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.

learn more… | top users | synonyms

0
votes
0answers
25 views

using convolution for power series solution method for DE's

Say I have a homogeneous linear differential equation of the form $y''+py'+qy=0$ and I want to solve it using the power series solution method. So I use the substitution ...
0
votes
0answers
44 views

Autocorrelation of Raised Cosine Function

Let us define the raised cosine function as follows: $f \left( x \right) = \dfrac{\left( 1 + \cos \left( x \right) \right)}{2}$, for $- \pi < x < \pi$. $f \left( x \right) = 0$, elsewhere. I ...
0
votes
0answers
35 views

Avoiding FFTs by reusing prior FFT results

Background From a mathematical point of view, the formulas similar to the following were produced: $F_1(f) = \mathcal{F}(T(t))$ $F_2(f) = \mathcal{F}(T(t)\times sin\Theta t)$ $F_3(f) = ...
1
vote
1answer
39 views

Convolution of Strictly Convex Function

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a $C^2$, strictly convex function, and $\theta_\epsilon$ the standard approximation to identity ...
1
vote
0answers
61 views

Absolute continuity and convolution

Suppose that $\mu$ is a finite Borel measure on the real line, $f, g\in L^1(\mu)$. Define $\nu=\mu\ast\mu$. Do I understand correctly that the convolution $f\mu\ast g\mu$ is absolutely continuous wrt ...
1
vote
1answer
36 views

What is the generalization for a convolution in $\mathbb C$?

Since the integration range of "the" convolution is $\mathbb R$, what is a sensible generalization in complex numbers? Would one still integrate over $\mathbb R$, or some other path, or over the ...
0
votes
0answers
31 views

Proof commutativity of (differential) convolution operater

I tried to proof a claim and I'm not sure if I did it right. It would be great if someone could have a look at it! First I give a definiton: Let $h : [0, \infty ) \rightarrow \mathbb{R}$. We define ...
1
vote
1answer
19 views

Show separability of discrete convolution.

Given two functions $I, H$ we define the discrete convolution as $$ I' (u,v) = I(u,v) \ast H(u,v) = \sum_{i = -\infty}^\infty \sum_{j = -\infty}^\infty I(u-i, v-j) H(i,j)$$ Now, I need to show that ...
2
votes
1answer
46 views

Fourier transform of a Laplace transform

Is there an easy way to find the Fourier transform of a Laplace transform of function? $$ F[L[f(t)]_{s}] $$ Where my $f(t)$ is $\sqrt{t}$. However, Before finding the Fourier transform I do the ...
2
votes
1answer
40 views

Laplace transform of a product of two functions

I have read questions and answers about this topic and i am still confused, using this formula we can calculate the Laplace transform of a product of two functions: $$ L[a_{(t)} b_{(t)}]={{1}\over{2 ...
1
vote
1answer
46 views

Bound on the derivative of a cut-off function

Let $\rho$ be a smooth function in $\mathbb R^n$ such that $0 \leq \rho \leq 1$ and $\rho$ is supported in the unit disk and let $\rho_\epsilon(x) = \epsilon^{-n}\rho(\epsilon^{-1}\|x\|)$. If $f$ is ...
1
vote
1answer
52 views

Convolution operator positive definite?

Let $\mu$ be a compactly supported Borel probability measure on $\mathbb{R}^n$. Consider the convolution operator $T: L^2(\mathbb{R}^n) \rightarrow L^2(\mathbb{R}^n)$ defined by $$ Tf = f \ast \mu $$ ...
2
votes
2answers
52 views

Is the convolution operation some kind of group operation?

I'm just curious but will the convolution operation be any sort of group operation? A motivating example would be to see that the natural exponential family of distribution functions are closed under ...
1
vote
1answer
47 views

Probability of a single variable from a Moment Generating Function

This is from the A/S/M study guide, and the answer is listed, I just don't understand how he's arriving at the answer... I'm sure it's something simple I am missing! I have two identically ...
2
votes
2answers
81 views

Density of sum of two uniform random variables

I have two uniform random varibles. $X$ is uniform over $[\frac{1}{2},1]$ and $Y$ is uniform over $[0,1]$. I want to find the density funciton for $Z=X+Y$. There are many solutions to this on this ...
0
votes
0answers
38 views

Re-expressing the Integral $\int h(t-\tau)x(t-\tau)x(\tau)^2\,\text{d}\tau$

Given the integral: $$ \int^{t}_{0} h(t-\tau)x(t-\tau)x(\tau)^2\,\text{d}\tau = \int^{t}_{0} h(t-\tau)F(\tau)\,\text{d}\tau + K $$ Can you find $F(\tau)$ so it is not a function of $t$ and find $K$ ...
0
votes
0answers
28 views

Variance reduction factor using gaussian filtering

I am currently trying to find the variance reduction ratio using gaussian filtering. For a simpler filter (as mean filtering for example), I am able to calculate it easily to find the well known ...
2
votes
0answers
50 views

Do asymptotically equivalent coefficients survive convolution at least in Θ?

This is a follow-up question to this one where I asked if asymptotic equivalence of coefficients carried over after convolution, resp. why this was not the case. Answerer Daniel Fischer proposed that ...
1
vote
0answers
52 views

What is $D\delta$ if $D$ is ordinary differential operator and $\delta$ is the Dirac distribution?

I'm reading some material about single-variable distribution theory. More specifically, I was checking some theorems of the convolution algebra $\mathcal{D}_+$, where $\mathcal{D}_+$ is the space of ...
3
votes
1answer
118 views

Why does convolution not maintain asymptotic equality of coefficients?

