Questions on the (continuous or discrete) convolution of two functions. It can also be used for questions about convolution of distributions (in the Schartz's sense) or measures.

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1answer
403 views

Non-linear Systems, Impulse Responses, and Convolution

In linear system theory, it is easy to find the particular solution to differential equation by means of convolving the system's impulse response with the forcing function. My question is why can we ...
2
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1answer
39 views

Norm of convolution

Let $f,g: \mathbb{R}^n \to R$. Let $|| \cdot ||_T$ be a translation invariant norm on functions on $\mathbb{R}^n$. How can I prove that $||f*g||_T \leq ||f||_1 ||g||_T$ (where $f*g$ means convolution ...
0
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0answers
15 views

Singular integral operator on a decay Hölder space

Denote $$E^{k,m}=\{f\in C^m(\mathbb{R}^3)\mid\sup_{x\in\mathbb{R}^3}(1+|x|)^{k+|\alpha|}|\partial^\alpha_xf(x)|<\infty\}.$$ Now given $f\in E^{2,m}$ for any fixed $m\in \mathbb{Z}^+$, and $\hat\...
2
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0answers
31 views

Sum of two logarithmic random variables

I would like to compute the PDF of the difference of the logarithms of two shifted Rayleigh laws ($Z$): \begin{equation} Z = \log{X_{1}} - \log{X_{2}} \end{equation} where $X_1 \sim R(\alpha_1, \...
4
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1answer
38 views

Is it possible to express an integral equation inside of a convolution

Given $$u(t) = \int_0^t y(\tau) d\tau$$ Consider a convolution type of integral $$W = \int_0^t\lambda^{t-\tau}y(\tau) d\tau$$ $\lambda$ a positive real number Is it possible to write $W = f(u(...
2
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2answers
746 views

Convolution of cosine with exponential

As part of an exercise, I'm trying to find the output of a cosine wave entering a low-pass filter by using a convolution integral. The impulse response of the filter is $h(t) = \frac{1}{RC}\exp\left({...
0
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1answer
480 views

Relation between Correlation and Convolution

We have two functions of time $f(t)$ and $g(t)$, for which convolution and correlation are defined as following: Convolution: $(f(t)\ast g(t))(\tau) = \int_{-\infty}^\infty{f(t)g(\tau-t)dt}$ ...
1
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3answers
47 views

How to derive through a convolution?

Let $f(t) = \alpha e^{-\beta t}$, where $\alpha, \beta$ are constants Let $g(t) = y(t)$ Then the resulting convolution $f\ast g$ is: $$f \ast g = \int_0^t \alpha e^{-\beta (t-\tau)} y(\tau) d\tau$$...
1
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1answer
29 views

Showing that a “convolution” operator is associative

I am dealing with the following operator $*$ : $(A*B)(t) = \inf\limits_{\tau\in\mathbb{R}} (A(\tau) + B(t-\tau))$. I would like to show that it is associative, i.e : $((A*B)*C)(t) = (A*(B*C))(t)$ . I'...
1
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1answer
20 views

Questions about the proof: Continuous function with weak derivatives $\Rightarrow$ $C^1$

For an open set $\Omega$ of class $C^1$, suppose we have $u \in W^{1,p}(\Omega)$ and that $u$ is continuous and all the partial derivatives of $u$ are continuous. I want to show that $u$ is $C^1(\...
1
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2answers
43 views

Convolution of two rectangular pulses

Determine the shape of the following function$$\int^\infty_{-\infty} \Pi(4\tau) \Pi(t-\tau) d\tau$$ Attempt: This function is a convolution of two rectangular functions. I know that the result has ...
0
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1answer
32 views

A proof of the fact that the Fourier transform is not surjective

Let $f_n = \mathbb 1_{[-n,n]}$ for all $n \in \mathbb{N}$ 1) Compute explicitly $f_n \star f_1$ for all $n \in \mathbb{N}$. 2) Show that $f_n \star f_1$ is the Fourier transform of $g_n = \...
1
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0answers
9 views

