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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0answers
42 views

Help with Fourier transform of product

I was reading this article in wikipedia, and I supposed $f,g \in L^1(\mathbb{R^n})$ such that their product $f \cdot g$ are in $L^1(\mathbb{R^n})$ too. So let $h=f \cdot g$, and $\hat{h}(x)=\int_{\...
3
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1answer
137 views

Convolution of tempered distributions where one has compact support.

For $u\in\mathcal E'(\mathbb R^n)$ and $v\in\mathcal S'(\mathbb R^n)$, we defined $u\ast v$ by $\langle u\ast v, \phi\rangle = \langle v, \check u \ast \phi \rangle$ for all $\phi\in\mathcal S(\mathbb ...
1
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1answer
44 views

Convolution with one of the variables is mixed and the other continuous

Suppose $X$ and $Y$ are independent random variables with CDF $F$ and $G$ and nonnegative support. If $X$ has a point mass $p$ at $0$ and otherwise some "density" $f$ (that is, $\Pr(0<X<a)=\...
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0answers
37 views

Integration of convolution

I'm trying to solve the following equation $$\int\limits_{-\infty}^t \,(f\ast g)(t')dt'.$$ $f$ could be a kind of $\delta$-function: $f(t) = \delta(t)$ but should not be limited to be one. $g$ is ...
0
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1answer
48 views

Simple case of Young's inequality

I have a question concerning Young's inequality stated as follows: $||a∗b||_{ℓ_q}≤||a||_{\ell_1}||b||_{ℓ_q},~~~~ 1≤q≤∞$. Here you can find something on $\ell_q\big(\mathbb{Z}\big)$: Young's ...
2
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2answers
121 views

When is convolution associative?

Convolution is associative on e.g. integrable function on $\mathbb{R},$ but not on distributions. What about the convolution of measures on an unimodular group $G$?
5
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1answer
69 views

Reversing results for sums of independent variables

Please let me use a specific example to illustrate the general title above. (1) It is well known that if $X$ and $Y$ are independent and $X,Y\sim N(0,1)$ then $$ Z\equiv X^2+Y^2\sim\chi_2^2 $$ where $...
1
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0answers
57 views

On the convolution of $f(x)=\sin x/x$ and $g(x)=1-|x|$

I am having trouble with computing the convolution of $f(x)=\sin x/x$ and: \begin{equation} g(x)=\begin{cases} 1-|x|,& -1 \leq x \leq 1 \\ 0, & x \notin [-1,1] \end{cases} \end{equation} I ...
0
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1answer
65 views

1D FFT on rotated image column by column

I am facing a problem: performing 1D FFT on a rotated column by column on a rotated image, described as following: Original Image: Rotated Image: What I have: original image convolution ...
3
votes
2answers
50 views

Steinhaus-like problem

I know there are similar problems on here, but I believe this is not a duplicate. Let $E \subset \mathbb{R}$ be a measurable set of positive finite measure. Define $f:[0,\infty) \rightarrow \...
2
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1answer
41 views

Convolution of various functions

There is asked in an example to do convolution $ h_1(t)*h_2(t) + h_3(t)*h_4(t) $ where $h_1(t) = e^{-2t}u(t)$ $h_2(t) = 2e^{-t}u(t) $ $h_3(t) = e^{-3t}u(t) $ $h_4(t) = 4\delta(t) $ and then the ...
2
votes
1answer
81 views

a Bound for functions in $L^p$ after convolution with a $G_\lambda$ almost a heat Kernel

The following questiion comes from the article of Stroock & Varadhan (Diffusion processes with continuous coefficients I - 1969 - pg 378 ) We consider the operator $G_\lambda$ $$G_\lambda f(s,x)...
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1answer
184 views

Convolution in Matlab with different “sampling”

I am trying to figure out how to "normalize" the convolution that Matlab does (using the "conv" operator). If I have a rect function with spacing T and I do the convolution of that function with ...
4
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2answers
103 views

Estimate of a convolution from a paper by Michael Christ

I don't understand Lemma 2 of the paper Hilbert transforms along curves, II: A flat case by Michael Christ. The situation is as follows. I slightly simplified it from the exact context in the source. ...
0
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1answer
30 views

Does the operand in a convolution have a particular name?

In a convolution: $$(f*g*h)(t) = \int f(x)g(y)h(z) \delta(t-x-y-z) dxdydz$$ do the operands $f,g,h$ have a specific name, besides the general "operand"?
2
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1answer
48 views

How can I find the limit?

Let $f\in$$L^1(\mathbb{R})$and $g\in$$L^1(\mathbb{R})$$\cap$$L^\infty(\mathbb{R})$. My question is how can I find the value $\lim_{x \to \infty}f\ast g (x)$ ? I know that $f\ast g $ is continuous and $...
0
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0answers
59 views

Transformation of function of three random variables

I have been doing exercises in basic probabilities, and there is a type of questions which I am never quite sure of the answer, so I wanted to get some opinion about the way I do it. The question is ...
3
votes
1answer
175 views

Convolution product, Lp spaces

I wonder how to prove the following statement, Let p,q be real numbers s.t $1\leq p \leq\infty$, $1\leq q \leq\infty$ and $ \frac{1}{p}+ \frac{1}{q}=1$ Let $f \in L^p(\mathbb R^n)$ and $g \in L^q(\...
1
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0answers
44 views

Why is polynomial convolution equivalent to multiplication in F[x]/(xn−1)?

