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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Calculate the convolution of the product of two simple functions. (5.6-12)

Synopsis: I cannot duplicate the answer given in a very reputable online symbolic integral calculator as shown in this link ($x$ is $\tau$) although my answer does appear very similar. This tells me ...
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0answers
9 views

How to get the output of a signal given the input and impulse

Here is my question So given the input and impulse response how would I go about getting the output? The output is given in solution (in teal) but I have no idea how to get it . thanks
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41 views
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Help with conditional expectation of a convolution of exponential random variables

I'm working through this paper, with lots of help from all the great people on this site. Obviously my statistics/probability is a lacking to follow all the mathematical steps. Currently, I'm trying ...
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0answers
4 views

Derivative of convolution w.r.t one of the functions themselves

Let $z(t)=(y*x)(t)$, or in the discrete case (for vectors), $z=Yx$ where $Y$ is the matrix operator performing convolution with a kernel $y$. I know the derivative $z'=y'*x=x'*y$, but what is the $\...
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1answer
380 views

difference between convolution of two densities and mixture density?

I am wondering about the difference of the convolution of two probability density functions and the mixture of those two. This is not the same right? But what is the difference and how can it be ...
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1answer
434 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 ...
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28 views

Fourier transform in complex analysis

What does convolution means in complex analysis? In particular, I want to calculate $\varphi \ast 1/z$, where $\varphi$ is the characteristic function of unit ball in $\mathbb{C}$, i.e. $\varphi= \...
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1answer
24 views

Convolution of a piecewise function with itself

Convolution of a function with itself I was going through this question. In the answer, the limits of the integral was transformed from (-infinity)-(+infinity) to 0 to x. Can anyone explain how this ...
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1answer
37 views

Convolution of a function with itself n times convergence to bell curve [duplicate]

If we have a piecewise function defined as $f(x) = \begin{cases} 1, & \text{0 $\le$ $x$ $\le$ 1} \\ 0, & \text{otherwise} \end{cases}$ Explain how the convolution of $f$ with itself for $n$ ...
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1answer
516 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: $(f(t)\ast g(t))(\tau) = \int_{-\infty}^\infty{f(t)g(\tau-t)dt}$ ...
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3answers
35 views

Does Gaussian convolution respects order?

Assume that we have two continuous integrable functions $f,g \in L^1(\mathbb{R})$ such that, for some $x_0 \in \mathbb{R}$, we have, $$f(x_0) \leq g(x_0) \; \; \; \; (1).$$ Now let us define the ...
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0answers
40 views

Bizarre binomial sum

It is many times that we need to compute discrete convolutions. Driven by this need we have discovered a following formula: \begin{equation} \sum\limits_{l=0}^k \binom{l+A_1}{A_2} \binom{-2 \beta (k-l)...
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0answers
14 views

Sum of Independent Levy RVs is Levy RV [closed]

I want to show that the summation of independent Levy random variables X and Y with scaling parameters a and b is a Levy random variable with scaling parameter c = (a^(1/2)+b^(1/2))^2 using ...
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2answers
30 views

Convolution of normal distribution not equal to product with constant?

Convolution of a normal distribution says: If, $X \sim \mathcal{N}(\mu, \sigma^2)$, then $X+X\sim\mathcal{N}(\mu+\mu, \sigma^2+\sigma^2)=\mathcal{N}(2\mu,2\sigma^2)$ However, Multiplication of a ...
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1answer
39 views

Prove that $u\leq v$ everywhere.

Let $u$ be a subharmonic function on an open set $U$ in $\mathbb{C}$, and let $v$ be an upper semicontinuous function on $U$ such that $u\leq v$ almost everywhere. Prove that $u\leq v$ everywhere. ...
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1answer
36 views

Laplace Transform

Question: Use Laplace Transform to solve the following differential equation $\ y''+y =sin(t); y(0)=1, y'(0)=-1 $ My try,where F(s) is the transform of f(t)=y(t) $F(s)= \frac{1}{(s^2+1)^2} + \frac{...
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1answer
33 views

Meaning of limits, $\int_{\max(0, t-5)}^{\max(0,t-3)} e^{-3s} \, ds $?

What does it mean to have $\max(0,t-3)$ and $\max(0,t-5)$ in the limits? Is it a abbreviation? $$ \int_{t-5}^{t-3} e^{-3s}u(s) \, ds = \int_{\max(0, t-5)}^{\max(0,t-3)} e^{-3s} \, ds $$ Source of ...
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0answers
27 views

convolution of two decaying polynomial

Show that if the functions $f,g$ defined on $\mathbb{R}^n$ satisfying $|f(x)| ≤ A(1+|x|)^{−M} $and $|g(x)| ≤ B(1+|x|)^{−N}$ for some $M,N > n$, then $|(f ∗g)(x)| ≤ ABC(1+|x|)^{−L} $, where $L = ...
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1answer
528 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 ...
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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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1answer
54 views

