All Questions
Tagged with convolution fourier-analysis
329
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
0
votes
1
answer
55
views
Is convolution theorem on $l^2(\mathbb{Z})$ valid?
I have a doubt about Fourier transform $F:L^2([0,2\pi])\to l^2(\mathbb{Z})$. If $f,g\in l^2(\mathbb{Z})$ then $f*g\in l^2(\mathbb{Z})$, then, $\mathcal{F}^{-1}(f*g)\in L^2([0,2\pi])$.
Question $\...
- 3,650
0
votes
0
answers
80
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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
1
vote
0
answers
25
views
Classifying bounded linear functions satisfying convolution identity
Find all bounded linear functionals $T$ on $Y= \{f \in W^{1,1}[0,1]: f(0)=0\}$ such that there exists $K>0$ such that $$\int_0^1|Tf(\cdot-t)|\ dt\le K \|f\|_1\qquad \forall\ f\in Y\tag{1}$$
My ...
- 373
1
vote
1
answer
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
3
votes
1
answer
137
views
Properties of fourier series on $SO(3)$
With standard fourier series we can use some identities like convolution theorem and Parseval's theorem:
(convolution theorem) Fourier series of the convolution of $f$ and $g$ is the point-wise ...
- 555
0
votes
0
answers
17
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Prove that the derivative of the mollification approaches the strong $L^p$ derivative
Let $g \in C_c(\mathbb{R})$ be such that $\int_\mathbb{R} g dx = 1$. Suppose that $f \in L^p(\mathbb{R}$ has a strong $L^p$ derivative, i.e. there is some $h$ such that $\displaystyle\lim_{y \to 0} \...
- 3,800
1
vote
1
answer
33
views
Complex exponential Fourier coefficients of a convolution involving the exponetial function
In the book "Elementary Classical Analysis", by Marsden, the following is proposed as a worked example:
Let $f:[0,2\pi]\to\mathbb{R},g:[0,2\pi]\to\mathbb{R}$ and extend by periodicity. ...
0
votes
0
answers
35
views
How to show the limit of a series of convolutions exist?
Suppose $\rho:\mathbb{R}^d\to\mathbb{R}$ is an even, smooth test function, compactly supported in the ball of radius $1$. Define $\rho^{(n)}(x):=2^{nd}\rho(2^nx),$ and
$$\rho^{(n,m)}:=\rho^{(n)}\ast \...
- 329
0
votes
0
answers
43
views
Solution to convolution integral equation
I have two positive definite kernels $k_1,k_2 : \mathbb R^d \to \mathbb R$ and a probability measure $\rho$ on $\mathbb R^d$. The kernels are of the form $k_i(x,y)=K_i(x-y)$ for some function $K_i:\...
- 182
0
votes
0
answers
40
views
Convolution of sinc and dirac comb
I just learned about the dirac comb
$$ 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
3
votes
1
answer
98
views
Summability of the Fourier Transform.
I was reading the notes "Introduction to Complex Analysis" by Michael E. Taylor, and I am stuck on the following exercise about the Fourier Transform and the space $\mathcal{A}(\mathbb{R})$ ...
0
votes
0
answers
78
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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
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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