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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8 views

Combining multiple 2D and 1D convolutions into a single 3D convolution

I have a 3D image (multiple 2D slices) and two kernel images (one 2D kernel and one 1D kernel). And I do the following operations: I convolve each 2D slice (x,y direction) of the 3D image with my 2D ...
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1answer
16 views

Laplace transform of convolution with no function of t

Instructions: Evaluate the given Laplace transform. Do not evaluate the integral before transforming. Problem Given: $\mathscr{L}\{\int_0^t e^{-\tau} cos\tau d\tau \}$ My Problem: To treat this as ...
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0answers
25 views

Convolving two functions

I'm trying to convolve two functions $f$ and $g$. $$f(x) = e^{-\frac{{(x-p_2)}^2}{2 q_2^2}}$$ $$g(x) = \left(i_1 e^{-\frac{(a-x)^2}{2 \sigma ^2}}+j_1 e^{-\frac{(b-x)^2}{2 \sigma ^2}}\right) \left(i_0 ...
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0answers
11 views

Min $+$ convolution is associative

Although the following question was encountered in a Communication Networking textbook, the problem is still one of algebraic and analytic manipulation. Define the (min,+) convolution of two real ...
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0answers
32 views

family of functions/sequences taken over reals instead of naturals

How does convergence for a sequence and a family of functions change when considering $n$ taken from $\mathbb{R}$ instead of the $\mathbb{N}$? for example, consider mollifiers which are defined as ...
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18 views

convolution and Fourier series [closed]

I have a $2 \pi$ periodic function, defined by: $f(x)=0$ if $x \in [-\pi,0)$ and $f(x)=1$ if $x \in [0, \pi)$. 1) I would like compute it's exponential series expansion. 2) Compute $f*f(x)$, here ...
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1answer
28 views

If $r_n\to r$ and $s_n\to s$, then $(r \star s)_M/M \to rs$.

I was going to ask this question, but I think I figured it out, so I thought I'd post my answer: In this question of mine, a user's answer makes the following claim: Suppose $r_n$ and $s_n$ are ...
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0answers
15 views

Relation between Dirichlet convolution and Bell series and convolution of functions and the Fourier transform?

We define the Dirichlet convolution of two arithmetical functions $a,b:\mathbb{N}\to\mathbb{C}$ to be $$ (a*b)(n)=\sum_{d\mid n}a(d)b\left(\frac{n}{d}\right). $$ Given a prime $p$, we define the Bell ...
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1answer
30 views

Mollification of a function continuous on $\mathbb{R}\backslash\{0\}$, need uniform convergence

We know that if $f:\mathbb{R} \to \mathbb{R}$ is continuous, then its mollification $f_\epsilon$ converges uniformly to $f$ on compact subsets of $\mathbb{R}$ as $\epsilon \to 0$. My question is, ...
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0answers
10 views

Average Orders and Convolutions

If I know the average order of an arithmetic function $f=I*g$, where $I$ is the identity function defined by $I(n)=n$, is there a way to find the average order of $g$?
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Details about Generalized Convolution (Number Theory - Apostol)

In "Introduction to analytic Number Theory" by Apostol there is chapter about generalized convolution. Let F denote a real or complex-valued function defined on the positive real axis such that ...
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1answer
32 views

Convolution of a continuous function and uniform continuity

Suppose $f$ is a continuous function, and let the convolution $f_n(x) := f \star \varphi_n(x)$ where $\varphi_n$ are smooth test functions. We know $f_n \in C^\infty$. We know that if $f$ is ...
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0answers
30 views

Power series of characteristic function

I read that if $f \in \ell^1 (\mathbb Z)$ and $w$ is the characteristic function of $\{1\}$ then $$ f(m) = \sum_{n \in \mathbb Z} f(n) w^n(m)$$ where $w^n = w \ast \dots \ast w$ is $n$ times the ...
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0answers
21 views

The derivative of convolution

For a compact supported continuous function $\rho$ in $R^3$, consider the convolution $f(y)=\int_{R^3}\frac{1}{|x-y|}\rho(x)d x$, did the following communicate of derivative and integral holds? $$ ...
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0answers
11 views

Is the autocorrelation of a function the same if one term is flipped on the y axis?

I have some questions about autocorrelation. They are very related, so I thought that one single post was appropriate for the topic. The first question is already illustrated in the subject: if I ...
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1answer
19 views

How to convolve two stair-case functions?