Joint distribution of sum and summand

Let $Z_1$ and $Z_2$ be independent random variables with known distributions $F(.;\theta_1)$ and $F(.;\theta_2)$ of the same convolution closed family. Then $Y = Z_1 + Z_2$ has distribution $F(.;\...
0
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0answers
21 views

Approximate n^th power convolution

What is the approximation for $n^{th}$ convolution power (n-fold convolution) $g(x)= \underbrace{p * p * p * \cdots * p * p}_n$ with respsect to $p(x)$, where $p(x)$ is a probability density function?...
3
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1answer
55 views

Eigenfunctions of non-uniform convolution

Consider a non-uniform ("generalized"?) convolution operator: $$ A_h[f](t) = \int f(x)h(x,t)dx $$ I would like determine the eigenfunctions. In the "stationary" case where $h(x,t) = h(x-t)$ we have ...
1
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1answer
509 views

Distribution of the sum of many lognormal random numbers from same distribution

In my application I have to sum up a lot (between 1000 and 2000) lognormally distributed random numbers and use their sum. All random numbers that I sum up follow the same distribution. The current ...
1
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2answers
42 views

Convolution and Fourier transform problem

I was struggling with this question, can use some help. given that $a\not=0$ $$f_a(x) =\frac{1}{x^2+a^2}$$ I'm trying to find k and c dependent on a and b $$(f_a ∗ f_b) (x) = kf_c(x) $$ I know ...
0
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1answer
31 views

Writing an operator $T$ defined by $(T f)(t) = \int_{-\pi}^\pi h(t − s)f(s)ds$ as $\sum_{n \in \mathbb Z} \mu_n \langle f, \varphi_n\rangle \varphi_n$

Let $h$ be a continuous function with period $2\pi$. Define $T : L_2[−\pi, \pi] \to L_2[−\pi, \pi]$ by $(T f)(t) = \int \limits _{-\pi}^\pi h(t − s)f(s)ds$. Let $\{\varphi_n(t) =\frac{1}{\sqrt{2\pi}} ...
0
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0answers
25 views

Green's function moving dipole source

I am trying to calculate $$\vec{F}(\vec{r},t) = \iint \underline{G}(\vec{r}-\vec{r}',t-t') \frac{\partial}{\partial t'} \left[ \vec{p}(t') \delta(\vec{r}'-\vec{r}_0(t'))\right] d\vec{r}'dt'$$ for $\...
0
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0answers
8 views

any way to simplify discrete convolution $g[n]x[n] *g[n]x[n]$ to the form $Gp(x[n])$

Is there any way to simplify discrete convolution $g[n]x[n] * g[n]x[n]$ to the form $Gp(x[n])$ where $G$ is a matrix and $p$ is a feature map which can be computed quite fast. Note $p(x[n])$ must be ...
2
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0answers
60 views

Convolution proof

If I have two functions in a convolution like $$X*Y=1$$ $$X*Z=1$$ then it means (trivially) $Y=Z$. Is this correct or are there subtleties in the convolution theorem where $Y=Z$ isn't always true?
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2answers
2k views

Undecimated Wavelet Transform (a trous algorithm) - how to determine 'anchor'/'center' of convolution filter

i am currently implementing the 'Undecimated Wavelet Transform' with the 'a trous' algorithm. See e.g. http://www.znu.ac.ir/data/members/fazli_saeid/DIP/Paper/ISSUE2/04060954_2.pdf, section II-A. As ...
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0answers
20 views

derivative of convolution integral

I'm confused for a derivation related to the derivative of convolution. Given that $$ C_{im}(x,t)=\omega e^{-\omega t}*C_m(x,t)+C_{im}(x,0)e^{-\omega t} $$ By taking derivative of the above equation ...
5
votes
2answers
390 views

Number Theoretic Transform (NTT) example not working out

I'm reading up on the NTT, which is a generalisation of the DFT. I'm working in $\mathbb{F}_5$ with primitive root $w=2 \mod 5$. Suppose I want to compute the NTT of $x=(1,4)$. So far I have obtained: ...
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0answers
41 views

