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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-1
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0answers
13 views

convolution and Fourier series [on hold]

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 ...
0
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1answer
17 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 ...
1
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0answers
14 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 ...
0
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1answer
28 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, ...
0
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1answer
31 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 ...
0
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0answers
9 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 $g$?
0
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0answers
14 views

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 ...
0
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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 ...
0
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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? $$ ...
0
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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 ...
0
votes
1answer
17 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\\ ...
3
votes
1answer
34 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 ...
0
votes
2answers
33 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.
1
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0answers
72 views

Approximate convolution of independent Beta variates?

Is there a way to approximate the convolution of Beta variables? Specifically, I am trying to find an approximation to $g(x_0)$: $$g(x_0) = \int \delta(x_0-\sum_{i=1}^{n} a_i x_i) \prod_{i=1}^{n} ...
0
votes
0answers
16 views

Sum of two independent random variable, Convolution.

I need the graphic of this two function to evaluate this correlation?
1
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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 ...
1
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1answer
55 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 ...
0
votes
1answer
20 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?
0
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1answer
371 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 ...
1
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0answers
42 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 ...
0
votes
0answers
42 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 ...
5
votes
3answers
305 views

The condition for Y to make $\mathbb{E}[\max\{X_1+Y,X_2\}] > \mathbb{E}[\max\{X_1, X_2\}]$

I would like to know the condition for a random variable Y in order to make $\mathbb{E}[\max\{X_1+Y,X_2\}] > \mathbb{E}[\max\{X_1, X_2\}]$, where $X_1$ and $X_2$ are iid. Any help would be ...
0
votes
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) ...
0
votes
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 ...
5
votes
1answer
46 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 ...
-1
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0answers
19 views

How to express convolution formula from integral of exponential

I have one formula that is defined: $E=\int g(y-x) q_0 e^{-2f(y)\over a} dy$ where $a$ and $q_0$ are const I want to express the above function as convolution form. How to do it. Thank you so much
12
votes
5answers
751 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 ...
0
votes
0answers
35 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 ...
1
vote
2answers
741 views

Is convolution in spherical harmonics equivalent to multiplication in the spatial domain?

Spherical harmonic convolution is defined as: $$ ( k \star f )^l_m = \sqrt{ \frac{ 4 \pi }{2l+1} } h^l_m f^l_m $$ I have a function with RGB values for every $(\theta,\phi)$ in the spatial domain. ...
0
votes
1answer
13 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?
1
vote
2answers
64 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 ...
2
votes
1answer
67 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 ...
0
votes
0answers
20 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}} ...
1
vote
0answers
21 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 $1/p+1/q=1$ ...
0
votes
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 ...
2
votes
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 ...
0
votes
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 ...
0
votes
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 ...
0
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0answers
21 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 ...
1
vote
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 ...
2
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0answers
33 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 ...
4
votes
1answer
61 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 ...
1
vote
2answers
94 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 ...
0
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1answer
48 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 ...
1
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0answers
13 views

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: ...
1
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0answers
36 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] ...
6
votes
1answer
136 views

Extend a function by convolution

Let $f \in \mathcal{C}^{\infty}(\mathbb{R})$ be a compactly supported function ($supp(f)\Subset\mathbb{R})$. I am wondering about the existence of a $g \in L^p(\mathbb{R})$, for some $p$, such that ...
1
vote
1answer
171 views

Should mean be subtracted before convolution?

Suppose I have a signal in the form of N real numbers. I am searching for a continuous pattern within this signal. The pattern is represented by M real numbers. (M < N) To find the pattern's ...
0
votes
1answer
33 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 ...
0
votes
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 ...