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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2
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237 views

Need help with the convolution of two complex functions

Could someone start me off with how to find the convolution of these two functions? Using the normal equation for convolution seems impossible as a common overlap interval is required for ...
2
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0answers
160 views

Show that convolution satisfies partial differential equation

Consider the equation $$\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2} + a\frac{\partial u}{\partial x}$$ for a function $u(x,t)$ with initial value $$u(x,0)=f(x).$$ Let ...
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0answers
55 views

Inverse Fourier transform to get convolution

Consider the equation $$\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2} + a\frac{\partial u}{\partial x}$$ for a function $u(x,t)$ with initial value $u(x,0)=f(x).$ Let ...
2
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0answers
577 views

How Heaviside step function changes limits of integration

This question involves the Laplace transform of the convolution of two functions. The derivation in my textbook has a step that really confuses me. First I'll lay out their argument. $$ f(t) = f_1(t) ...
2
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0answers
242 views

Proof of a method to find the points of maximum slope

According to method described in a paper [1] if we want to find points of maximum slope in a signal $f(t)$, then one has to do following Convolve $f(t)$ with $g(t)$ where $g(t)=-cos(\omega ...
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0answers
324 views

Convolution of compactly supported functions

Let $f,g : \mathbb R \rightarrow \mathbb R$ continuous and compactly supported. I want to show that $f*g$ is continuous and compactly supported. I am 100% sure how to do it. I began as follows: ...
2
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0answers
85 views

Solving Dirichlet problem by means of potential theory

Let $\Omega\subset\mathbb{R}^N$ be a bounded smooth domain and consider the Dirichlet problem with $f\in H^{-1}(\Omega)$ $$\tag{1}-\Delta u=f$$ Is there a way to solve this problem by using ...
2
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0answers
387 views

How to express multiplication of two spherical harmonics expansions in terms of their coefficients?

Consider a spherical harmonics expansion/series like this: $$f(\theta,\varphi)=\sum_{\ell=0}^\infty \sum_{m=-\ell}^\ell f_\ell^m \, Y_\ell^m(\theta,\varphi)$$ Presumably if we take two functions on ...
2
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1answer
62 views

Asymptotics at the origin of the convolution with an approximation to the identity.

In short, I am trying to find sufficient conditions for an approximation to the identity function $K_h$ so that, for $h$ small enough and fixed, the asymptotics at the origin of an $L^1 \cap L^2$ ...
2
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1answer
143 views

The differentiability of convolutions

Yes, again, this type of question. Similar ones this and this. I come with another variant. Let $f\in\mathcal{S}$, i.e. Schwartz function, and $g\in L^{p}(\mathbb{R}^d),p\in[1,\infty]$. The following ...
2
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2answers
180 views

What if the cauchy product of two series in $\mathbf{Z}$ is null

I have a problem I do not find a solution. Given two series $\left(a_n\right)_{n \in \mathbf{Z}}$ and $\left(b_n\right)_{n \in \mathbf{Z}}$ which have a cauchy product $\left(c_n\right)_{n \in ...
2
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1answer
134 views

On using fourier transforms to solve the root of a convolution

In continuation of Lower bounds of laplace transform of characteristic functions. My question is: Can anyone point out where i'm going wrong in the derivation below. It's been a while ...
2
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0answers
129 views

General approach to prove the smoothness of convolution

Consider $\mathbb{R}^d$ and $D_i = \partial /\partial x_i$, in many cases $$D_i(f*g) = D_if*g,$$ given one of $f,g$ is smooth and the other is $L^p$ integrable. I am wondering if there is a general ...
2
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1answer
94 views

clarification asked for 'difference between convolution and crosscorrelation?'

I don't understand answer formulated in ways like this "Thus, $p\ast q$ is the distribution of $X+Y$. The cross-correlation $p\circ q$ is the distribution $c=(c_n)_n$ defined by ...
2
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2answers
190 views

convolution computation involving $e^{-x^2}$

In working a problem involving convolution, I have arrived at the following integral, but do not know how to compute it: $$2\int_0^{\infty}e^{-a(x-y)^2-by^2}dy$$ I thought that this integrand did not ...
2
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2answers
214 views

Understanding convolution

Take: $$ (u*v)(k) = \sum_{i=-\infty}^\infty u(i)v(k-i). $$ The $k$ is there, it's because you want to define $$ \ldots\ldots, (u*v)(-3), (u*v)(-2), (u*v)(-1), (u*v)(0), (u*v)(1), (u*v)(2), ...
2
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0answers
99 views

Convolution and Smoothness Conditions

Suppose $f(x),g(x)\in L_1(\mathbb{R})$, with both $|f(x)| \leq 1$, $|g(x)| \leq 1$ and $|f(x)| \rightarrow 0$, $|g(x)| \rightarrow 0$ for $|x| \rightarrow \infty$. Given that we have two other ...
2
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0answers
67 views

