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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64 views

What is $D\delta$ if $D$ is ordinary differential operator and $\delta$ is the Dirac distribution?

I'm reading some material about single-variable distribution theory. More specifically, I was checking some theorems of the convolution algebra $\mathcal{D}_+$, where $\mathcal{D}_+$ is the space of ...
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1answer
139 views

Why does convolution not maintain asymptotic equality of coefficients?

Assume I have four (generating) functions $f$, $f'$, $g$ and $g'$. If that is interesting, we can assume that they all share the same radius of convergence $\rho > 0$. In addition, we know that ...
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2answers
134 views

Intuition behind convolution identity for Laplace transforms

Convolutions, relatively speaking, are fairly straightforward for simple systems (from an applied perspective), but I cannot, at all, find the intuition behind the Laplace identity for convolutions. ...
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1answer
132 views

convolution of compactly supported continuous function with schwartz class function is again a Schwartz class function?

Suppose $f$ is continuous function on $\mathbb R$ with compact support; and $g\in \mathcal{S}(\mathbb R),$ (Schwartz space) My Question is: Can we expect $f\ast g \in \mathcal{S(\mathbb R)}$ ? ...
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0answers
33 views

Change of Variables for two level Guassian model

I have a multivariate Gaussian distribution from which two variables, u and v, are drawn. The next variables, U and V, are U = 1/(u^2+5) + N(0,sig_U) and V = v^3 + N(0,sig_V). U and V are known, ...
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1answer
56 views

Using Laplace transforms to solve a convolution of two functions

Hi I have this problem where I need to take the convolution of functions and I am not sure if I got the right answer or something close so any advice or help would be very appreciated. So here is the ...
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1answer
31 views

Apparent paradox commuting this convolution: where is the mistake?

Starting with some vector $x$, I am performing two operations: First, I convolve $x$ with another vector $g$ to compute $x*g$, where $~*~$ denotes convolution. Second, I pointwise multiply the result ...
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1answer
37 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
66 views

Operations on Random Variables

It is known that the equivalent resistance of a parallel combination of two resistors is equal to \begin{align*} R = \frac{R_1R_2}{R_1+R_2} \end{align*} which could be also written as ...
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0answers
28 views

What is correlation kernel and compare with gaussian kernel

I read a paper that said about correlation kernel that defined: $$W(x-y)=(α/1+d(|y − x|))$$ where $α =  (\int(1+d(y − x)dy)^{-1}$, $(d(|y − x|))$ is spatial Euclidean distance from the central ...
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2answers
102 views

Easy way to compute $Pr[\sum_{i=1}^t X_i \geq z]$

We have a set of $t$ independent random variables $X_i \sim \mathrm{Bin}(n_i, p_i)$. We know that $$\mathrm{Pr}[X_i \geq z] = \sum_{j=z}^{\infty} { n_i \choose j } p_i^j (1-p_i)^{n_i -j}.$$ But is ...
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0answers
56 views

Partial derivative in frequency domain when only time domain function is known

I want to calculate $$ \frac{\partial F_p(X(\omega))}{\partial X(\omega)} $$ So $F_p$ operates in some way on $X(\omega)$ but I know the analytical form only in time domain, represented by $f_p$. ...
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1answer
74 views

Question about transformations and sums on uniformly distributed random variables.

I'm looking into a few problems as a hobby of mine, and found myself with the following problem: let $X$ be a random variable uniformly distributed on $[0,1]$. What is the probability that after $N$ ...
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1answer
39 views

Convolution module

I am trying to apply GCC-PHAT algorithm here to process audio files and find delay between them. Im coding using Android and Java with the help of this library, and comparing the results with Matlab. ...
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1answer
114 views

$L^1$ norm of convolution

Let $f_{\lambda} = \frac{\lambda}{2}e^{-\lambda |x|}$. Prove that $||f_{\lambda} \ast g - g || \to 0$, when $\lambda \to \infty$, where $g \in L^1$
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0answers
115 views

Expectation and convolution question.