For the life of me, I haven't been able to grasp convolution for functions with multiple pieces. For example, $$ h(\lambda) = \left\{ \begin{array}{l l} 2 & \quad \ 0\leq \lambda < 1\\ ...
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1answer
35 views

Separate the variables of the function $\frac{x^2}{\sqrt{x^2+y^2}}$

Is there a way to express the function $\frac{x^2}{\sqrt{x^2+y^2}}$ as the product of two functions: $f(x)\cdot g(y)$, i.e. one in each variable? This is becasue I want to apply a convolution whose ...
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2answers
35 views

convolution product of characteristic functions

Consider the characteristic function $f(x)=1_{[0,r]}(x)$. How to compute $(f*f)(x)$ where $r \in [0,1[$ and by definition $(f*g)(x)=\int_{\mathbb R} f(y)g(x-y) \ dy$ ? thanks.
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0answers
16 views

Sum of two independent random variable, Convolution.

I need the graphic of this two function to evaluate this correlation?
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0answers
35 views

Is this function monotonically non-decreasing?

I am wondering if the function $L[n]$ defined on $n=0,1,2,\ldots,N$ below is "monotonically" non-decreasing in $n$. I put monotonically in quotes because the function is not continuous and I am not ...
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1answer
21 views

Difference between two independent geometric random variables

Let $\xi_1$ and $\xi_2$ be independent random variables: $\xi_1 \simeq Geom(1/2), \xi_2 \simeq Geom(1/6)$. How do you find the probability mass function of $\eta=\xi_1-\xi_2$ using convolution?
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1answer
58 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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0answers
45 views

What is the energy of sinc(t)^2?

So the energy would be sinc(t)^2 * sinc*(t)^2. Which will be two rect functions convolved convolved with each other. So it'll be two triangles convolved with each other. And then you would take the ...
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0answers
43 views

Convolution is continuous map

I can prove this when $f$ is assumed as continuous function but without assuming continuity i got confused. Suppose $ p \in (1, \infty) $ and $q$ is its conjugate exponent. Prove that if $f\in ...
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1answer
47 views

Intuitive basis of Mobius inversion?

If we're given $f(n)= \sum_{d|n}g\left(\frac{n}{d}\right),n \in \mathbb{N},$ then Mobius inversion gives $$g(n)=\sum_{d|n}\mu \left( d\right) f \left( \frac{n}{d}\right).$$ Also, the generalised ...
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1answer
29 views

Convolution of fraction function

I know that convolution is defined: $$f*g=\int f(x-y)\cdot g(y) \, dy $$ How to develop below functions to convolution equation $$\int {f(x-y) \over g(y)} \, dy =\text{ ???}$$ and $$\int {f(x-y) ...
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0answers
22 views

Find optimize of gaussian regression

I have one gaussian regression and I want to find parameters to optimize the LOSS function: $E(\sigma,\mu,b_0)=\int K(x-y).||f-f_i||^2$ where $f=\epsilon+b_0 ;$ $\epsilon$ ~$N(\sigma,\mu)$ is ...
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0answers
38 views

Conversion of covariance matrix from Cartesian to Spherical coordinates for integration

I have to perform a convolution of a function in polar coordinates $\rho(\textbf{x}) = \rho(r,\theta,\phi)$ with a function $P(\textbf{x}) = P(x,y,z)$ in cartesian coordinates. $\int ...
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1answer
14 views

The signal $\cos(2 \pi t )$ is an eigenfunction of every LTI system?

for $\sin(2 \pi t)$: Apparently that it's not an eigenfunction real-valued impulse response $h(t)$ but it's a eigenfunction for real-valued and even impulse response $h(t)$ What gives?
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5answers
754 views

Definition of convolution?

Why do we use $x - y$ rather than $x + y$ in the definition of the convolution? Is it just convention? (If we are thinking of convolutions as weighted averages, for instance against "good kernels," it ...
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2answers
65 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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1answer
70 views

$g, f, \hat {f} \in L^{1}(\mathbb R)\cap L^{2}(\mathbb R) \cap C_{0}(\mathbb R) \implies \widehat{(fg)}= \hat{f} \ast \hat{g} ? $

Let $f, g\in L^{1}(\mathbb R)$ and it Fourier transform of $f$, $\hat{f} (y) = \int _ {\mathbb R} f(x) e^{-2\pi i x \cdot y} dx, \ (y\in \mathbb R)$ and the convolution of $f $ and $g$; $f\ast g ...
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0answers
21 views

convolution periodic function and compactly supported function.