Is the convolution integration zero?

$$F(\omega)=\frac{\sin \omega}{\omega}$$ $$G(\omega)=\frac{\sin \omega}{\omega}e^{-j2\omega}$$ \begin{align} F(\omega )*G(\omega)&=\int^{+ \infty}_{-\infty} \frac{\sin \tau}{\tau}\frac{\sin (\...
-1
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1answer
23 views

Convolution of the cumulative normal distribution and the uniform distribution [closed]

What is the resulting function of convolving the cumulative normal distribution and the continuous uniform distribution?
4
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4answers
86 views

Trivial or not: Dirac delta function is the unit of convolution.

My task is to prove that the Dirac delta function is the unit of convolution and all I find always is this formula but no further explanation: $$[f*\delta](t)=\int_{-\infty}^{\infty}f(t-\sigma)\delta(...
0
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1answer
17 views

Uniform convergence of a sequence of functions given as product and convolution.

Suppose we have, for an open bounded set $\Omega \subset \mathbb{R}^n$: A function $u \in L^p(\mathbb{R}^n) \cap C(\mathbb{R}^n)$. A sequence of mollifiers $(\rho_n) \subset C_c^{\infty}(\mathbb{R}^...
2
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1answer
102 views

Show that if $\,u \in W^{1,p}\left(I\right) \bigcap C_c\left(I\right)$, then $\,u \in W_{0}^{1,p}\left(I\right)$.

I want to show the following statement ($1 \leq p < \infty$), for an open interval $I$: If $u \in W^{1,p}\left(I\right) \bigcap C_c\left(I\right)$ then $u \in W_{0}^{1,p}\left(I\right) $. $W^...
1
vote
1answer
455 views

Convolution: $ f (-)*g = g(-)* f$ does this mean both $f$ and $g$ have to be even functions?

Assuming $f$ and $g$ are different functions, does $ f (-)*g = g(-)* f$ mean both $f$ and $g$ have to be even functions? In fact, this is equivalent to $f\star g = g \star f$ (i.e., cross-correlation ...
1
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0answers
19 views

Fourier Convolution Inversion

Consider a Fourier convolution $f(x) = (g * h)(x)$, where $g$ and $h$ are arbitrary but known functions with reasonable properties. Is there any possibility to determine the inverse function of this ...
1
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2answers
73 views

Solving convolution problem with $\delta(x)$ function

Suppose we had the functions: $$g(t)=\theta(t)(e^{-t}+2e^{-2t})+2\delta(t)$$ and $$u(t)=2(\theta(t)-\theta(t-2))$$ Then we have $$u*g=\int_{-\infty}^{\infty}g(\tau)u(t-\tau)d\tau=2\int_{t-2}^{t}(e^{-\...
2
votes
2answers
42 views

Convolution of square function with itself

I have the square function $$f(x) = \begin{cases} 1, & 0 \leq x \leq 1 \\ 0, & \text{otherwise}\end{cases}$$ and I am trying to calculate the convolution $(f * f)(t)$ using the definition of ...
1
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1answer
29 views

Reverse of convolution theorem

If I have a convolution $$z(t) = x(t) * y(t)$$ where I know $x(t)$ and $z(t)$, is there a way to determine $y(t)$? Is there a "reverse" convolution theorem for this? I know there are numerical ...
3
votes
1answer
121 views

Convolution: How to construct it for a given function?

