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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1answer
19 views

Approximations of $L^p$ functions, convolutions, mollifiers, etc. (resource needed)

What is a good resource in which I can read about mollifiers, basic theorems regarding convolutions, smooth approximations of $L^p$ functions and the like? (the presence of exercises would be great, ...
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1answer
19 views

Compute $1_{[0,n]} * 1_{[0,n]}$

$n$ is a natural number. I want to find the convolution of $f = 1_{[0,n]}$ with itself ($1$ is for indicator). Is my work correct $$(f *f)(x) = \int_0^n 1_{[0,n]}(x-y)dy = 1_{[0,2n]}(x)$$ thanks
3
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3answers
65 views

Compute $e^{-x^2} * e^{-x^2}$

How to compute the convolution of $e^{-x^2}$ with itself? $$e^{-x^2} * e^{-x^2} = \int_{\mathbb R} e^{-(x-y)^2} e^{-y^2}dy = e^{-x^2}\int_{-\infty}^{\infty} e^{2xy - 2y^2} dy$$ I can't solve it. I ...
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1answer
35 views

Fourier Transform Proof $ \mathcal F(f(x)g(x))=(\frac{1}{2\pi})F(s)*G(s)$

I need to prove this: $$ \mathcal F(f(x)g(x))=(\frac{1}{2\pi})F(s)*G(s)$$ So far, I believe I have to use the Fourier transform standard equation $$ \mathcal ...
3
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1answer
30 views

convolution: how can I show that $(y*f)'(t) = (y'*f)(t) + y(0)f(t)$

I have the following math problem from my intro to dif. eq. class: (so don't just give an answer) If the convolution $$ (y*f)(t) = \int_0^t y(t-v)f(v)\,dv$$ then show that $$ (y*f)'(t) = ...
1
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1answer
38 views

Laplace Transform: $g(x)=a\sin(x)+\int_0^x \sin(x-u)g(u) du$ [closed]

$$g(x)=a\sin(x)+\int_0^x \sin(x-u)g(u)du$$ I need to find $g(x)$ I believe I need to use Laplace Transform with this in mind (Convolution Thm): $$(f*g)(x)= \int_0^x f(x-t)g(t)dt$$ However I don't ...
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0answers
41 views

Convolutional codes in matlab

I'm trying to construct a convolutional code in Matlab and encode some random data. However the length of the codeword are not as expected. This is the problem information: ...
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0answers
9 views

Simple convolution between 2 signals

I have these two signals: $x_1(t) = 2 rect(\frac{t}{4})$ $x_2(t) = 2 u(t-1)$, where u is the $u(t)$ is the Heaviside function So, $x_1(t) (conv) x_2(t) = ...
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2answers
36 views

Help with integral $\int_0^y x^{-\alpha} (y-x)^{-\alpha} dx$

How should I proceed to work out following convolution integral: $\int_0^y x^{-\alpha} (y-x)^{-\alpha} dx$ for real $\alpha$ > 0. It is the convolution of a powerlaw decaying impulse response ...
3
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0answers
35 views

Equation of convolution of measures

Let $\mu_1,\mu_2$ be two locally finite complex regular Borel measures on $[0,+\infty)$ and $\delta_x$ be the Dirac measure at point $x\in[0,+\infty)$. Suppose that for all $x\in(0,+\infty)$ ...
1
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1answer
26 views

Prove that if $f\in L^p(\mathbb{R_d})$ and $\phi\in\mathbb{S^d}$ then $f*\phi\in\mathbb{C^\infty}$

Show that if $f\in L^p(\mathbb{R^d})$ and $\phi\in\ S(\mathbb{R^d})$ then $f*\phi\in\mathbb{C^\infty}$, where $S(\mathbb{R^d})$ is the Schwartz class. How does one prove this rigorously? I have ...
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0answers
15 views

Approximate 2D convolutions as a sum of separable convolutions

Just like this 3D question, but for 2D. I have a set of 2D convolution kernels, not separable. Is there a good methods to approximate them as a sum of a relatively small number of separable arrays? ...
1
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1answer
31 views

Asymptotic of a convolution integral

$f(x) \ge 0$, $g(x) \ge 0$ are defined on $[0,\infty)$ and $f(x) \sim x^{-a}, \ x \to \infty$, where $a>1$. The integrals $\int_0^\infty f(x)dx<\infty$ and $\int_0^\infty g(x) dx<\infty$. ...
2
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0answers
42 views

Find a function that's convolution with itself is a given function

I would like to solve this equation for $f(x)$: $$ \int_{-\infty}^{\infty} f(z)f(t-z) dz = g(t). $$ Are there any standard ways to solve such problems? $g(t)$ can be assumed to be continuous, but may ...
1
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1answer
65 views

How can I find this sum?

