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Questions tagged [convolution]

Questions on the (continuous or discrete) convolution of two functions. It can also be used for questions about convolution of distributions (in the Schwartz's sense) or measures.

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How does the solution $U∗z(t)$ work for renewal equation?

This post discusses Example 3.5.4 in Resnick's book on page 203. However, when I try to understand the next example 3.5.5, I still have some doubts. Example 3.5.5 If $F(dx) = xe^{-x}dx$, then we ...
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If $F_1$ and $F_2$ are distributions with compact support , how to well define the convolution of them , and how to show $F_1*F_2=F_2*F_1$

Suppose $F_1$ and $F_2$ are given distributions with $F_2$ having compact support , then we define the convolution $F_1*F_2$ as the distributions $(F_1*F_2)(\varphi)=F_1(F_2^{a}*\varphi)$ where $F_2^a(...
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1answer
28 views

Convolution theorem with respect to Laplace transforms

I have recently been learning about Laplace transforms and how to use them for solving ODEs. I understand how to calculate and use both $\mathcal{L}$ and $\mathcal{L}^{-1}$ however I am struggling to ...
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1answer
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Convolution example

Let $f(x)$ be an even function on $(-\infty , \infty)$ and let $g(x) = \sin(ax)$, $a >0$. Show that $$(f \ast g)(x) = \int_{-\infty}^{\infty} f(t)\sin(a(x-t))\,dt= \sin(ax)\hat{f}(a).$$ I ...
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How does this convolution algorithm work mathmatically

I stumbled across this code which describes how you can construct smooth functions with compact support. Unfortunately, I'm not familiar with the programming language used so I can only guess what ...
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An odd-power identity involving convolution - reference request

Let be a power function $f_{r,M}(s)$ defined for every $s$ within the finite set $M$ as follows $$ f_{r,M}(s)= \begin{cases} s^r, \ &s\in M,\\ 0, \ &\mathrm{otherwise}. \end{cases} $$ Let a ...
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1answer
89 views

Find function $f$ such that $(p*f)(x) = xf(x)$

Question: The function $p(x)$ is defined as $p(x) = e^{-x}$ for $x>0$ and $p(x) = 0$ for $x<0$. Find the Fourier Transform of $p(x)$ and use the convolution theorem to find $f(x)$ such that:...
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1answer
24 views

Elements reduction in a general case

If $G$ is a group, then each element has an inverse and $\forall x, y, z \in G, xy = xz \Rightarrow x^{-1} \cdot xy = x^{-1} \cdot xz \Rightarrow 1 \cdot y = 1 \cdot z \Rightarrow y = z$ However, we ...
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16 views

Efficient way to compute the integral $\int_0^{\infty}d\tau f(\tau)\int_0^{\tau}d\theta g(\tau-\theta)h(\theta)$

I would like to know if there exists a way in which the double integral $$\int_0^{\infty}d\tau f(\tau)\int_0^{\tau}d\theta g(\tau-\theta)h(\theta)$$ can be computed (numerically) efficiently. It is ...
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Pick $2$ numbers from $[-1,1]$,what is the probability that their sum is greater than $1$?

Pick 2 numbers from $[-1,1]$, what is the probability that their sum is greater than 1? It is equal to the probability that the sum of 2 uniform random variables on $[-1,1]$ is greater than 1? so ...
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1answer
36 views

Convergence of convolution under $\| \cdot \|_\infty$ with a weaker kernel

Let $(K_\delta)_{\delta > 0}$ be a family of integrable functions so that there is a constant $C \in (0 ,\infty)$ such that $\int K_\delta = 1$, $\int | K_\delta | \leq C$ for every $\delta &...
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1answer
64 views

Proof that Good Kernels are Approximations of Identity in $L^p(\mathbb R^d)$

$\textbf{The Problem:}$ Suppose that $(K_{\delta})_{\delta>0}$ is a family of integrable functions such that there exists a constant $C\in(0,\infty)$ such that $\int K_{\delta}=1,\int\vert K_{\...
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Convolution of bounded Lipschitz function with heat kernel solves the heat equation

Let $\phi_t(x)=\Phi(x,t)=(4\pi t)^{-1/2}\exp(\frac{-x^2}{4t})$ be the heat kernel and let $f$ be a bounded Lipschitz function. Don't ask about the two different $\phi,\Phi$ notations for the heat ...
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Exponential decay of convolution with gradient of heat kernel in the half-space

I am wrestling with the decay of this integral: $$I := \int_{\mathbb{R}^3_+} \nabla_x^k \Gamma(x-y,\frac{1}{2})(\xi(y)f(y))dy,$$ where $f:\mathbb{R}^3_+ \rightarrow \mathbb{R}^3$ is a smooth vector ...
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1answer
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Counter example of $\|f_{\epsilon} -f \|_{\infty}\rightarrow 0$ as $\epsilon \rightarrow 0$

Theorem: Let $f_\epsilon = f* K_{\epsilon}$, where $K\in L^1(\mathbb R^n)$ and $\int_{\mathbb R^n} K =1$. If $f\in L^p(\mathbb R^n)$, then $\|f_{\epsilon} -f \|_p\rightarrow 0$ as $\epsilon \...
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How to prove $f_{\epsilon}\rightarrow f$ as $\epsilon \rightarrow 0$ at each point of contimuity of $f$?

