which is of dimensions $(2K)\cdot(2T)$, which ...
- 1,999
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
2
votes
1
answer
96
views
If $f \in L^p(\Bbb R)$ for $1 < p\le 2$, then $f\ast \frac{\sin \pi t}{\pi t} \in L^p(\Bbb R)$.
I'd like help with the following problem:
Let $f\in L^p(\Bbb R)$ for $1 < p\le 2$. Show that $f(t)\ast \frac{\sin \pi t}{\pi t}$ is defined, and $f(t)\ast \frac{\sin \pi t}{\pi t} \in L^p(\Bbb R)$....
- 14.2k