I'm doing some examples related to convolution (digital signal processing). I post my problem here because it is actually mathematics problem. I have to calculate this sum: $$\sum_{k\ = \ n-5}^{n+5} ...
0
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1answer
31 views

Calculating the convolution of $\sin(t)/t$

So how would I go about calculating the convolution of: $$1* \sin(t)/t$$ This will be a simple looking integral, however $\sin$ or $\cos$ are not defined for infinity and negative infinity.
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0answers
64 views

Integral Equation from zero to infinity

Is there anyone could help to solve the following problem: Suppose $\,h\left(x\right)\,$ is a known function and $\,y\left(x\right)\,$ is unknown, you may assume these two are nice functions. I am ...
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0answers
23 views

Calculating a multiple convolution with variables bounded both individualy and by total

I am trying to find a closed form or a transformation which simplify the numerical treatment of this multiple integral $$\int_0^{U_1} \cdots \int_0^{U_N} \delta(U,\sum_g u_g) \prod_g u_g^{n_g} \, du_1 ...
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0answers
23 views

Find $ \mathbb P \{ \max(X, Y) - \min(X, Y) \gt 0.2 \} $.

Let $X$ and $Y$ be iid random variables for which $X \sim \text{Expo}(2)$ and $Y \sim \text{Expo}(3)$. Find $ \mathbb P \{ \max(X, Y) - \min(X, Y) \gt 0.2 \} $. SOLUTION: There is a clever way to ...
0
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1answer
17 views

Convolution bounds

For $t\geq0$, let $g_\beta(t)=e^{-t}\sin(\beta t)$, where $\beta$ is a real number, and for $t<0$, $g_\beta(t)=0$. Find $h*g_\beta(t)$ for all $t\geq0$, where ...
0
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1answer
29 views

Convolution of a signal with the butterworth filter.

Let $f(t)$ be a signal that is $0$ when $t<0$ or $t>1$. Show that, for the Butterworth filter, one has $$Ae^{-\alpha t}\int_{0}^{\min\{t,1\}}e^{\alpha\tau}f(\tau)d\tau$$ My attempt: ...
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0answers
6 views

Gaussian Smoothing Kernel Sigma from x-axis

I already asked a question and got great swift answers: Gaussian smoothing kernel with different sigma values However I have come across another problem. As I understand it, Gaussian kernels used ...
0
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1answer
17 views

Proof involving convolutions

Suppose that $f(t)=0$ for all $|t|\geq a>0$ and that $g(t)=0$ for all $|t|\geq b>0$. Show that $f*g(t)=0$ for all $|t|\geq a+b$. Without loss of generality, assume $b\leq a$. Then ...
1
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1answer
24 views

Convolution of Measures, several definitions

I'm familiar with this definition of convolution : $$f*g(x)=\int f(x-y)g(y) dy$$ But anyone help me see the link between this one and the two definitions hereafter? : Definition 1 : ...
1
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2answers
26 views

Convolution of two piecewise functions

$$\phi(t)=\begin{cases}1&t\in[0,1)\\0&\text{otherwise}\end{cases}$$ and $$\psi(t)=\begin{cases}1&t\in[0,1/2)\\-1&t\in[1/2,1)\\0&\text{otherwise}\end{cases}$$ I know that ...
0
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1answer
26 views

Convolution of $h(t) = u(t+2) - u(t-2)$ and $ f(t) = tu(t) - tu(t-2)$.

Could someone please explain how I perform the convolution? My professor only taught me how to use the table, but I have been teaching myself from the book. I know that convolution is associative and ...
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0answers
39 views

Calculating a function from its auto-correlation

How do I calculate a function if I know its auto-correlation? To be more specific, I have a function of one variable, let's call it $g(x)$, and I know it's the cross-correlation of a function $f(x)$ ...
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0answers
19 views

Classes of binary operations between functions

Let $f,g : D\to \mathbb{R}$ be two functions defined from a domain $D\in \mathbb{R}$ to $\mathbb{R}$. I am looking for classes of binary operations $\circ$ between $f$ and $g$ that produce an ...
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0answers
37 views

Proof verification for the Inverse Fourier transform of a Convolution

Below is a word by word copy from my textbook of a certain derivation for the Fourier transform of a Convolution: Let $g_1(\alpha)$ and $g_2(\alpha)$ be the ...
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0answers
29 views

Proving that the Laplace Transform is an isomorphism with convolution

My question is primarily more about the convolution integral/theorem than the proof in question, but I wanted to give some idea of why I'm asking. The Laplace transform of the convolution $$(f\star ...
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0answers
18 views

finding image and kernel from convolution result

given a one dimension image/(signal) I and three kernels K,L,J. given the results of the convolution of I with each kernel, for example \begin{matrix}I*K =[ ...
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0answers
20 views

Can any integral of this form be written as a sum of convolutions?