Let $f_{\epsilon} = f * K_{\epsilon}$, where $f\in L^{p}(\mathbb R^n)$, $p>1$, $K\in L^{1}(\mathbb R^n)\cap L^{\infty}(\mathbb R^n)$,$ \int_{\mathbb R^n} K = 1$, and $K(x) = o(|x|^{-n})$ as $|x|\...
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Why we need this assumption of Kernel to get aprroximtion of identity pointwise at every continuous points?

Let me rephrase the question more precisely fist. It is theorem 9.9 on Zygmund's measure and Integral. It assumed that $K$ lies in $L^1(\mathbb{R}^n)\cap L^{\infty}(\mathbb{R^n})$, but in the proof, ...
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DFT modulo $p$: how to find the primitive root $\omega_n$.

On complex numbers: Suppose that we want to find the DFT of the polynomial $A$ given in coefficient form $$a = (a_0, ..., a_{n-1})$$ where $n$ is the length $a$. What we do is to Find the $n$th ...
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Multiplications between four convolutions (discrete time). With partial solution.

This part of the thesis, PAM4 signal filtering. p - convolution noise x - input signal (PAM4) $$\ E(p^4[n]x^2[n]) = $$ Where E - is Expected Value and p[n] = x[n]*e[n], * - denotes convolution. ...
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Condition to the convolution product well defined

i'm lost in the following question: why $(1*\delta')*H$ and why $1*(\delta' * H)$ are well defined? Where $*$ is the product of convolution and $H$ is the function of Heaviside. Thank you in advance
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Integrate / Convolution of a dirac Delta function from 0 to t

I have a question about the convolution integral resulting due to an inverse Laplace transform. Considering one has a multiplication of 2 functions in the laplace Domain $G(s) F(s)= \exp(-a s) F(s)$ ...
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1answer
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Convolution of Delta Functions with a pole

I've been experimenting with a lot of ideas to resolve: Decomposing functions to Taylor-Fourier series That is given a complex function $f: \mathbb{C} \rightarrow \mathbb{C}$ expressed as $$f(x)= \...
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38 views

Integrals involving Dirichlet kernel and sinc function

I am looking to evaluate the following definite integral involving Dirichlet kernel and sinc function with phase. \begin{equation}I(d) = \int_{-\infty}^{\infty} \frac{\sin[N(k^{"}-k^{'})]}{\sin(k^{"}-...
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1answer
19 views

What is the Fourier transform of the product of two shifted-input functions?

I have the following expression $$f(x+\delta)g(x+\delta)$$ I need to find the Fourier Transform. I know for $$f(x+\delta)$$ the Shift Theorem states that the transform is $e^{ik\delta}\hat{f}(k)$, ...
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How to simplify repeated convolution and Hadamard multiplication

I’ve determined that the following expression gives me the correct answer in a programming challenge: $$ (A \circledast M) \times H) \circledast M) \times H) \circledast M) \times H) \circledast M) \...
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Reference request: Symmetry of convolution $(f*f)(n)$

In wikipedia, Convolution, at the right side there are visualizations of various convolutions of pair of functions $f,g$. In case if $f=g$, e.g convolution takes the form $(f*f)(n)$, it is shown that ...
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Burnside convolution

Let $G$ be a group. Say that an orbit is a nonempty transitive $G$-set. Let $\Xi$ be a set of finite orbits such that each finite orbit is isomorphic to exactly one element of $\Xi$. If $X,Y,Z\in\Xi$,...
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Confusion about convolution. Are the integrals equivalent?

The formula for a convolution is given by $$f*g(t,x)=\int_{\mathbb{R}}f(t-u)g(u) \ du. \tag1$$ My question is, is the following equally correct? $$f*g(t,x)=\int_{\mathbb{R}}f(u)g(t-u) du.\tag2$$ I'...
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Understanding scaling and zero padding in 1D convolution

I am trying to understand discrete convolution using scaling $S$ and zero-padding of a vector with the formula given according to: $$ R_{t} = \left\{ \begin{array}{ll} \frac{1}{S}\sum_{i=0}^{k-...
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Convolution of $L^{1}$ functions is well-defined

Let $f,g\in L^{1}(\mathbb R^d).$ The convolution of $f$ and $g$ is the function $f\ast g$ defined by $$(f\ast g)(x)=\int_{\mathbb R^d}f(x-y)g(y)\,dy.$$ Show that $(f\ast g)(x)$ is well-defined for a....
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32 views

Fast Fourier Transform with Negative Integer Exponent

Given $f(x)=ax+b+\frac{c}{x}$ and $N$, I'd like to ask how to calculate $\sum_{i=1}^{N}f(x)^i$ efficiently using fast Fourier transform?
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PDF of the time at which the second raindrops hits

Raindrops hit you at a rate or 1 raindrop per second, what is the PDF of the time at which the second raindrop hits you? Clearly, we have to use exponential random variable. Also we are asked to use ...
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Solving a Partial Differential Equation using Fourier transform and Convolution

