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
votes
1answer
39 views

Norm of convolution

Let $f,g: \mathbb{R}^n \to R$. Let $|| \cdot ||_T$ be a translation invariant norm on functions on $\mathbb{R}^n$. How can I prove that $||f*g||_T \leq ||f||_1 ||g||_T$ (where $f*g$ means convolution ...
0
votes
0answers
15 views

Singular integral operator on a decay Hölder space

Denote $$E^{k,m}=\{f\in C^m(\mathbb{R}^3)\mid\sup_{x\in\mathbb{R}^3}(1+|x|)^{k+|\alpha|}|\partial^\alpha_xf(x)|<\infty\}.$$ Now given $f\in E^{2,m}$ for any fixed $m\in \mathbb{Z}^+$, and $\hat\...
2
votes
0answers
31 views

Sum of two logarithmic random variables

I would like to compute the PDF of the difference of the logarithms of two shifted Rayleigh laws ($Z$): \begin{equation} Z = \log{X_{1}} - \log{X_{2}} \end{equation} where $X_1 \sim R(\alpha_1, \...
4
votes
1answer
38 views

Is it possible to express an integral equation inside of a convolution

Given $$u(t) = \int_0^t y(\tau) d\tau$$ Consider a convolution type of integral $$W = \int_0^t\lambda^{t-\tau}y(\tau) d\tau$$ $\lambda$ a positive real number Is it possible to write $W = f(u(...
1
vote
3answers
47 views

How to derive through a convolution?

Let $f(t) = \alpha e^{-\beta t}$, where $\alpha, \beta$ are constants Let $g(t) = y(t)$ Then the resulting convolution $f\ast g$ is: $$f \ast g = \int_0^t \alpha e^{-\beta (t-\tau)} y(\tau) d\tau$$...
1
vote
1answer
20 views

Questions about the proof: Continuous function with weak derivatives $\Rightarrow$ $C^1$

For an open set $\Omega$ of class $C^1$, suppose we have $u \in W^{1,p}(\Omega)$ and that $u$ is continuous and all the partial derivatives of $u$ are continuous. I want to show that $u$ is $C^1(\...
1
vote
2answers
43 views

Convolution of two rectangular pulses

Determine the shape of the following function$$\int^\infty_{-\infty} \Pi(4\tau) \Pi(t-\tau) d\tau$$ Attempt: This function is a convolution of two rectangular functions. I know that the result has ...
0
votes
1answer
32 views

A proof of the fact that the Fourier transform is not surjective

Let $f_n = \mathbb 1_{[-n,n]}$ for all $n \in \mathbb{N}$ 1) Compute explicitly $f_n \star f_1$ for all $n \in \mathbb{N}$. 2) Show that $f_n \star f_1$ is the Fourier transform of $g_n = \...
1
vote
0answers
9 views

Joint distribution of sum and summand

Let $Z_1$ and $Z_2$ be independent random variables with known distributions $F(.;\theta_1)$ and $F(.;\theta_2)$ of the same convolution closed family. Then $Y = Z_1 + Z_2$ has distribution $F(.;\...
0
votes
0answers
21 views

Approximate n^th power convolution

What is the approximation for $n^{th}$ convolution power (n-fold convolution) $g(x)= \underbrace{p * p * p * \cdots * p * p}_n$ with respsect to $p(x)$, where $p(x)$ is a probability density function?...
3
votes
1answer
55 views

Eigenfunctions of non-uniform convolution

Consider a non-uniform ("generalized"?) convolution operator: $$ A_h[f](t) = \int f(x)h(x,t)dx $$ I would like determine the eigenfunctions. In the "stationary" case where $h(x,t) = h(x-t)$ we have ...
0
votes
1answer
31 views

Writing an operator $T$ defined by $(T f)(t) = \int_{-\pi}^\pi h(t − s)f(s)ds$ as $\sum_{n \in \mathbb Z} \mu_n \langle f, \varphi_n\rangle \varphi_n$

Let $h$ be a continuous function with period $2\pi$. Define $T : L_2[−\pi, \pi] \to L_2[−\pi, \pi]$ by $(T f)(t) = \int \limits _{-\pi}^\pi h(t − s)f(s)ds$. Let $\{\varphi_n(t) =\frac{1}{\sqrt{2\pi}} ...
0
votes
0answers
25 views