Why is polynomial convolution equivalent to multiplication in $F[x]/(x^n-1)$? From this, I still can not understand how to get this $$ \begin{align} &f*g +(x^n-1)\sum_{k=0}^{n-1}\sum_{i+j=k+n}...
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0answers
55 views

Shifting a function for convolution

We have $f(t)$, $h(t)$ and we want to compute the convolution of these two functions. So we will have a dummy variable τ for the product: f(τ)h(t-τ) as we know... Why we are not permitted to ...
0
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0answers
29 views

Graphical transformation : reflect and shift

I know that x[-n] will be reflection of x[n] along y-axis and x[n+k] will shift x[n] to left by k points. Now if I take x[n] 1. x'[n]=x[-n] should reflect along y axis 2. x'[n+k]=x[k-n] should shilf ...
3
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1answer
110 views

Convolution of two functions is a constant

I have the convolution of two functions $f(t)$ and $g(t)$ is a constant $K \in \mathbb{R}$, i.e., $(f*g)(t) = K$ for all $t \in \mathbb{R}$. Taking Fourier transform of the convolution yields $$ F(s)...
2
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0answers
106 views

Laplace transform of inverse error function

I want to calculate the convolution of a function with the inverse error function. Therefore I chose to try to first find an integral transform of the inverse error function like the laplace transform:...
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0answers
26 views

Convolution notations and some miscellanous convolution questions

Convolution between distributions over the real number line, as it is mentioned in the optics course in my uni and also in wolfram is defined as: $$f(t) * g(t)=\int_{-\infty}^{\infty}f(\tau)g(t-\tau)...
0
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1answer
82 views

use laplace transform to solve the given integral equation

use Laplace transform to solve the given integral equation I don't know how start because it differences on other Laplace question I see before
0
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1answer
65 views

Convergence of $u * \eta_\epsilon$

Let $\eta \in C_c^\infty(B(0,1)), \eta \ge 0, \eta$ radially symmetric and $\int_{\mathbb{R}^n} \eta d\mathcal{L}^n = 1$. $\eta_r := r^{-n} \eta(\frac{x}{r}) \in C_c^\infty(B(0,r))$. Integral of $\...
0
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0answers
69 views

Young's inequality for convolutions for functions of bounded support

If $$f\in L^P(\mathbb{R}^d), g\in L^q(\mathbb{R^d}), \; \frac{1}{p}+\frac{1}{q}=1+\frac{1}{r},$$ then Young's inequality for convolutions states $$\|f*g\|_{L^r}\leq\|f\|_{L^p} \|g\|_{L^q}.$$ In ...
0
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1answer
67 views

Branch points of functions defined as convolution integrals

I am studying sets of equations containing convolution integrals of the following type: $$ u\mapsto \int_D dz g(z) f(z-u), $$ where $g$ is analytic, but $f$ has a pole at the origin (so colloquially $...
1
vote
1answer
143 views

Fourier transform of a product of two rect functions

I am trying to evaluate the following expression $$\mathcal{F}\{\mathrm{rect}_{L_{x}}(x)\mathrm{rect}_{L_{y}}(y)\}$$ which denotes the 2-dimensional Fourier transform (reciprocal variables $k_x$, $k_y$...
1
vote
1answer
22 views

integral over domain $1_{(x+y≥100)}$

The problem is: Compute: $\frac {1}{40^2}\int_{40}^{80}\int_{40}^{80} (100-x)1_{(x+y≥100)}dxdy$ my attempt: $$=\frac {1}{40^2}\int_{40}^{80}\int_{100-x}^{80} (100-x) dydx = \frac {1}{40^2}\int_{40}^...
1
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0answers
147 views

Multiple convolutions

Let $\phi(x)=1$ on [0,1] and 0 anywhere else. Is there a was to say what the support of the n-times convolution of phi with itself, that I wann to denote by $B_n$, is? In especially is it possible to ...
4
votes
1answer
133 views

A convolution involving binomials

Given $$f(i)\gt0,\:g(i)>0,\:i =0,1,2,3,...\:$$and$$\sum_{i=0}^{\infty}f(i) = 1,\sum_{i=0}^{\infty}g(i) = 1$$Prove that, if$$\frac{g(l-k)f(k)}{\sum_{i=0}^{l}f(i)g(l-i)}=\binom{l}{k}p^k(1-p)^{l-k}\: ...
0
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4answers
84 views

Laplace of $\int_0^t \frac{sinx}{x}dx$

What is the Laplace transform of $\int_0^t \frac{\sin x}{x}dx$ I'm thinking about approaching it as a convolution but I am not sure how. Could I define it as the convolution of $1$ and $\frac{\sin ...
1
vote
1answer
61 views