Does the second equality always hold? $$ I(x) \equiv \int dy F(y,x-y) = \sum_{i=1}^{N}\int dy f_i(y)g_i(x-y) $$ Motivation: The first integral is not obviously a convolution that I could calculate ...
0
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2answers
59 views

Find the integral $\int\limits_{-\infty}^{+\infty}{\frac{dt}{(1+(x-t)^2)(1+t^2)}}$

My problem is to calculate auto-convolution of $f(x) = \frac{1}{1+x^2}$. I know that $$(f \star f)(x) = \int\limits_{-\infty}^{+\infty}{\frac{dt}{(1+(x-t)^2)(1+t^2)}} = \frac{2 \pi}{x^2+4},$$ but I ...
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0answers
44 views

Solve a function consist of two functions in a sigma summation

I'm still new in here. I have been struggling to solve an equation. The equation is like: f(T)= $\sum_{i=0}^t [g(T-i)*d(i)]$ It is an equation describing the relationship between f(T), g(T) and ...
1
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1answer
31 views

Convolution form of the renewal density

In Asmussen's Applied Probability and Queues, Proposition 2.7 makes a claim about the form of the renewal density in terms of the density of the interarrival distribution $F$, namely: The renewal ...
2
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0answers
21 views

Estimating Self-Convolution of Surface Measure on Sphere

Let $n\geq 2$, and let $\sigma$ denote the standard surface measure on the $n$-sphere $S^{n-1}$, normalized to have total measure $1$. According to the exercise at the end of Terence Tao's blog post ...
1
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1answer
15 views

Interchanging integral and derivative operations in the context of Duhamel's formula

I'll give you the whole context: In solving the heat equation $u_t = ku_xx$ with bounds $u(x,0)=0, u(0,t)=0, u(l,t)=f(t)$, let $v(x,t)$ be the solution for the special case $f(t)=1$. Use the Laplace ...
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0answers
29 views

Help understanding the math of a basic convolution

We are learning about convolutions and I am totally totally lost. I think I understand the idea behind them, basically you're summing the output of the two signals at every point in time. Its the math ...
2
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1answer
40 views

Convolution of convex function and Gaussian is convex

Let $f:\mathbb{R}\to\mathbb{R}$ be a convex function, i. e., for all $x_1, x_2 \in \mathbb{R}$ and $t \in [0, 1]$, $$ \qquad f(tx_1+(1-t)x_2)\leq t f(x_1)+(1-t)f(x_2).$$ I want to prove that the ...
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0answers
39 views

How to calculate the volume of intersection of two cylinders which have radius $r$ and $R$ $(r\le R)?$

The cylinders have their axis meet at right angle. I try to use integral to derive the equation but I have a hard time visualizing the edge of intersection. Could someone help?
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0answers
34 views

Additive combinatorics: Switching $\mathbb{Z}/N\mathbb{Z}$ with $\mathbb{Z}$

For a positive integer $N$, denote $\mathbb{Z}_N = \mathbb{Z}/N\mathbb{Z}$. Now let $F$ be a field and let $f_1, \ldots, f_s : \mathbb{Z}_N \to F$. Can we find functions $\tilde{f_1}, \ldots, ...
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0answers
45 views

Transform with tensor product

I'm new to Laplace and Fourier transforms when convolution is involved, and I've never seen an example involving a tensor product. I'd like to see how the Fourier transforms of the following would ...
0
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1answer
27 views

Problem with convolution, insecure

$$f(t)= t^2\cdot u(t),\quad g(t)=t^4\cdot u(t)$$ I know that I need to use convolution theorem to solve this problem, but I really don't know what to do with step functions. Do I need to include ...
1
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1answer
32 views

Convolution using Integration

Using integration, how would I solve f(t) convolve g(t) given that $$f(t)=u(t)-u(t-5)$$ and $$g(t)=2[u(t)-u(t-1)]$$ I know it should be $$\int_0^6 f(\tau) \ast g(t-\tau)~ d\tau = ...
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0answers
24 views

Square root of Bezier curve via deconvolution

I calculate the product of two Bezier curves via convolution as described in Sanchez-Reyes 2003. I would also like to calculate the square root of a Bezier curve (I have not seen this published ...
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0answers
24 views

Approximation of Sobolev function with convolution

As a homework exercise in Sobolev spaces course we have the following: Suppose $u \in W^{1,p}(\mathbb{R}^n_+)$ and $u_\varepsilon(x)=u(x+2\varepsilon e_n)$, where $e_n=(0,\dots, 0,1)$ and ...
1
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1answer
69 views

Convolution $f * g$

Assume that $f$ is in $L^1 (\mathbb{R})$ and $g(x)= e^{2iπx}$. Compute $f * g$ I just need a hint and not the entire answer. How can I compute the convolution when I don't know what $f$ is?
1
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1answer
48 views

Graphical Convolution

For the first part of the above problem, I copied an example from my book and I got the answer to be $$t(t-1)+t(t-2)=t^2-3t$$ considering that the integral is the sum of the area of the rectangles ...
0
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1answer
24 views

Why is the inf-convolution of lower semicontinuous functions continuous?

I'm confusing now about the continuity of inf-convolution. I understand that the inf-convolution of lower semicontinuous functions is semiconcave and so it's locally Lipschitz continuous (in ...
0
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0answers
26 views

convolution properties of distributions

Let $f,g,h \in D'(R^n)$. How we define the convolution of these functions? I'm trying to show some properties of convolutions such as $\delta\ast f=f$ $(f\ast g)' = f'\ast g=f\ast g'$ $(f\ast g) ...