Green's function moving dipole source

I am trying to calculate $$\vec{F}(\vec{r},t) = \iint \underline{G}(\vec{r}-\vec{r}',t-t') \frac{\partial}{\partial t'} \left[ \vec{p}(t') \delta(\vec{r}'-\vec{r}_0(t'))\right] d\vec{r}'dt'$$ for $\...
0
votes
0answers
8 views

any way to simplify discrete convolution $g[n]x[n] *g[n]x[n]$ to the form $Gp(x[n])$

Is there any way to simplify discrete convolution $g[n]x[n] * g[n]x[n]$ to the form $Gp(x[n])$ where $G$ is a matrix and $p$ is a feature map which can be computed quite fast. Note $p(x[n])$ must be ...
1
vote
2answers
42 views

Convolution and Fourier transform problem

I was struggling with this question, can use some help. given that $a\not=0$ $$f_a(x) =\frac{1}{x^2+a^2}$$ I'm trying to find k and c dependent on a and b $$(f_a ∗ f_b) (x) = kf_c(x) $$ I know ...
2
votes
0answers
60 views

Convolution proof

If I have two functions in a convolution like $$X*Y=1$$ $$X*Z=1$$ then it means (trivially) $Y=Z$. Is this correct or are there subtleties in the convolution theorem where $Y=Z$ isn't always true?
1
vote
0answers
20 views

derivative of convolution integral

I'm confused for a derivation related to the derivative of convolution. Given that $$ C_{im}(x,t)=\omega e^{-\omega t}*C_m(x,t)+C_{im}(x,0)e^{-\omega t} $$ By taking derivative of the above equation ...
1
vote
0answers
41 views

Is the convolution integration zero?

$$F(\omega)=\frac{\sin \omega}{\omega}$$ $$G(\omega)=\frac{\sin \omega}{\omega}e^{-j2\omega}$$ \begin{align} F(\omega )*G(\omega)&=\int^{+ \infty}_{-\infty} \frac{\sin \tau}{\tau}\frac{\sin (\...
-1
votes
1answer
23 views

Convolution of the cumulative normal distribution and the uniform distribution [closed]

What is the resulting function of convolving the cumulative normal distribution and the continuous uniform distribution?
4
votes
4answers
86 views

Trivial or not: Dirac delta function is the unit of convolution.

My task is to prove that the Dirac delta function is the unit of convolution and all I find always is this formula but no further explanation: $$[f*\delta](t)=\int_{-\infty}^{\infty}f(t-\sigma)\delta(...
1
vote
1answer
29 views

Showing that a “convolution” operator is associative

I am dealing with the following operator $*$ : $(A*B)(t) = \inf\limits_{\tau\in\mathbb{R}} (A(\tau) + B(t-\tau))$. I would like to show that it is associative, i.e : $((A*B)*C)(t) = (A*(B*C))(t)$ . I'...
0
votes
1answer
17 views

Uniform convergence of a sequence of functions given as product and convolution.

Suppose we have, for an open bounded set $\Omega \subset \mathbb{R}^n$: A function $u \in L^p(\mathbb{R}^n) \cap C(\mathbb{R}^n)$. A sequence of mollifiers $(\rho_n) \subset C_c^{\infty}(\mathbb{R}^...
1
vote
0answers
19 views

Fourier Convolution Inversion

Consider a Fourier convolution $f(x) = (g * h)(x)$, where $g$ and $h$ are arbitrary but known functions with reasonable properties. Is there any possibility to determine the inverse function of this ...
2
votes
2answers
42 views

Convolution of square function with itself

I have the square function $$f(x) = \begin{cases} 1, & 0 \leq x \leq 1 \\ 0, & \text{otherwise}\end{cases}$$ and I am trying to calculate the convolution $(f * f)(t)$ using the definition of ...
1
vote
1answer
29 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
votes
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
votes
0answers
42 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
votes
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
votes
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
votes
1answer
42 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
59 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
votes
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
votes
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
votes
2answers
28 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
19 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
121 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
votes
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
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
43 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
24 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
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
vote
1answer
26 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
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
votes
1answer
42 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
14 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, ...