I am learning in an image processing course, and the professor did the following: As part of a derivation, has this: What I do not understand, is how he was able to remove $r(i,j)$ to the ...
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1answer
31 views

Holder continuity of $\frac{x}{|x|^3} \ast f$ with $f \in C^1_0$ in $\mathbb{R}^3$

Ok, so I need to show that for $f \in C^1_0(\mathbb{R}^3)$ the convolution with $k(x) := \frac{x}{|x|^3}$ is Holder continuous. The exponent doesn't matter much as long as I can bound it using ...
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1answer
58 views

Prove or disprove: $e^{-nG(x)}$, normalized, is an approximation to the identity for $G(x)$ strictly convex

We are given the sequence of functions $$ \phi_{n} = \frac{e^{-nG(x)}}{\int_{\mathbb{R}}e^{-nG(x)}dx}$$ for a nonnegative, strictly convex function $G$ (that is, $G'' \geq c$ for some $c>0$) that ...
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0answers
55 views

Laplace transform $y''''+37y''+36y=g(t)$

Hey this problem is making me insane so have at it and let me know what I keep screwing up. Express the solution of the initial value problem in terms of a convolution integral: ...
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1answer
26 views

if $f$ is in weak $L^p$ and $\phi$ is $C_0^{1}$ then $f \ast \phi$ is in weak $L^p$

Okay, so I'd like to know if what I wrote in the title is true. Suppose that $f \in L^{p,\infty}(\mathbb{R}^n)$ (weak $L^p$ space) and $\phi \in C_0^1(\mathbb{R}^n)$ [or even $C_0^{\infty}$ if it ...
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0answers
46 views

convolution of 3 uniform random variables

Please help. X,Y,Z are uniformly distributed random variates over the closed interval [3,5] independently/ The sum, S, of any of the two random variates X,Y,Z has a triangular distribution with pdf: ...
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1answer
26 views

Using convolution to impose differentiablilty.

If I had a function $g$ that was not differentiable at a known point, is it possible to convolute it with say a $C^{\infty}$ function $f$, resulting in a differentiable function? Thanks in advance!
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0answers
87 views

convolution of three functions of two variables

Give three functions of two variables $a(x,y),b(x,y),c(x,y)$ one can construct the following convolution like integral: $y(x,y) = \int dx' dy' a(x',y')b(x-x',y') c(x-x',y-y')$ which I have a hard ...
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1answer
69 views

Proof of a corollary about associativity of (differential) convolution operater

I m working on the proof the following corollary for ages... I would appreciate any help!! Cor: Let $h : [0, \infty ) \rightarrow \mathbb{R}$. We define the convolution operator $*$ for the ...
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0answers
15 views

does convolution with the derivative of sinc give the derivative?

If we convolve a sampled waveform, which is sampled according to the Nyquist rate, with the sampled derivative of the sinc function do we get the sampled derivative of the original waveform? Is this ...
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0answers
40 views

Deconvolution vs convolution.

This is now a second time I am attempting to ask this very important but simple question here. What I want to know is can you do deconvolution by convolving a signal. It is often stated that, for ...
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1answer
344 views

convolution of function with itself 4 times

I have to compute the convolution of $ f(t) = \frac{1}{\pi}\frac{1}{t^2 + 1} $ with itself 4 times, i.e. $$ f \star f \star f \star f $$ I slightly doubt that doing it in steps, i.e. taking $f \star ...
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1answer
78 views

What is the pdf of $Z=X/\max(X,Y)$ with $X,Y$ exponentials of lambda parameter?

Given $X,Y$ 2 independent r.v.'s both distributed as $\exp(λ)$, what is the pdf of $Z=X/\max(X,Y)$?
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2answers
42 views

estimating $L^p$ norm of $\frac{x}{|x|^3} \ast (-\text{div}_x \ldots)$

I've been having problems with following an argument in a paper I'm reading, I hope someone can help me understand the point. Suppose $f(t,\cdot,\cdot)$ is a $C_0^1 (\mathbb{R}^6)$ function for all ...
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1answer
79 views

Convolution of ring Delta function

Assume $f(r)=\delta(r-R)$ where $\delta(\cdot)$ is a ring delta function. In other word, $f$ is a circular delta function on a circle with radius $R$. I want to do the convolution of $f$ with itself ...
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2answers
91 views

Derivative of a convolution

I need to find the derivative of the following equation, which I do think is a convolution: Could anybody give me a hint on how to find the derivative of V(x)? Many thanks in advance!
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2answers
95 views

Pdf of $Z=(XY)^{1/2}$. with X,Y independent r.v. with the same distribution (iid) [closed]

Let be $X,Y$ two independent random variables having the same distribution (the following is the density of this distribution) $$f(t)= \frac{1}{t^2} \,\,\, \text{for $t>1$}$$ Calculate the ...
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2answers
158 views

What's the density of $Z=\max(X,Y)-\min(X,Y)$ with $X,Y$ exponentials of parameter $\lambda$?