Solving the ODE $y^{\prime\prime}(x)-y(x)=g(x)$ using the Fourier transform, without missing solutions

I'm supposed to solve the ODE $y^{\prime\prime}(x)-y(x)=g(x)$ using the Fourier transform and then explain if I got the most general solution. First of all, I don't know what "solve" means here ...
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1answer
24 views

Convergence of composition of functions in $L^p$

I am dealing with the proof of proposition $9.5$ given in Haim Brezis' Functional analysis, Sobolev Spaces and Partial differential equations. I quote it here: How does one conclude $G \circ ...
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0answers
19 views

How to evaluate chirp transform in O(nlgn) time? [duplicate]

The question says to evaluate chirp transform in O(nlgn) time using the equation in the hint. But I'm unable to get any idea on how to prove the chirp transform from it. Any help is appreciated.
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2answers
56 views

estimate a probability

Let $X_1....X_{48}$ be independent random variables, each follows a uniform probability distribution over [0,1]. What is the best way to estimate P($\Sigma_{i=1}^{48} X_i > 20)$?
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5answers
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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 ...
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0answers
11 views

Properties of the mollification operator (Sohr - Navier-Stokes equations)

My question refers to Sohr - The Navier Stokes Equations p.66/67. I am trying to understand the statement (1.7.16). Since p.66 is not available via Google-books I will formulate the problem in the ...
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1answer
1k views

Convolution is uniformly continuous and bounded

Suppose $f\in L^\infty(\mathbb{R})$ and $K\in L^1(\mathbb{R})$ with $\int_\mathbb{R}K(x)dx=1$. Show that the convolution $f\ast K$ is a uniformly continuous and bounded function. The definition of ...
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4answers
93 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(...
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1answer
30 views

Approximation estimates in Sobolev spaces

Let's consider a bounded domain $\Omega \subset \mathbb{R}^d$, $d =2,3$, and let $\varphi$ be in $H^1(\Omega) \cap W^{1,\infty}(\Omega)$. Is it there a smooth (at least $W^{2,4}(\Omega)\cap W^{1,\...
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1answer
478 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 ...
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2answers
42 views

The convolution of two functions is L1

I have to proof the following corollary: $ \text{Let } 1 \leq p \leq \infty \text{, } f \in L^{1} \text{, } g\in L^{p} \text{. Then } f \ast g \in L^p \text{ and } \Vert f \ast g \Vert _{L^{p}} \...
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0answers
23 views

Convolution is continuous

Convolution is continuous Let $f,g\in L^2\left(\mathbb T,\mathbb C\right)$ (Hilbert space of $1$-periodic functions) then $f*g$ should be continuous by Young's inequality (the map is $(f,g)\mapsto ...
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23 views

Is the convolution algebra a *-algebra?

Let $G$ be a finite abelian group with $n$ elements. Consider the convolution algebra $C^*(G) \subset l^2(G)$, with multiplication: $$(a * b )(g) = \frac{1}{n}\sum_{x\in G} a(x)b(g-x)$$ Is there a ...
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36 views

Integration bounds for convolution

Given a joint density $f_{XY}(x,y)$ on triangle with vertex $(5,0),(5,1),(4,1)$ I can't understand how to find integration bounds for convolution of $Z=X+Y$. I know that the solution should be $$F_z(...
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0answers
16 views

Is convolution with Gaussian distribution a stationary process

So might be a silly question, but if I have $ X(t) $ a strict sense stationary process and $ h(t) $ is a Gaussian density function with mean zero, then can we guarantee the convolution process $ (X*h)(...
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0answers
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Is $\pi^1:C_c(G)\rightarrow \operatorname{End}(H)$ a homomorphism of the convolution algebra when $G$ is not unimodular?

Let $G$ be a Hausdorff locally compact group and $H$ a Banach space. Let $\pi:G\rightarrow \operatorname{GL}(H)$ be a representation and define $$\pi^1(\phi)v = \int_G\phi(x)\pi(x)vdx$$ for $v\in H$ ...
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18 views

Singular integral operator on a decay Hölder space

Denote $$E^{k,m}=\left\{f\in C^m(\mathbb{R}^3)\mid\sup_{x\in\mathbb{R}^3}(1+|x|)^{k+|\alpha|}|\partial^\alpha_xf(x)|<\infty\right\}.$$ Now given $f\in E^{2,m}$ for any fixed $m\in \mathbb{Z}^+$, ...
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1answer
46 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 ...
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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, \...
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1answer
39 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(...
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2answers
783 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({...
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3answers
49 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: $$f \ast g = \int_0^t \alpha e^{-\beta (t-\tau)} y(\tau) d\tau$$...
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1answer
31 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'...
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1answer
24 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(\...
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2answers
59 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 ...
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1answer
33 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 = \...
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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(.;\...
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0answers
26 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?...
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1answer
57 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 ...
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2answers
43 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 ...