0
votes
0answers
18 views

What is correlation kernel and compare with gaussian kernel

I read a paper that said about correlation kernel that defined: $$W(x-y)=(α/1+d(|y − x|))$$ where $α =  (\int(1+d(y − x)dy)^{-1}$, $(d(|y − x|))$ is spatial Euclidean distance from the central ...
-2
votes
0answers
56 views

Convolution Integral

Does someone know how to solve this convolution integral? $$V(x)=c_1\int\limits_{-\infty}^\infty \left(r(\tilde x)+\dfrac{c_2}{n(\tilde x)}-n(\tilde x)\right)\left(\sqrt{c_3+(x-\tilde x)^2}-|x-\tilde ...
3
votes
1answer
35 views

$L^1$ norm of convolution

Let $f_{\lambda} = \frac{\lambda}{2}e^{-\lambda |x|}$. Prove that $||f_{\lambda} \ast g - g || \to 0$, when $\lambda \to \infty$, where $g \in L^1$
1
vote
0answers
40 views

Laplace transform $y''''+37y''+36y=g(t)$

Hey this problem is making me insane so have at it and let me know what I keep screwing up. Express the solution of the initial value problem in terms of a convolution integral: ...
1
vote
1answer
16 views

if $f$ is in weak $L^p$ and $\phi$ is $C_0^{1}$ then $f \ast \phi$ is in weak $L^p$

Okay, so I'd like to know if what I wrote in the title is true. Suppose that $f \in L^{p,\infty}(\mathbb{R}^n)$ (weak $L^p$ space) and $\phi \in C_0^1(\mathbb{R}^n)$ [or even $C_0^{\infty}$ if it ...
0
votes
0answers
37 views

First variation of convolution of two nonlinear functions, how to reexpress $\left[x \delta x * x^2 \right]$?

A new variational principle is presented in this paper: Mixed Convolved Action When trying to derive something like the equation of motion of a Duffing oscillator, I take the following approach: Set ...
1
vote
0answers
31 views

Question about the principal value of some integral

So here is my problem, is it possible that $$\int_{[0,1]}f(y)\cot(\pi(x-y))dy= p.v \int_{[0,1]}f(y)\cot(\pi(x-y))dy$$ I see that the left integral is singular for $x=0$ but since I never worked with ...
1
vote
0answers
32 views

Convolving two functions

I'm trying to convolve two functions $f$ and $g$. $$f(x) = e^{-\frac{{(x-p_2)}^2}{2 q_2^2}}$$ $$g(x) = \left(i_1 e^{-\frac{(a-x)^2}{2 \sigma ^2}}+j_1 e^{-\frac{(b-x)^2}{2 \sigma ^2}}\right) \left(i_0 ...
1
vote
1answer
88 views

Convolution $f*g$ is continuous

Statement: Let $f,g: \mathbb{R}^d \rightarrow \mathbb{R}$ be Lebesgue measurable functions such that $f\in L^1(\mathbb{R}^d)$ and $g\in L^\infty(\mathbb{R}^d)$. The convolution $f*g:\mathbb{R}^d ...
0
votes
1answer
32 views

Convolution of fraction function

I know that convolution is defined: $$f*g=\int f(x-y)\cdot g(y) \, dy $$ How to develop below functions to convolution equation $$\int {f(x-y) \over g(y)} \, dy =\text{ ???}$$ and $$\int {f(x-y) ...
1
vote
2answers
73 views

Fourier transform of function

What is Fourier transform of $$f(x)=\frac{e^{-|x|}}{\sqrt{|x|}}?$$ I tried to calculate it using $$F(e^{-|x|})=\sqrt{\frac{\pi}{2}}e^{-|a|}$$ and $$F(\frac{1}{\sqrt{|x|}})=\frac{1}{\sqrt{|a|}}$$ and ...
2
votes
2answers
61 views

How can this integral be rewritten with convolutions?