Hello I am having an issue with the solution I have obtained for a problem versus the problem given in a book. The PDE is : $$u_t = \alpha u_{xx}$$ with initial condition: $$\phi = e^{-x^2}$$ where ...
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2answers
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Asymptotic behaviour of Convolution power

We have a function $f(t): [0,\infty) \to \mathbb{R}$. The convolution of $f(t)$ with itself is: \begin{equation} (f*f)(t) = \int\limits_0^t \! \mathrm{d}\tau \; f(\tau) f(t-\tau) \end{equation} We ...
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real valued functions that are invariant under convolution

Consider the function $f(x) = e^{-x^2}$, notice that $f$ convoluted with itself is of the form $a\cdot f(bx+c)$ for reals $a, b$ and $c$. Another way of saying this is that the shape of the function $...
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1answer
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Convolutions of two functions

I am having trouble understanding how you take the convolution of two functions. For example, if $f_1(x) = x$, with x ranging from [0,X] how do I solve for $f_2(x)$ when $$f_2(x) = \int_{0}^{\infty} ...
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Local Convergence when differentiating mollifiers

Let $\Omega$ be some domain (connected, open) in $\mathbb{ R}^n$ so that $u$ is defined on it. If we mollify an $L^1(\Omega)$ function $u$ by convolving it with a bump function, i.e taking $\phi_\...
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Integral equation with convolution kernel and non-stationary kernel

I have encountered this kind of integral equation: $$\int f(t')g(t-t')dt'=\int f(t')h(t,t')dt'\quad ,$$ where $f,g,h$ are smooth functions and $f$ is arbitrary. Can $h$ be related directly to $g$ ...
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Find the coefficient of $x^{13}$ in the convolution of two generating functions

Four thiefs have stolen a collection of 13 identical diamonds. After the theft, they decided how to distribute them. 3 of them have special requests: One of them doesn't want more than ...
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1answer
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Use convolution and conditional probability method to find $f_Z(z)$

Exercise $\boldsymbol{10.44}$. Let $X$ and $Y$ be independent continuous random variables with probability density functions $f_X(x)$ and $f_Y(y)$. Compute $P(X+Y \leq z)$ by first conditioning on $Y=...
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Is the rectangular function a convolution of $L^1$ functions?

Do there exist functions $f,g$ in $L^1(\mathbf{R})$ such that the convolution $f \star g$ is (almost everywhere) equal to the indicator function of the interval $[0,1]$ ?
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How to compute DT convolution?

I am confused as to what to do when we have a DT convolution, like the following: $x[n]=u[n]; h[n]=2^nu[-n]$ What I tried was: $y[n]=\sum_{k=-\infty}^\infty u[k]2^{-n+k}u[n-k]=2^n\sum_{k=0}^\infty (...
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1answer
26 views

Laplace Transform of an integral function of a convolution

Making suitable assumptions wherever necessary, what is the Laplace Transform $\mathcal{L}(S(t))$ where $S(t)=\int_{0}^{t}\int_{0}^{t}f(t-s_1,t-s_2)g(s_1)h(s_2)ds_1ds_2$. I tried using the Double ...
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2answers
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convolution of nonvanishing function

I guess function $f$, defined by $$f(x)=\int_{\mathbb{R}^2}\log|x-y|\exp(-|y|^2)dy,$$ is 'not' vanishing at infinity. But I don't know how to prove rigorously.
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Convolution notation.

If $s(\cdot,\cdot),\frac{\partial s}{\partial x}(\cdot,\cdot) \in C(\mathbb{R}\times\mathbb{R^{+}}), K(\cdot)\in C(\mathbb{R}) $ are integrable. The convolution between $\frac{\partial s}{\partial x}(\...
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72 views

constant function under convolution with 3 *

$p$ is prime Can someone show the intermediate steps, I don't understand the $1$st step even with the definition of a convolution in front of me. Thank you :)
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2answers
64 views

Show that convolution of two $L^1(\mathbb{R})$ functions is continuous

Suppose $f, g \in L^1(\mathbb{R})$. I want to show that their convolution is continuous. I can show continuity if one of the functions were in $L^\infty(\mathbb{R})$. I have tried to approximate ...
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0answers
29 views

Understanding/calculating the fourier coefficients of multiplied functions

I am hoping to get some clarifications/help on dealing with coefficients of a multi-dimensional Fourier series. First, I apologize for any mistakes or notations that may be off, I know just enough ...
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0answers
27 views

Log concavity of repeated convolution

Let $f\in C^\infty(\mathbb{R})$ be a smooth and positive function with support in $[-1,1]$, satisfying $\int f(t)\,dt = 1$. Define $g_1 = f$ and $g_{k+1} = f\ast g_k$. That is, $g_k$ is the $k$-fold ...
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0answers
50 views

Is integration by parts used in this equality?

The starting point is this convolution $$ \frac{\partial v_0}{\partial G}(t) = \int_0^t v_G(\xi) \Psi^{(0)}_G(t-\xi) \,d\xi. $$ Applying the product rule for differentiation \begin{align*} \frac{\...