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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61 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, ...
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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 ...
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25 views

How to calculate the volume of revolution around $x=3$ for the region bounded by x, y axis and $\sqrt{x}+\sqrt{y}=1$?

So I intend to do it in both shell and disk ways. Let's first use shell method (formula: $\int 2\pi x(f(x)-g(x))dx$): Since $x=3$ is the major axis, and $y=(1-\sqrt{x})^2$, we have $V=\int_0^1 2\pi (...
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46 views

What does closed under convolution mean in Probability Theory?

I understand what does it mean for a set to be closed under addition or multiplication, i.e. the sum/product of elements in a set, is still in a set. Now, I am a little bit confuse when it says the ...
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1answer
26 views

Laplace and unit step- multiplication vs convolution

Please be gentle if the question is stupid. When using the laplace transform, you often multiply the function of interest by a shifted unit step function to operate on the positive portion of the ...
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1answer
23 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
20 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
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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
36 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 F(f(x))=\frac{1}{2\pi}\int_{-\infty}^\...
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1answer
31 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) = (y'*f)(...
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1answer
39 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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55 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: https://i.imgur.com/...
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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) = 4\int_{-\infty}^{\infty}rect(\frac{\tau}{4}...
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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 ...
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0answers
36 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 $x\in(0,+\infty)$ $$\...
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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
18 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? ...
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1answer
32 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 $\int_0^\infty g(x) dx<\infty$. ...
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0answers
43 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 ...
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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} ...
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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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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
26 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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25 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 ...
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1answer
18 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 $h(t)=\begin{cases}1/d&t\in[0,d]\\...
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1answer
31 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: \begin{...
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7 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 ...
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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 $$f*g(t)=\...
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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 : $$f*g(A)=\...
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2answers
33 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 $\mathcal{...
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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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40 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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22 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 $h:=f\...
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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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32 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 g)...
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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 =[ 1/3&2/3&1&1&1&1&1&...
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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 ...
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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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45 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 d(t)....
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1answer
34 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 ...
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22 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 ...
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1answer
19 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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36 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 ...
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1answer
45 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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41 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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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, \tilde{...
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46 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 ...
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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 ...
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1answer
36 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 = \int_0^6(u(\...
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34 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 ...