I've got $f:\mathbb{R}\rightarrow\mathbb{R}$ bounded and I'm trying to write `$\mathtt{f}$,' a discrete version of $f$, where each element in the domain takes on the average of the corresponding ...
0
votes
0answers
107 views

Convolution of piecewise function

I would like to compute the convolution of piece wise function Following is the piecewise function $$ C_a(t) = \begin{cases}0& t\leq t_d\\ ...
4
votes
0answers
97 views

Symbolic math engines barf on this ostensibly tractable integral.

$$\frac14 \int_{-M\pi}^{N\pi - s} \cos(tu/M) \cos((t+s)u/M)(1-\cos(t/M))(1-\cos((t+s)/N))\space \mathrm d t$$ with integer $u$. Alpha runs out of time. Maxima gives a tremendous result that can ...
2
votes
1answer
165 views

Dirac delta convolution with function

I've come into a bit of a snag, and thought some more talented mathematicians could maybe help. I am trying to do the following integral: $$S(x,t) = \int I(z)\delta(x-G(z,t)) \mathrm{d}z,$$ where ...
1
vote
0answers
33 views

Asymptotics of a convolution

For $r>1$ define the functions $$f(x)=|x|^{-1/2}\chi_{[-1,1]\setminus\{0\}}\quad\text{and}\quad g(x)=|x|^{-1/2r}(-\chi_{[-1,0)}+\chi_{(0,1]}).$$ I am interested in the asymptotic behavior of ...
1
vote
2answers
210 views

Evaluating the convolution integral of two sine functions

This is a homework problem, so I'm not looking for a worked out solution, merely to be pointed in the right direction. Convolve x(t) with h(t) where: $$ x(t) = sin(t) \\ h(t) = e^{-.1t}sin(2t)u(t) ...
1
vote
1answer
113 views

$f$ is bounded and continious $\Rightarrow$ the convolution integral $\int f(\tau)g(x-\tau)\text{ d}\tau$ is bounded and continuous

Let $g\in L^1(\mathbb{R}^n)$ and $f:\mathbb{R}^n\to\mathbb{R}$ be bounded and continuous. Why is the convolution integral $$f*g:\mathbb{R}^n\to\mathbb{R}\;,\;\;\;\int f(\tau)g(x-\tau)\text{ d}\tau$$ ...
3
votes
1answer
89 views

Show that for any $f\in L^1$ and $g \in L^p(\mathbb R)$, $\lVert f ∗ g\rVert_p \leqslant \lVert f\rVert_1\lVert g\rVert_p$.

I write the exact statement of the problem: Show that for any $g \in L^1$ and $f ∈ L^p(\mathbb{R})$, p $\in (1, \infty)$, the integral for f ∗g converges absolutely almost everywhere and that $∥f ∗ ...
2
votes
1answer
139 views

How to show that the difference of two Gumbel distributed random variables follows a Logistic distribution?

How can you show that when you have two random variables $X,Y\sim\text{Gumbel}[0,1]$ , then $X-Y\sim\text{Logistic}[0,1]$ . I tried to use the convolution formula ...
0
votes
1answer
91 views

How to calculate the threefold convolution $f*f*f$

Somehow this convolution is driving me crazy. I am trying to calculate for the indicator function $f:=1_{[0,1]}$ the threefold convolution $$f*f*f$$ But honestly, it does not work somehow. ...
0
votes
0answers
196 views

How to bring f(x) from denominator to numerator?

Is it possible to rewrite the following ratio in a way that $f(x)$, its powers or derivatives appears only as numerator. $$\frac{1}{\int_{0}^{c}(c-x)^{2}f(x)dx}$$ $c>0$ is a constant. $f(x)$ is ...
1
vote
0answers
77 views

Adding truncated normals: calculating convolutions

Problem: Suppose that $X$, $Y$, and $Z$ are independent standard normal random variables. What is the probability of: \begin{equation} P\{ X+Y+Z+\Delta>0 \, | \, Z+\Delta>0, \, ...
0
votes
1answer
133 views

Green's Function vs. Fundamental Solution

From the texts I've used, the Green's function is of a problem is $G(x,y)$ such that $LG(x,y) = \delta(x-y)$. The fundamental solution is u(x) such that $Lu(x)=\delta(x)$. They seem to be used for the ...
1
vote
1answer
78 views

Any clue how to solve this convolution integral?