Let be $X,Y$ two independent exponential random variables with parameter $\lambda$. What is the pdf of $Z=\max(X,Y)-\min(X,Y)$? Thanks for your help.
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1answer
78 views

Decay of a Convolution

Let $f, g \in L^1\cap L^\infty(\mathbb{R}^d)$ be probability distributions on $\mathbb{R}^d$, and suppose at large $|x|$, $f$ decays like $|x|^{-\alpha}$ while $g$ decays like $|x|^{-\beta}$, with ...
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3answers
87 views

Convolution of maximum and minimum of uniform random variables

Let $X_1,\ldots, X_n$ be $n$ independent random variables uniformly distributed on $[0,1]$. Let be $Y=\min(X_i)$ and $Z=\max(X_i) $. Calculate the cdf of $(Y,Z)$ and verify $(Y,Z)$ has independent ...
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0answers
107 views

Power spectral density of convolution of stochastic processes

I was wondering what it is the result of convolving two WSS processes in terms of power spectral densities. I know that, the output $Y(t)$ of a generic linear time invariant system with impulse ...
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1answer
475 views

Probability Density of Convolution of Two Random Processes or Variables

Suppose that we have two stationary random processes $x(t)$ and $y(t)$ with probability density functions $f_{x}(x)$ and $f_{y}(y)$ respectively. Now suppose we form: $z(t) = x(t) \ast y(t)$ What is ...
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2answers
131 views

Convolution, indicator function

I need to calculate $(f*f)(x)$ of $f(x) = 1_{[0,1]}(x)$, which is the indicator function defined with Calculating the integral $(f*f)(x) = \int_{0,}^{x}1_{[0,1]}(t) \cdot1_{[0,1]}(x-t) dt$ gives ...
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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
100 views

Fast convolution with striding

I want to convolve two discrete functions $f$ and $g$ using convolution stride size $a$ to get the result as $s_{a, i}$: $$s_{i,a} = \sum_i g_k f_{ai-k}$$ I know that simple convolution with $a=1$ ...
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1answer
109 views

Bound on uniform norm of convolution of $L^p$ functions

This is Proposition 8.8 in Folland's Real Analysis: If $p$ and $q$ are conjugate exponents, $f \in L^P$, and $g \in L^q$, then $f*g(x)$ exists for every $x$, $f*g$ is bounded and uniformly ...
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1answer
57 views

Question about mollifiers.

So here is my problem, Let $\rho \in C^\infty (\mathbb{R}^n,R)$ with $\rho\geq 0$, $\rho(x)= 0 \; \forall \|x\|\geq 1$ and $\int_{\mathbb{R}^n}\rho(x)dx=1$. Further, consider the linear map ...
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1answer
34 views

Differential Question about Laplace/Delta/Convolution

I need help understanding a part of this question. Let $a.) y''+4y = \delta (x)$, $y(0)=y'(0)=0$. and $b.) y'' + 4y = f(x)$, $y(0)=y'(0)=0$ where $f(x)$ is some continuous function of finite ...
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1answer
46 views

proof that$ L^1 (G)$ is a subspace of $M(G)$

Let G be a locally compact group, and let $M(G)$ be the space of complex Radon measures on G. I identify the function f with the measure $f(x) \rm dx$ . but How do I prove this inclusion?؟ . .
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2answers
95 views

Solving a differential equation using the laplace transform involving convolution

The problem is the following The thing that puzzles me here is the integral on the right hand side, so: How to take the laplace transform on the right hand side? Any help to get me going would be ...
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0answers
45 views

Question about the principal value of some integral

So here is my problem, is it possible that $$\int_{[0,1]}f(y)\cot(\pi(x-y))dy= p.v \int_{[0,1]}f(y)\cot(\pi(x-y))dy$$ I see that the left integral is singular for $x=0$ but since I never worked with ...
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3answers
260 views

Laplace transforms: Convolution

Find $$1*1*1*\cdots*1\quad n\,\,\text{ factors}$$ that is, a function $f(t)=1$ convolution with itself for a total of $n$ factors. Would anyone mind helping me? I have no idea what I should do. ...
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1answer
217 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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1answer
69 views

convolution -questions

I'm lost, can you help me please. How compute the product convolution between two distributions $T$ and $S$? (we suppose that $T * S$ exist)? How we compute the product convolution between an ...
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1answer
38 views

Convolution computing

How we can compute the convolution product $$\Big(\sum_{n=0}^{+\infty} \delta_n^{(n)}\Big) \star \Big(\sum_{n=0}^{+\infty} \delta_n\Big)$$ where $\delta$ is Dirac distribution? Thank's for the help