Show compactness of an evolution operator

Consider the heat equation $$ u_{t}=u_{xx},~~~~~u_0(x)=u(0,x)$$ with $u\colon [0,T]\times\mathbb{R}\to\mathbb{R}, (t,x)\mapsto u(t,x)$ and the evolution operator $E(T)$ with $E(T)u_0=u(T,x)$. 1.) ...
2
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0answers
241 views

Poisson exponentiation distribution family and convolution

Assume $\xi_i \sim \mathbb{F}_{\lambda_i}(x)$ are random variables from Poisson distribution. Consider random variables $\eta_i \sim \tilde{F}_{\lambda_i,t}(x)$, where $\tilde{F}_{\lambda_i,t}(x) = ...
2
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0answers
96 views

bound on Hilbert transform

Consider $\widehat{Tf(\xi)}=m(\xi)\hat{f}(\xi)$, where $m(\xi)=(1-\vert\xi\vert)1_{[-1,1]}$, i.e. $T$ is the operation of taking Fourier transform and multiplying with the function $m(\xi)$. I am ...
2
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1answer
327 views

Convolution converges in infinity norm?

Assume $\phi$ to be a nonnegative continuous function on the real line with compact support. Also assume that integral of $\phi$ over $\mathbb{R}$ is normalized to $1$. Let $\phi_e(x) = ...
2
votes
2answers
490 views

Convolution of irreducible characters of a finite group

If $\chi^{\lambda}$ and $\chi^{\mu}$ are the characters of two irreducible representations $V^{\lambda}$ and $V^{\mu}$ of a finite group $G$, is there a simple way of proving that : $$ \chi^{\lambda} ...
2
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0answers
262 views

FFT signal post processing

This is more a "post a suggestion" topic rather than a question. And thank you if you are willing to read this whole. I've been studing the code in the Nvidia Cuda SDK regarding how to operate a ...
2
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1answer
43 views

Calc $\frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty -\frac{t}{1+t^2}(\delta (\omega-t-\pi)-\delta(\omega-t+\pi))dt$

The answer to this integral:$$\frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty -\frac{t}{1+t^2}(\delta (\omega-t-\pi)-\delta(\omega-t+\pi))dt$$ is ...
1
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1answer
360 views

What's the difference between convolution and crosscorrelation?

What's the difference between convolution and crosscorrelation? So why do you use '-' for convolution and '+' for crosscorrelation? Why do we need the "time reversal on one of the inputs" when doing ...
1
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2answers
178 views

Convolution with sign function

I am having some trouble calculating the convolution $ (f*g)(t) $ between these two functions: $$ f(t)=e^{-t}1(t) $$ where $1(t)$ is the unit step function, and $$ g(t)=\mathrm{sgn}(t) $$ Using ...
1
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1answer
59 views

Solving $ F_{n} = \sum_{i=1}^{n-1} (F_{i}\cdot F_{n-i}) $?

I need to find $F_{n}$ in : $$ F_{n} = \sum_{i=1}^{n-1} (F_{i}\cdot F_{n-i}) , F_0 = 0 , n>=2 $$ This equation screams convolution , I think , but I find it as a quite long solution sometimes. ...
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3answers
3k views

Exact deconvolution of two matrices using numerical techniques

Suppose that I am given two $n \times m$ matrices $\bf{A}$ and $\bf{C}$, and let $\bf{B}$ be a matrix that is convolved with $\bf{A}$, such that: $\bf{A} * B = C$ In the above, $*$ is the ...
1
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2answers
35 views

convolution of $\delta(t+4) \ast \delta(t-1)$?

How does one solve convolution of $\delta(t+4) \ast \delta(t-1)$ where $\delta$ is dirac delta function? In ordinary function convolution, tricks are obvious but does dirac delta function share the ...
1
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1answer
38 views

What is the generalization for a convolution in $\mathbb C$?

Since the integration range of "the" convolution is $\mathbb R$, what is a sensible generalization in complex numbers? Would one still integrate over $\mathbb R$, or some other path, or over the ...
1
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1answer
156 views

Convolution $f*g$ is continuous

Statement: Let $f,g: \mathbb{R}^d \rightarrow \mathbb{R}$ be Lebesgue measurable functions such that $f\in L^1(\mathbb{R}^d)$ and $g\in L^\infty(\mathbb{R}^d)$. The convolution $f*g:\mathbb{R}^d ...
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2answers
451 views

How can you do convolution graphically?

I don't exactly remember whether I should get the common area under the curves of the functions being convolved or I should multiply them and get the area under the resulting curve.
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2answers
48 views

Convolution of a function with itself

Function $\phi (x)$ is defined as: $$\phi(x) = \begin{cases} 1 & \text{ if } 0 \leq x \leq 1\\0 & \text{otherwise} \end{cases} $$ How do I find the convolution of $\phi(x)$ with itself? I ...
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3answers
32 views

Convolution Product $\sin t *\sin t$ by complex replacement.