With other words: find a (closed) expression for $\;\overline{\mbox{sinc}}(x)$ . $$ \overline{\mbox{sinc}}(x) = \int_{-\infty}^{+\infty} \frac{\sin(\omega\xi)}{\omega\xi} ...
0
votes
2answers
147 views

What does triple convolution actually look like?

I have to prove associativity of the convolution of three functions. I'm having trouble picturing how the variables will look. For periodic functions $f$, $g$, and $h$, I have the definition $$ (f * ...
1
vote
0answers
39 views

Integral of convolution difference approaches zero

Let $u(x,t)=f(x)\ast\left(\dfrac{1}{2\sqrt{\pi t}}e^{-\dfrac{(at+x)^2}{4t}}\right)$, and suppose that $f\in L^1$. Show that $$\lim_{t\rightarrow 0^+}\int_{-\infty}^\infty|u(x,t)-f(x)|dx=0$$ How ...
0
votes
0answers
28 views

Convolution of complex-valued probability distributions

This may be an elementary question, but I am wondering: suppose that I have two complex-valued random variables $X$ and $Y$ with corresponding density functions $f_X(x)$ and $f_Y(y)$. Obviously ...
2
votes
0answers
280 views

How Heaviside step function changes limits of integration

This question involves the Laplace transform of the convolution of two functions. The derivation in my textbook has a step that really confuses me. First I'll lay out their argument. $$ f(t) = f_1(t) ...
2
votes
1answer
65 views

Does negative distributive property of convolution over cross correlation holds?

Let $\star$ denote convolution binary operation and $\otimes$ denote cross correlation binary operation between two functions. Let $f,g,h$ be functions. Does this negative distribution property ...
3
votes
1answer
191 views

Evaluating the convolution using the convolution integral

I am having trouble evaluating the convolution of two signals using the convolution integral.I want to find the convolution of two signals x and h where, $$ x(t) = \begin{cases} e^{-at} ...
1
vote
1answer
379 views

Integral of repeated convolution of the unit step function

Background Let $\theta$ be the unit step function: $$\theta(x) = \begin{array}{ll} \left\{ \begin{array}{ll} 0 & x \lt 0 \\ 1 & x\ge 0. \end{array}\right. \end{array} $$ Further, the ...
-1
votes
2answers
88 views

Integral of an integral with variable limits

I'd like to prove the following but not sure where to start: ...
2
votes
0answers
195 views

Proof of a method to find the points of maximum slope

According to method described in a paper [1] if we want to find points of maximum slope in a signal $f(t)$, then one has to do following Convolve $f(t)$ with $g(t)$ where $g(t)=-cos(\omega ...
1
vote
1answer
53 views

Recovering function from convolution with a square function

Let $m : \mathbb{R} \to \mathbb{R}$ be a continuous function of compact support. Given $M$ as: $$ M(x) = \int_{\mathbb{R}} m(t) (t+x)^2\ \mathrm{d}t,$$ is it possible recover $m$? I thought it ...
0
votes
1answer
31 views

Convolution and integrating over G(t)

I'm struggling with the following expression in a statistics script: $$H(x) = \int_{-\infty}^\infty F(x-t) dG(t)$$ What does the dG(t) mean exactly? I've never seen that notation before. Background: ...
2
votes
2answers
59 views

Evaluate $\int_{-\infty}^{\infty} \chi_{[0,1]}(x-y) \chi_{[0,1]}(y) \, \mathrm{d}y$

I'm trying to evaluate the integral $$\int_{-\infty}^{\infty} \chi_{[0,1]}(x-y) \chi_{[0,1]}(y) \, \mathrm{d}y$$ where $\chi_{[0,1]}(x)=1$ is the characteristic function, i.e. equals $1$ for $x \in ...
0
votes
1answer
40 views

