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.

learn more… | top users | synonyms

0
votes
1answer
26 views

Convolution of Gaussian and error function

I am trying to evaluate the following integral: $$ \int_{-\infty}^{\infty}e^{-\frac{x^2}{2}}\Phi(x-t)dx $$ where $$ \Phi(y) = \frac{1}{2} + \frac{1}{2}erf\left(\frac{y}{\sqrt{2}}\right) $$ I have ...
1
vote
1answer
28 views

The convolution is in $L^1$

According to my notes: The convolution is in $L^1$. Indeed $$\left| \int_{\mathbb{R}^n} \left( \int_{\mathbb{R}^n} f(y) g(x-y) dy\right) dx\right| \leq \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} |f(y) ...
-1
votes
0answers
7 views

How would I draw (sketch) the result of DT convolution sum

This picture shows the result of a convolution sum. PICTURE IS HERE! The question is how can I draw $y[n]$. I would appreciate any hints to start
0
votes
2answers
19 views

Alternative integration limits in a Laplace transform

The unilateral Laplace transform of $f(t)$ is $\int_0^\infty e^{st} f(t) \mathrm{d}t$. If we define the transform as $\int_{a}^\infty e^{st} f(t) \mathrm{d}t$, would it conserve all the nice ...
0
votes
1answer
33 views

Continuity of characteristic function

Problem: Let $G$ be an open subset of $\mathbb{R}$. Show that $\chi_G$ is continuous on $G\cup(\mathbb{R}\backslash\overline{G})$. Consequently, $\chi_G$ is continuous a.e. on $\mathbb{R}$. My ...
0
votes
2answers
23 views

DFT and windows

I am using DFT with windows. The way I understand how a window makes the DFT "look" better, is that multiplication in time domain is convolution in frequency domain. Therefore a window with following ...
0
votes
0answers
11 views

Can 2d convolution been represented as matrix multiplication?

Discr. convolution on a discrete periodic signal can be represented as multiplication of input with matrix M. Where M is presented a special case of Toeplitz matrices - circulant matrices. The ...
3
votes
0answers
69 views
+50

Convolution: How to construct it for a given function?

While working on my thesis my advisor handed me an unfinished paper which states the following: First, define the operators \begin{align*} A_i &:= -\operatorname{div}(\sigma_i\nabla) \\ A_e ...
1
vote
1answer
62 views

Static self-gravitating gas spherically symmetric?

Let me state my problem in $\mathbb{R}^3$: Suppose we have an open set $U\subset \mathbb{R}^3$ and a positive $C^2$ density $\rho:U \to (0,+\infty)$ such that its total mass is finite: ...
0
votes
0answers
15 views

Verify the Green's function for Helmholtz equations

It is well known that $$ G(x)=\frac{1}{4\pi}\frac{\exp(ik|x|)}{|x|} $$ is the Green's function for Helmholtz equation $$ (\Delta+k^2)f=0 $$ in $\mathbb{R}^3$. My question is, given $v\in ...
0
votes
0answers
26 views

Laplace transform of a convolution-like function

Is there a way to calculate the Laplace transform of the following function? $$ \sum_{k=1}^{+\infty}f(t-(g(t)-\theta_k))h(g(t)-\theta_k), \qquad t>0. $$ Thanks in advance.
2
votes
1answer
42 views

Convolution - Hölder inequality

I wonder if you guys can help me out with a question(not homework). I have $\phi(x)=\int_\mathbb{R} |f(t)g(x-t)|dt$ where $f \in L^1(\mathbb{R}) $ and $g \in L^p(\mathbb{R})$ and p and p' are ...
0
votes
1answer
21 views

Calculating convolution of binomial distribution using moment generating function

I have two independent random variables $X_{1}, X_{2}$ on the same probability space. $X_{1}$ is bin bin(n, p) and $X_{2}$ ís bin (m, p) with n, m natural numbers and p in the interval [0,1]. I need ...
1
vote
0answers
13 views

Can't integrate when solving this convolution question

Suppose we have $f(t) = 1$ and $g(t) = 100\cos(20t)$. Find $p(t) = f(t)*g(t)$. Solution $$(f*g)(t) = \int f(t-x)g(x)\,dx$$ $$f(t-x)=1$, $g(x)=100\cos(20x)$$ $$=100 \int \cos(20x)\,dx$$ $$p(t) = ...
1
vote
0answers
22 views

Calculation of renewal function $R(t) = \sum{F^n(t)}$?