Conditional distribution of the sum of k, independent ordered draws

Suppose I make k independent draws from the same distribution (say uniform). The distribution of the sum of these order statistics should equal the distribution of the sum of k independent random ...
1
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1answer
28 views

Convolution, g(t)=sin(t)

I have two functions: $f(t)=(t+\pi)\theta(t+\pi)-2t\theta(t)+(t+\pi)\theta(t-\pi)$, (looks like $-|x|+1, -\pi < x < \pi$ ) and $g(t) = \sin{(t)}$ Could someone please point me in the right ...
-3
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1answer
85 views

Fourier Analysis question on convolution [closed]

Let $f(x) = \operatorname{sinc}(x)^2$. Find $(f*f)(x)$? This is what I tried $f(x)=\mathrm{sinc}(x)^2$ $$ \begin{align} f\ast f(x) &=\int f(u)\cdot f(x-u)\,\mathrm{d}u\\ &=\int\mathrm{sinc}(...
1
vote
1answer
102 views

Convolution of Two Random Variables

I have been working a few hours on this particular problem. Please excuse my lack of formatting. This is the question: Let $X$ and $Y$ be random variables with density function $f(x) = 2x$ on $[0, 1]...
1
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1answer
17 views

Convolution of derivatives

When transforming nonlinear PDE to its Fourier space, I encounter the following problem: Consider the equation $u_t=(u^3-u)_{xx}$. Then, when transforming to Fourier space we get \begin{equation*} ...
1
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1answer
34 views

What is this Toeplitz like matrix called and how do I represent it as a convolution?

I have a matrix that is used to represent the Green's function in a popular class of fast E & M solvers (CG-FFT). The matrix represents distances, that are later filled in using the appropriate ...
0
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0answers
27 views

Convolution of two equations f*g(t)

I am trying to calculate the convolution of: $$ f*g(t) $$ where $f(t) = \delta''(t) -\delta(t) $ and $ g(t)=e^{-t^2}+e^{-t} \theta (t) $ Maybe someone can give me a solution or point me in the ...
0
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0answers
66 views

How to convert infinite intergral to sum

How to convert Wiener filter formulas from integral to sum? They are for images therefore it must be possible to convert them to sums. Any help will be appreciated: I could not find much info on ...
2
votes
1answer
41 views

Is convolution a coercive bilinear form in $L^2$ -space?

This is one of the problems in functional analysis course I'm having. Suppose $f,g \in L^2(0,10)$. Then define a bilinear form $$ B:(f,g)\mapsto \int_0^{10} f(x)g(10-x) dx. $$ Now I have to find out ...
7
votes
2answers
1k views

Proving the sum of two independent Cauchy Random Variables is Cauchy

Is there any method to show that the sum of two independent Cauchy random variables is Cauchy? I know that it can be derived using Characteristic Functions, but the point is, I have not yet learnt ...
1
vote
2answers
67 views

Sums of independent random variables (more than two) [closed]

I read that the convolution of two iid random variables is $$(f * g) (z) = \int f(z-y) g(y) dy$$ What is the general formula for more than two RVs? For example, for three RVs.
0
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1answer
40 views

Graphs of functions defined by convolution

A sequence of functions on the real line is defined as $$f_0=\chi_{[-1,1]},\qquad f_{n+1}=f_n*f_0, n=0,1,2,\dots $$ Here * means convolution. I tried to draw the graphs of the functions and see what ...
0
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1answer
247 views

Sum of two uniform independent random variables

I would like to find the cdf of $Z=X_1+X_2$, with $X_1\sim U(0,1) $, $X_2\sim U(0,2)$ I always prefer to find the cdf instead of the pdf with convolution, and this time I am having troubles with the ...
0
votes
1answer
78 views

Convolution of an integrable function an $L^\infty$ function [duplicate]

Let $f$ be an integrable function on $\mathbb{R}$, and $g$ be an $L^\infty$ function on $\mathbb{R}$. Then, the convolution $f*g$ is said to be continuous and bounded on R. I managed to show that it ...
0
votes
1answer
233 views

The maximum value (peak) of multiple self-convolution of rectangular function

In Multiple self-convolution of rectangular function - integral evaluation, formula for self-rectangular function of rectangular function seems to have been derived. How do we prove that this formula ...
0
votes
1answer
38 views

Discrete convolution equation

Let $x_1 = (x_1^k)_{k =-\infty}^{+\infty}$, $x_2 = (x_2^k)_{k=-\infty}^{+\infty}$, $x_3 = (x_3^k)_{k=-\infty}^{+\infty}$ be three sequences of real numbers such that $x_j^k = 0$ for $k < -m_j < ...
3
votes
1answer
55 views

Given $f\in L^1(\mathbb{R})$ with $\|f\|_1<\infty$ and $g_n=\sqrt{n/2\pi}e^{-nx^2/2},f_n=g_n\ast f$, show that $\lim\|f_n-f\|_1=0$

Given $f$ a Lebesgue integrable function on $\mathbb{R}$ with finite $L^1$-norm, I am asked to show that $\lim_{n\to\infty} \|f_n - f\|_1 = 0$, where $f_n = f \ast g_n$ and $g_n = \sqrt{\frac{n}{2\pi}}...