Function with $|f(x)-\int^{\delta}_{-\delta}f(x+u)du|<\epsilon$

I am looking for a function $f:\mathbb{R}\to \mathbb{R}$ and $\epsilon>0$ such that there is no $\delta>0$, for him any $x\in\mathbb{R}$: $|f(x)-\int^{\delta}_{-\delta}f(x+u)du|<\epsilon$ ...
-2
votes
2answers
60 views

Lebesgue conduct integral

I. Suppose $f\in \mathcal{L^1}(R^n),g\in \mathcal{L^1}(R^n)$, then conduct integral $f*g$ is defined as $f*g(x)= \int_{R^n}f(x-y)g(y)dy$ for all $x$. My task is to prove following statements. (1) ...
0
votes
1answer
123 views

Integration function spherical coordinates, convolution

How can I calculate the following integral explicitly: $$\int_{R^3}\frac{f(x)}{|x-y|}dx$$ where $f$ is a function with spherical symmetry that is $f(x)=f(|x|)$? I tried to use polar coordinates at ...
4
votes
0answers
417 views

Infinite self-convolution for a function

I have a mathematical problem that leads me to a particular necessity. I need to calculate the convolution of a function for itself for a certain amount of times. So consider a generic function $f : ...
0
votes
1answer
52 views

Need help with convolution problem.

I'm just learning about the convolution integral and am stuck on this example problem: Given $x_1(t) = \left \{\begin{array}{lr} 1 : 0 < x < 1 \\ 0: \text{elsewhere} \end{array} ...
1
vote
0answers
71 views

Fubini theorum for integrating 1 dimension of a 3d convolution

I have 3D volume that is convoluted with a 3D blur function. Both are positive and integrate to a finite value. I can see experimentally (meaning playing with matlab) that this is true: $\int_{-a}^{a ...
0
votes
1answer
1k views

Finding limits of integration in convolution

I am struggling to fully get how to choose proper limits of integration when calculating convolutions. Right now I am stuck on a problem where I have to show that when taking the Fourier transform of ...
0
votes
0answers
185 views

What is the purpose and usage of convolution?

I am curious of what the purpose and usage of convolution are. Why is convolution created? In layman's term (and in mathematical term), what defines convolution?
1
vote
1answer
196 views

Integrating a function within a convolution, variable substitution

Let $f(x)= \begin{cases} 1 & |x|\lt 1 \\ 0 & |x|\ge 1 \end{cases}$ I want to find the convolution $f(x) \ast f(x) = \int\limits^{\infty}_{-\infty}f(y)f(x-y)\,\mathrm{d}y$ I started out by: ...
4
votes
1answer
246 views

How can I compute $\int_{-\infty}^\infty f(x)f(y-x)\, \mathrm dx$

If $f(x)=\text{arccot}(x)$ for non-negative $x$ and $0$ otherwise, how can I calculate $$\int_{-\infty}^\infty f(x)f(y-x)\, \mathrm dx$$ for $y\in\mathbb{R}$?
1
vote
1answer
275 views

analytically calculate value of convolution at certain point

i'm a computer science student and i'm trying to analytically find the value of the convolution between an ideal step-edge and either a gaussian function or a first order derivative of a gaussian ...
0
votes
1answer
740 views

Integration Limits for calculating Convolution

to calculate the convolution.. I know how to divide it into intervals. but i don't know how to set the integration limits. i 've been working on it for 10 hours and become wrong results. can anyone ...
0
votes
1answer
149 views

Verify this distribution convolution: $E(t,x)\ast (g(x)\delta(t)) = t\int_{\omega\in S^2}{\frac{g(x-t\omega)}{4\pi}dS(\omega)}$

In our class notes we are asked to verify the following equality: $$E(t,x)\ast (g(x)\delta(t)) = t\int_{\omega\in S^2}{\frac{g(x-t\omega)}{4\pi}dS(\omega)}$$ where ...