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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1answer
120 views

An issue of applying Fubini's theorem in Fourier transform on Schwartz space.

Let $\hat{f}$, $\hat{g}$ be the the Fourier transform of $f$ and $g$ respectively where $f$ and $g$ are the members of the Schwartz space $\scr{S}{(\mathbb{R}^{N})}$. Then in the process of ...
1
vote
1answer
61 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 ...
1
vote
0answers
37 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 ...
0
votes
1answer
85 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 ...
1
vote
1answer
49 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 ...
0
votes
1answer
35 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
vote
0answers
59 views

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

We define the Dirichlet convolution of two arithmetic 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
votes
1answer
213 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
votes
0answers
16 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
votes
1answer
36 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
votes
1answer
95 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
votes
1answer
33 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
52 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
109 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.
2
votes
0answers
71 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
vote
1answer
211 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?
1
vote
1answer
179 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 ...
1
vote
0answers
88 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
1answer
42 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
1answer
58 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?
12
votes
5answers
885 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 ...
1
vote
2answers
81 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
343 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 ...
1
vote
0answers
34 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 $1/p+1/q=1$ ...
2
votes
2answers
78 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
30 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
84 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 ...
1
vote
1answer
45 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
votes
0answers
308 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
199 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 ...
0
votes
1answer
58 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
vote
0answers
17 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
vote
0answers
1k 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] ...
1
vote
2answers
135 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 ...
1
vote
2answers
541 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
55 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
votes
2answers
1k 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 ...
7
votes
1answer
165 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
240 views

Can convolution of two radially symmetric function be radially symmetric?

For example, take $x\in R^3$ and let $f(x)$ and $g(x)$ be radially symmetric. Can we prove that $f\ast g$ is also symmetric?
-1
votes
1answer
42 views

Prove that $f \ast g$ is continuous and bounded if $f\in L^1(R^n)$ and $g\in L^\propto (R^n)$ [duplicate]

My Engliah is no so good and it is my first time to use this website, so I apologize for it if I didnot make myself clearly:)
0
votes
1answer
45 views

Use of convolutions to compute the distribution of the sample mean?

Let's consider N i.i.d continuous random variables from some arbitrary distribution. Why do we have to approximate the distribution of the sample mean using the CLT? Why can't we explicitly compute ...
1
vote
0answers
21 views

Convolution of $\mathbb{1}_{\mathbb{Q}}$

Is it possible to compute the convolution of $\mathbb{1}_{\mathbb{Q}}$ with $e^{-1/(1-t)^2}$?
4
votes
0answers
106 views

Symbolic math engines barf on this ostensibly tractable integral.

$$\frac14 \int_{-M\pi}^{N\pi - s} \cos(tu/M) \cos((t+s)u/M)(1-\cos(t/M))(1-\cos((t+s)/N))\space \mathrm d t$$ with integer $u$. Alpha runs out of time. Maxima gives a tremendous result that can ...
1
vote
1answer
73 views

What is the solution of this differential equation? / How to solve it?

I have the following problem : $$m\ddot{x} + c\dot{x} + kx = f_f\delta(t-t_0) + f_c \sin(\omega t) + f_h \theta (2t_0-t)$$ where $x(t)$ is a function of time, $t>0$ and $t_0>0$ and where ...
1
vote
0answers
110 views

Convolution of a product with focal kernel

Consider the following convolution of a product of two functions $f(x)$ and $g(x)$: $\int f(x')g(x')K_n(x-x') dx'$ where the kernel $K_n$ is a sequence of functions that approach a Dirac delta ...
5
votes
2answers
88 views

Is the convolution of a function $f(x)$ and a polynomial $p(x)$ always a polynomial?

After reading the following question: How do I prove a convolution is a polynomial? I want to ask if that is always the expected result, that is to say, does the following holds? A convolution ...
3
votes
2answers
655 views

Dirac delta convolution with function

I've come into a bit of a snag, and thought some more talented mathematicians could maybe help. I am trying to do the following integral: $$S(x,t) = \int I(z)\delta(x-G(z,t)) \mathrm{d}z,$$ where ...
3
votes
1answer
328 views

Differentiating a convolution integral

I'm trying to turn the integro-differential equation $\phi'(t) + \phi(t) = \int_0^t \sin{(t - \xi)} \, \phi(\xi) \, \mathrm{d} {\xi}$ into the differential equation $\phi'''(t) + \phi''(t) + ...
1
vote
1answer
82 views

Sums of normal CDF's

This is my following problem: $$ CDF_A:F_A(x)=\Phi(x)^2 $$ $$ CDF_B:F_B(x)=1-\Phi(-x)^3 $$ $$ Defining: X=A+(-B) $$ I have those two CDF's and I want to calculate the probability that X is smaller ...
1
vote
0answers
40 views

Asymptotics of a convolution

For $r>1$ define the functions $$f(x)=|x|^{-1/2}\chi_{[-1,1]\setminus\{0\}}\quad\text{and}\quad g(x)=|x|^{-1/2r}(-\chi_{[-1,0)}+\chi_{(0,1]}).$$ I am interested in the asymptotic behavior of ...