I am trying to understand why the convolution between a compactly supported function (let say $f$) and a periodic function (let say $g$) exists. By convolution I mean $(f*g)(x)=\int_{\mathbb{R}} ...
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0answers
22 views

how to prove this limit, convolution?

I am wondering how to prove that $\vert (f*g)(x) \vert \rightarrow 0 $ when $\vert x \vert \rightarrow \infty$ if we assume $f \in L^{p}(\mathbb R)$ and $g \in L^{q}(\mathbb R)$ where $1/p+1/q=1$ ...
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0answers
45 views

It is possible to find such functions?

I would like to know if it is possible to find a couple of functions $f,g$ such that $f*g$ and $g*f$ exists and such that $f*g\ne g*f$ ? if not it would mean that the convolution product commutes ...
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2answers
55 views

How can this integral be rewritten with convolutions?

I've got $f:\mathbb{R}\rightarrow\mathbb{R}$ bounded and I'm trying to write `$\mathtt{f}$,' a discrete version of $f$, where each element in the domain takes on the average of the corresponding ...
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1answer
13 views

Fidelity of measurement using conditional probabilities

Let me begin by saying that I'm not entirely sure if this is the correct forum, or if Cross Validated would be more suitable. The problem I'm about to describe is statistical in nature, but I believe ...
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0answers
22 views

Require help with the convolution of two complex conjugates

I need to find the convolution of the following two functions: When rationalizing the denominator, the numerators become complex conjugates of each other. I have tried obtaining the Fourier ...
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1answer
53 views

Convolution of two sums (fourier transform)

This question is from the book "Advanced Engineering Mathematics" by Stroud. I can't seem to get the required answer for this. I've derived the two Fourier transform equations for them. . U and ...
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1answer
38 views

Is $(g \ast f ) '= g'\ast f$ true?

Take $ f \in L^{1} (\mathbb{R})$, and $ g \in L^{\infty}(\mathbb{R})$, with $g$ almost everywhere differentiable and such that $g' \in L^{\infty}(\mathbb{R})$. Prove or disprove: $(f \ast g) \in ...
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0answers
37 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 ...
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1answer
64 views

Convolution of a probability measure with a smooth function

If $f\in L^1(\mathbb{R}^n)$ and $g\in L^p(\mathbb{R}^n)$ then by Young's convolution inequality we have the estimate: $$ \|f*g\|_{L^p}\leq \|f\|_{L^1}\|g\|_{L^p}.$$ Question: Let $\mu$ be a ...
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1answer
49 views

Help with a question on convolution?

I need help solving this convolution question for an assignment. I need to find the convolution of the two functions. I've searched online for a way to approach this question, but this was the ...
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0answers
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Are the definitions of convolution here contradict each other?

Here are two definitions from a lecture slides file from the internet: It looks strange that the first uses x-u and the second uses x-u+1. Are they both correct? I am confused. Link to file: ...
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0answers
41 views

Discrete Convolution of unit step functions

convolution of the following functions? (u[n] - u[n-5]) * (u[n] - u[n-5]) In order to solve it I said: u[n] - u[n-5] = δ[n] + δ[n-1] + δ[n-2] + δ[n-3] + δ[n-4] and the answer is δ[n] ...
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2answers
95 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
37 views

How to prove that convolution on real sequences is associative?

Given two real sequences $\{ a_n \}$ and $\{ b_n \}$, where $n \ge 0$, the convolution operation (denoted $\ast$) is defined as $\{ c_n \} = \{ a_n \} \ast \{ b_n \}$, where $c_n = \sum_{k=0}^{n} a_k ...
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0answers
38 views

Convolution of rectangular pulse and delta function

$\newcommand{\rect}{\operatorname{rect}}$ I am working on evaluating the convolution between two functions: $$ \rect\left(t-\frac{a}{a}\right) * \delta(t-b) $$ I understand that the definition of ...
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0answers
43 views

Does there exist a nontrivial cumulative distribution function $F$ on $\mathbb{R}_+$ for which $(F^2)'= (F')^{\ast k}$ for some $k > 0$?

This question arises from the context of computing the distribution of total execution time with an underlying graph of tree. For instance, we can model the execution times of all the nodes on the ...
4
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1answer
121 views

Convolute exponential with a gaussian

I have data measuring an exponential decay that is convoluted by a gaussian response function. I have the measured shape of the gaussian, and want an analytical expression for the exponential ...