While working on my thesis my advisor handed me an unfinished paper which states the following: First, define the operators \begin{align*} A_i &:= -\operatorname{div}(\sigma_i\nabla) \\ A_e &...
2
votes
0answers
34 views

Result of a decay condition

Assuming that a function g is such that $ g(x) \leq C ( 1 + |x|)^{(-1 - \varepsilon)}$ for some $\varepsilon > 0$ , then how can we prove that $ \sum_{n = - \infty}^{n = + \infty} | g(x- k - \frac{...
2
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0answers
42 views

Convolution of a function and its inverse

I want to calculate the convolution of a function and its inverse, $$f(t) * f^{-1}(t)$$ e.g. $f(t)=1/(t-2i)$ I've heard that the answer can be a delta function. What requirements are necessary for $...
0
votes
4answers
41 views

Convolution: Give a proof that $f_T(t)=\int_{-\infty}^{\infty}f_X(x)f_Y(t-x)dx$ where $f_T(t)$ is the PDF of random variable T

Here is the question: Let $X$ and $Y$ be independent, continuous r.v.s with PDFs $f_X$ and $f_Y$ respectively, and let $T=X+Y$. Find the join PDF of $T$ and $X$, and use this to give a proof that $...
0
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0answers
19 views

Convolution involving the inverse Fourier transform

Suppose $$F(k) = \frac{1}{2\pi}\int f(x) e^{ikx} dx$$ and $$G(k) = \frac{1}{2\pi}\int g(x)e^{ikx} dx$$ Where $F(K),G(K)$ are Fourier transforms. Then how can I write the convolution of $F$ and $G$ ...
1
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1answer
59 views

The convolution is in $L^1$

According to my notes: The convolution is in $L^1$. Indeed $$\left| \int_{\mathbb{R}^n} \left( \int_{\mathbb{R}^n} f(y) g(x-y) dy\right) dx\right| \leq \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} |f(y) ...
0
votes
1answer
42 views

Convolution of Gaussian and error function

I am trying to evaluate the following integral: $$ \int_{-\infty}^{\infty}e^{-\frac{x^2}{2}}\Phi(x-t)dx $$ where $$ \Phi(y) = \frac{1}{2} + \frac{1}{2}erf\left(\frac{y}{\sqrt{2}}\right) $$ I have ...
0
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2answers
23 views

Alternative integration limits in a Laplace transform

The unilateral Laplace transform of $f(t)$ is $\int_0^\infty e^{st} f(t) \mathrm{d}t$. If we define the transform as $\int_{a}^\infty e^{st} f(t) \mathrm{d}t$, would it conserve all the nice ...
0
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2answers
28 views

DFT and windows

I am using DFT with windows. The way I understand how a window makes the DFT "look" better, is that multiplication in time domain is convolution in frequency domain. Therefore a window with following ...
0
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1answer
35 views

Continuity of characteristic function

Problem: Let $G$ be an open subset of $\mathbb{R}$. Show that $\chi_G$ is continuous on $G\cup(\mathbb{R}\backslash\overline{G})$. Consequently, $\chi_G$ is continuous a.e. on $\mathbb{R}$. My ...
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0answers
19 views

Can 2d convolution been represented as matrix multiplication?

Discr. convolution on a discrete periodic signal can be represented as multiplication of input with matrix M. Where M is presented a special case of Toeplitz matrices - circulant matrices. The ...
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0answers
19 views

Verify the Green's function for Helmholtz equations

It is well known that $$ G(x)=\frac{1}{4\pi}\frac{\exp(ik|x|)}{|x|} $$ is the Green's function for Helmholtz equation $$ (\Delta+k^2)f=0 $$ in $\mathbb{R}^3$. My question is, given $v\in C^0_b(\...
17
votes
4answers
3k views

Why convolution regularize functions?

There is a tool in mathematics that I have used a lot of times and I'm still not confortable with. In fact I can't figure out (by this I mean that I cannot understand it geometrically) why does ...
0
votes
0answers
26 views

Laplace transform of a convolution-like function

Is there a way to calculate the Laplace transform of the following function? $$ \sum_{k=1}^{+\infty}f(t-(g(t)-\theta_k))h(g(t)-\theta_k), \qquad t>0. $$ Thanks in advance.
2
votes
1answer
43 views

Convolution - Hölder inequality

I wonder if you guys can help me out with a question(not homework). I have $\phi(x)=\int_\mathbb{R} |f(t)g(x-t)|dt$ where $f \in L^1(\mathbb{R}) $ and $g \in L^p(\mathbb{R})$ and p and p' are ...