My textbook defines the renewal function $R(t) = E[N_t] = \sum_{n=0}^\infty F^n(t)$, where $F^n(t)$ appears to be the n-fold convolution of $F$ with itself. $F$ is the distribution of the interrenewal ...
1
vote
1answer
25 views

Convolution with Uniform and Exponential Random Variables

If $X$ Unif~$[2, 5]$ and $Y$ Exp~$(4)$ are independent, what is the probability density function of $X + Y$ ? I'm a bit confused about what the limits of integration should be to find the ...
1
vote
0answers
18 views

Negative Binomial convolution

I've seen a couple of questions where some users provide some help on how to calculate the convolution of two independent variables $X\sim NB(r,p)$ and $Y\sim NB(s,p)$ link 1, link 2. However they ...
0
votes
1answer
39 views

Fourier transform of square function

I am solving a problem about calculating the Fourier transform of the following quadratic function: $$f(x) = \frac{x^2 + 6x + 9}{16}$$ I tried to solve it directly by taking the transform of each ...
0
votes
3answers
24 views

Convolution of $te^{2t}$ and $\delta_1-\delta_2$?

I seek to find $f*g$ where $f=te^{2t}$ and $g=\delta_1-\delta_2$ and $\delta_a(t)= \displaystyle \lim_{\epsilon \to 0^+}d_{a,\epsilon}(t)$; i.e. $\delta$ is the Dirac Delta function. We have learned ...
1
vote
1answer
36 views

A Function of a Convolution (Laplace)

A paper I am reading makes the following claim: Assume that $a_n$ is a series of of positive, distinct, real numbers. Assume that $\epsilon_n$ are independent random standard exponential variables. ...
0
votes
0answers
9 views

How to compute this sum over values of the derivative of the sinc function?

If $g(t)=\frac{sin(\pi t)/T}{\pi t/T}$ and $g'(t) = \frac{\partial}{\partial t}g(t)$, then how to compute this sum? $$SUM = \sum_i a_i \sum_m h_m g'(kT - iT - \tau_k -mT),$$ where $\{a_i\} \in \{\pm ...
0
votes
1answer
11 views

How does multiple integral change into terms multiplying each other in convolution theorem of Laplace?

In the steps of the proofs highlighted below, how does a multiple integral changes in to multiplication of two integral. This is only possible if V is independent of u, but as it turns out V = t - ...
0
votes
0answers
14 views

Convolution to establish Gaussian process

A Gaussian process $z(s)$ can be established by convolving a gaussian white noise process $x(s)$ with a smoothing kernel $k(s)$ http://ftp.stat.duke.edu/WorkingPapers/01-03.pdf $$\\z(s)=\int_{S}^{} ...
0
votes
0answers
28 views

Convolution Finite vs Infinite Support

It is known that the convolution of two Gaussian function is also a scaled Gaussian function. This convolution is taken from $–\infty$ to $\infty$ since the Gaussian function has infinite support. ...
0
votes
0answers
11 views

Upper bound for ratios of “nearly negative binomial” probabilities

Let $\lambda\in(1/2,1)$, and define an iid sequence of nonnegative random variables $\{X_i\}$ which are "nearly" geometric, in that their distribution behaves similarly to the geometric distributions ...
-1
votes
0answers
20 views

How do solve the integral $\int_{-\infty}^{\infty} exp(-a|t|)exp(-a|t-\tau|)dt$?