I want to compute $$\sin t * \sin t=\int_0^t \sin u \sin (t-u) \, d u.$$ I have already tried complex replacement: $$\int_0^t \sin u \sin (t-u) \, \mathrm d u = Im{\int_0^t e^{iu} e^{i(t-u)}\,d u ...
1
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1answer
51 views

How to obtain the convolution directly (not graphical) of the two functions $e^{-t}u(t)$ and $e^{-2t}u(t)$?

I'm having trouble solving this convolution integral graphically. I don't understand where I stop sliding my function $h(t-\lambda)$ since $x(t)$ doesn't have a boundary as lambda approaches infinity ...
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3answers
187 views

Combining two convolution kernels

Is it possible to combine two convolution kernels (convolution in terms of image processing, so it's actually a correlation) into one, so that covnolving the image with the new kernel gives the same ...
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1answer
55 views

Differential equation with fourier transform and convolution

We have differential equation $3s(t)-2s''(t)=r(t)\,$ and $s(t)$ is convolution $s=g*r\,$ where $g(t)=ae^{-b\left | t \right |}\,$ $\\a,b\in\mathbb R+$ Solve constans a and b. I tried to solve ...
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2answers
38 views

Computing a messy convolution

Consider the functions $$ x(t) = u(t - \frac{1}{2}) - u(t - \frac{3}{2}) $$ and $$ h(t) = tu(t) $$ where $u(t) = 1$ if $t \geq 0$ and $u(t) = 0$ if $t < 0$. I'm trying to compute $$ (x*h)(t) ...
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1answer
63 views

Convolution of Strictly Convex Function

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a $C^2$, strictly convex function, and $\theta_\epsilon$ the standard approximation to identity ...
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1answer
35 views

Another proof of the iniectivity of a linear operator

Let $g(x)= \chi_{[-\frac{1}{2}, \frac{1}{2}]}(x) $, and $ T \colon L^2(\mathbb{R}) \longrightarrow L^2 (\mathbb{R})$ , $Tf= g \star f$. I was asked to prove that $T$ is injective, and I succedeed ...
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1answer
56 views

Inverse Laplace transform of a given function

1) The Laplace transform of f(t) is $\overline{f}(p)=\frac{1}{p}$ when $f(t)=1$ 2) The Laplace transform of $f(at)$ is $\frac{1}{a}\overline{f}(\frac{p}{a})$ 3) The Laplace transform of the ...
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1answer
214 views

The issue of treating an inverse Fourier transform in terms of a tempered distribution.

Consider the wave equation $$ u_{tt}=\Delta{u} \quad u(x,0)=f(x) \quad u_t(x,0)=g(x) \tag{*} $$ A solution to this equation is given by $$ u(.,t)=f*\partial_t\Phi_t+g*\Phi_t \tag{**} $$ where ...
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2answers
79 views

Fourier transform of function

What is Fourier transform of $$f(x)=\frac{e^{-|x|}}{\sqrt{|x|}}?$$ I tried to calculate it using $$F(e^{-|x|})=\sqrt{\frac{\pi}{2}}e^{-|a|}$$ and $$F(\frac{1}{\sqrt{|x|}})=\frac{1}{\sqrt{|a|}}$$ and ...
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2answers
129 views

Convolution of finite measures

I am puzzled by the following (maybe very stupid) question I stumble upon in the course of a project: let $p$ be a probability measure on some abelian group $E$ (actually, $E=\mathbb{Z}_n$ with its ...
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1answer
85 views

Any clue how to solve this convolution integral?

With other words: find a (closed) expression for $\;\overline{\mbox{sinc}}(x)$ . $$ \overline{\mbox{sinc}}(x) = \int_{-\infty}^{+\infty} \frac{\sin(\omega\xi)}{\omega\xi} ...
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1answer
366 views

Convolution of distributions.

We are given with distributions $f,g \in D'(\Bbb R)$. If $suppf\subset (-\infty,a)$ and $supp(g)\subset(b,\infty)$ then prove that $f*g$ is well defined distribution. where $a$ and $b$ are real ...
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1answer
313 views

Convolution Laplace transform

Find the inverse Laplace transform of the giveb function by using the convolution theorem. $$F(x) = \frac{s}{(s+1)(s^2+4)}$$ If I use partial fractions I get: $$\frac{s+4}{5(s^2+4)} - ...
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1answer
186 views

How to compute the convolution of two functions which diverge at infinity?

How to compute the convolution of two functions which diverge at infinity? e.g. $e^{x^2}*e^{x^4}$ We can't directly write as $\int_{-\infty}^\infty e^{t^2}e^{(x-t)^4}~dt$ or $\int_{-\infty}^\infty ...
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3answers
369 views

convolution of signals

I'm finding in trouble trying to resolve this exercise. I have to calculate the convolution of two signals: $$y(t)=e^{-kt}u(t)*\frac{\sin\left(\frac{\pi t}{10}\right)}{(\pi t)} $$ where $u(t)$ is ...
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2answers
466 views

A convolution identity (revision of the question “Is this convolution identity known?”)

I have deleted the content of the original post. The following exercise is inspired by answers to the questions this and this, but no knowledge of probability theory is required. The solution is ...