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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1answer
30 views

Reverse of convolution theorem

If I have a convolution $$z(t) = x(t) * y(t)$$ where I know $x(t)$ and $z(t)$, is there a way to determine $y(t)$? Is there a "reverse" convolution theorem for this? I know there are numerical ...
2
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0answers
34 views

Result of a decay condition

Assuming that a function g is such that $ g(x) \leq C ( 1 + |x|)^{(-1 - \varepsilon)}$ for some $\varepsilon > 0$ , then how can we prove that $ \sum_{n = - \infty}^{n = + \infty} | g(x- k - \frac{...
2
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0answers
43 views

Convolution of a function and its inverse

I want to calculate the convolution of a function and its inverse, $$f(t) * f^{-1}(t)$$ e.g. $f(t)=1/(t-2i)$ I've heard that the answer can be a delta function. What requirements are necessary for $...
0
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4answers
41 views

Convolution: Give a proof that $f_T(t)=\int_{-\infty}^{\infty}f_X(x)f_Y(t-x)dx$ where $f_T(t)$ is the PDF of random variable T

Here is the question: Let $X$ and $Y$ be independent, continuous r.v.s with PDFs $f_X$ and $f_Y$ respectively, and let $T=X+Y$. Find the join PDF of $T$ and $X$, and use this to give a proof that $...
0
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0answers
19 views

Convolution involving the inverse Fourier transform

Suppose $$F(k) = \frac{1}{2\pi}\int f(x) e^{ikx} dx$$ and $$G(k) = \frac{1}{2\pi}\int g(x)e^{ikx} dx$$ Where $F(K),G(K)$ are Fourier transforms. Then how can I write the convolution of $F$ and $G$ ...
0
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1answer
48 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
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1answer
61 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) ...
0
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2answers
23 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
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1answer
35 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
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2answers
29 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
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0answers
23 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
1answer
126 views

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 &...
0
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0answers
19 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 C^0_b(\...
0
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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
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1answer
44 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
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1answer
25 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
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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
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0answers
24 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
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1answer
27 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
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0answers
19 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
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1answer
46 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
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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
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1answer
37 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
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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
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1answer
17 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 - u, ...
0
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0answers
16 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}^{} \...
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0answers
37 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
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0answers
13 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 ...
0
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0answers
31 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 & \mbox{else}\end{...
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
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1answer
24 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
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0answers
19 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
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1answer
31 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 $L^p(...
3
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1answer
106 views

Show that if $\,u \in W^{1,p}\left(I\right) \bigcap C_c\left(I\right)$, then $\,u \in W_{0}^{1,p}\left(I\right)$.

I want to show the following statement ($1 \leq p < \infty$), for an open interval $I$: If $u \in W^{1,p}\left(I\right) \bigcap C_c\left(I\right)$ then $u \in W_{0}^{1,p}\left(I\right) $. $W^...
0
votes
3answers
65 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
41 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
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0answers
19 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) = \dfrac{1}{\sigma\sqrt{2\...
1
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0answers
18 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
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0answers
54 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
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1answer
17 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
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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
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1answer
27 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
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0answers
36 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
58 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
50 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
43 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
74 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 $$u*g=\int_{-\infty}^{\infty}g(\tau)u(t-\tau)d\tau=2\int_{t-2}^{t}(e^{-\...
0
votes
0answers
29 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.