How do you determine the autocorrelation of $g(t) = exp (-a|t|)?$ Plugging it into the equation $ \int_{-\infty}^{\infty} g(t)g(t-\tau) dt$ would result into something like $ \int_{-\infty}^{\infty} ...
0
votes
0answers
23 views

Discrete convolution of function with itself

Can anyone confirm if I am getting this right. I need to compute discrete convolution of a function with itself. $$f_i = \begin{cases} 1/2 & \mbox{if } i \in \{0,1\} \\ 0 & ...
-1
votes
0answers
18 views

Convolution - Trigonemtry Manipulation

Please see image, I'm just wondering how you get from 1st to the 2nd line? Thanks enter image description here
3
votes
1answer
50 views

Help in finding a function in $L^p$ such that $f*g=||g||_1 f$ with $g\in L^1(\mathbb{R})$ non negative and fixed

I consider a non negative function $g\in L^1(\mathbb{R})$. I want to find a function $f\in L^p(\mathbb{R})$ such that $$ f*g=||g||_1 \:f ,$$ where $*$ is the convolution. I would be very thankful if ...
1
vote
1answer
20 views

Bounds on derivatives of harmonic functions on unit ball

Let $u$ be a harmonic function on the unit ball in $\mathbb{R}^n$. Show that $$\sup_{B_{1/2}} \lvert \nabla u \rvert \leq C(n) \sup_{\partial B_1} \lvert u \rvert$$ More generally, show that ...
0
votes
1answer
18 views

Calculate basic convolution

I'm not totally sure I understand the concept, maybe an easy example will help me understand it. Let f be $ f(x) = 1 $ if $ 0 \le x \le 1 $ and $f(x) =0$ elsewhere. So the convolution is defined ...
0
votes
0answers
17 views

Convolution of two gaussian functions

I want to calculate the convolution $F * G$ of two Gaussian functions without resorting to Fouritertransforms: $F(t) := \exp(-at^2), G(t) := \exp(-bt^2) \qquad a,b>0$ But intuitively I expected ...
1
vote
1answer
26 views

If $f \in L^p(\Omega)$, then $(\rho_n *f) \to f $ in $L^p(\Omega)$, for a sequence $(\rho_n)$ of mollifiers.

I am reading Functional Analysis, Sobolev Spaces and Partial Differential Equations, by Haim Brezis. There it is shown that: If we have $f \in L^p(\mathbb{R}^n)$, then $(\rho_n *f) \to f $ in ...
0
votes
3answers
52 views

Convolution of sine and cosine.

So I came across this question while studying for the GRE Subject Exam, and I am not really sure how I am supposed to handle it. Let $$ f(x) = \int _0 ^{\pi} \sin t \cos (x+t) dt $$ I am to find where ...
0
votes
0answers
32 views

Convolution of Gaussian and parabolic function.

What is convolution of $\exp(-x^2)$ i.e Gaussian function and $2x^2$? I don't have any idea,as I have found that Laplace transform of Gaussian function involves complementary error function,so inverse ...
0
votes
0answers
13 views

Convert a landau function to a gauus function

Assume the Landau Distribution $$p(x) = \dfrac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}\left(x+e^{-x}\right)}$$ What I would like to do is "convert" it to a gauss function $$g(x) = ...
0
votes
0answers
14 views

Iterated convolutions w.r.t. different variables of a function

I do not understand a claim from a paper: Let $b:[0,T] \times \mathbb{R}^n \rightarrow \mathbb{R}$ be a bounded function and let $$b^{n} (t,x) = b(t,x) \ast \psi_n(t) \ast\phi_n(x), $$ where ...
0
votes
0answers
51 views

Square root of dirac delta function

Is there a measurable function $ f:\mathbb{R}\to \mathbb{R}^+ $ so that $ f*f(x)=1 $ for all $ x\in \mathbb{R} $, i.e $$\int\limits_{-\infty}^{\infty} f(t)f(x-t) dt=1 $$ for all $ x\in \mathbb{R} $.
2
votes
1answer
15 views

Uniqueness result for convolution

I have seen the convolution operator in different settings, and I was wondering about the following: Suppose $h=f\ast g$ for an unordered pair of functions $(f,g)$. Does there exist a pair of ...
1
vote
0answers
35 views

Convolution of two distributions

Consider the convolution product: $$H(x)\ast\operatorname{Pf}\dfrac{H(x)}{x},$$ where $\operatorname{Pf}$ denotes pseudo function. This means, that $\operatorname{Pf}\dfrac{H(x)}{x}$ is, as defined ...
1
vote
1answer
22 views

The product of a uniform probability density function and -1

What happens when you multiply a uniform probability density function between -1 and 1 by -1? Does the new uniform distribution become -1/2 between -1 and 1? I am asking because I am trying to find ...
0
votes
0answers
34 views

Finding a function given as a part of a convolution integral

I am trying to solve the following equation for the function $f$. $$t^{-\alpha} \exp{ \left(- \beta x^2 t^{-2 \alpha} \right)} = \int_0^t \frac{f\left(x, s\right)}{t - s}ds$$ where $\alpha$ and ...
3
votes
1answer
53 views

Why is a convolution of an $L^1$ and an $L^p$ functions well-defined?

I was reading the wikipedia article about convolution and I found this one: If $f \in L^1(\mathbb{R}^n)$ and $g\in L^p(\mathbb{R}^n)$, then $||f\ast g||_p \leq ||f||_1||g||_p$. But to $f\ast g$ ...
0
votes
3answers
46 views

what is the convolution of $sin(bx)$ and $e^{-a|x|}$?

my textbook says that $F[f*g] = F[f] \cdot F[g]$ but what is $F[sinbx]$? It doesn't exist right? So how should I solve it? $F$ is fourier transform
0
votes
2answers
38 views

Show that these two differential equations have the same solution

Question: Show that the problems $ax'' + bx' + cx = f(t); x(0) = 0, x'(0) = v_0$ and $ax'' + bx' + cx = f(t) + av_0 \delta(t); x(0) = x'(0) = 0$ have the same solution for $t \gt 0$. Thus the effect ...
1
vote
2answers
60 views

Solving convolution problem with $\delta(x)$ function

Suppose we had the functions: $$g(t)=\theta(t)(e^{-t}+2e^{-2t})+2\delta(t)$$ and $$u(t)=2(\theta(t)-\theta(t-2))$$ Then we have ...
0
votes
0answers
28 views

Convolution of a Pareto and a Uniform distribuion

I would like to convolve a Pareto and a Uniform Distribution. Pareto's PDF: $f_X(x)=\begin{cases} 0 & x < c \\ b\frac{c^b}{x^{b+1}} & x\geq c \end{cases}$ The $[0,a]$ ...
-2
votes
1answer
35 views

How to show $ |f*g|_{1} \le |f|_{1}|g|_{1}$ [closed]

Well, as it is stated in the titel. I have to show $ \|f*g\|_1 \le \|f\|_1\|g\|_1$. thank you already now for your help.
0
votes
1answer
53 views

Proof of Rudin's Theorem 8.14, RCA

In Rudin's proof of Theorem 8.14, which states that convolutions of Lebesgue integrable functions over the real line are Lebesgue integrable, he first proves the result for Borel measurable functions, ...
0
votes
0answers
28 views

How to solve the laplace transform of $f_m(t_m)$ = $f_1(t)$ $\int_{0}^{\alpha} f_2(\tau) d\tau$ + $f_2(t)$ $\int_{0}^{\alpha} f_1(\tau) d\tau$.

Could you please help me to solve the following : if $t_m$ = min($t_1$,$t_2$) The probability density function $t_1$ is $f_1(t_1)$ and $t_2$ is$f_2(t_2)$ then $f_m(t_m)$ is the probability density ...