Assume I have four (generating) functions $f$, $f'$, $g$ and $g'$. If that is interesting, we can assume that they all share the same radius of convergence $\rho > 0$. In addition, we know that ...
2
votes
2answers
118 views

Intuition behind convolution identity for Laplace transforms

Convolutions, relatively speaking, are fairly straightforward for simple systems (from an applied perspective), but I cannot, at all, find the intuition behind the Laplace identity for convolutions. ...
2
votes
1answer
77 views

convolution of compactly supported continuous function with schwartz class function is again a Schwartz class function?

Suppose $f$ is continuous function on $\mathbb R$ with compact support; and $g\in \mathcal{S}(\mathbb R),$ (Schwartz space) My Question is: Can we expect $f\ast g \in \mathcal{S(\mathbb R)}$ ? ...
0
votes
0answers
32 views

Change of Variables for two level Guassian model

I have a multivariate Gaussian distribution from which two variables, u and v, are drawn. The next variables, U and V, are U = 1/(u^2+5) + N(0,sig_U) and V = v^3 + N(0,sig_V). U and V are known, ...
0
votes
1answer
46 views

Using Laplace transforms to solve a convolution of two functions

Hi I have this problem where I need to take the convolution of functions and I am not sure if I got the right answer or something close so any advice or help would be very appreciated. So here is the ...
1
vote
1answer
28 views

Apparent paradox commuting this convolution: where is the mistake?

Starting with some vector $x$, I am performing two operations: First, I convolve $x$ with another vector $g$ to compute $x*g$, where $~*~$ denotes convolution. Second, I pointwise multiply the result ...
1
vote
1answer
33 views

Another proof of the iniectivity of a linear operator

Let $g(x)= \chi_{[-\frac{1}{2}, \frac{1}{2}]}(x) $, and $ T \colon L^2(\mathbb{R}) \longrightarrow L^2 (\mathbb{R})$ , $Tf= g \star f$. I was asked to prove that $T$ is injective, and I succedeed ...
0
votes
1answer
65 views

Operations on Random Variables

It is known that the equivalent resistance of a parallel combination of two resistors is equal to \begin{align*} R = \frac{R_1R_2}{R_1+R_2} \end{align*} which could be also written as ...
0
votes
0answers
24 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 ...
4
votes
2answers
98 views

Easy way to compute $Pr[\sum_{i=1}^t X_i \geq z]$

We have a set of $t$ independent random variables $X_i \sim \mathrm{Bin}(n_i, p_i)$. We know that $$\mathrm{Pr}[X_i \geq z] = \sum_{j=z}^{\infty} { n_i \choose j } p_i^j (1-p_i)^{n_i -j}.$$ But is ...
0
votes
0answers
27 views

Partial derivative in frequency domain when only time domain function is known

I want to calculate $$ \frac{\partial F_p(X(\omega))}{\partial X(\omega)} $$ So $F_p$ operates in some way on $X(\omega)$ but I know the analytical form only in time domain, represented by $f_p$. ...
1
vote
1answer
64 views

Question about transformations and sums on uniformly distributed random variables.

I'm looking into a few problems as a hobby of mine, and found myself with the following problem: let $X$ be a random variable uniformly distributed on $[0,1]$. What is the probability that after $N$ ...
0
votes
1answer
31 views

Convolution module

I am trying to apply GCC-PHAT algorithm here to process audio files and find delay between them. Im coding using Android and Java with the help of this library, and comparing the results with Matlab. ...
3
votes
1answer
62 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 $g \in L^1$
2
votes
0answers
67 views

Expectation and convolution question.

I am learning in an image processing course, and the professor did the following: As part of a derivation, has this: What I do not understand, is how he was able to remove $r(i,j)$ to the ...
1
vote
1answer
25 views

Holder continuity of $\frac{x}{|x|^3} \ast f$ with $f \in C^1_0$ in $\mathbb{R}^3$

Ok, so I need to show that for $f \in C^1_0(\mathbb{R}^3)$ the convolution with $k(x) := \frac{x}{|x|^3}$ is Holder continuous. The exponent doesn't matter much as long as I can bound it using ...
1
vote
1answer
54 views

Prove or disprove: $e^{-nG(x)}$, normalized, is an approximation to the identity for $G(x)$ strictly convex

We are given the sequence of functions $$ \phi_{n} = \frac{e^{-nG(x)}}{\int_{\mathbb{R}}e^{-nG(x)}dx}$$ for a nonnegative, strictly convex function $G$ (that is, $G'' \geq c$ for some $c>0$) that ...
1
vote
0answers
47 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
20 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
30 views

convolution of 3 uniform random variables

Please help. X,Y,Z are uniformly distributed random variates over the closed interval [3,5] independently/ The sum, S, of any of the two random variates X,Y,Z has a triangular distribution with pdf: ...
1
vote
1answer
22 views

Using convolution to impose differentiablilty.

If I had a function $g$ that was not differentiable at a known point, is it possible to convolute it with say a $C^{\infty}$ function $f$, resulting in a differentiable function? Thanks in advance!
1
vote
0answers
45 views

convolution of three functions of two variables

Give three functions of two variables $a(x,y),b(x,y),c(x,y)$ one can construct the following convolution like integral: $y(x,y) = \int dx' dy' a(x',y')b(x-x',y') c(x-x',y-y')$ which I have a hard ...
0
votes
1answer
52 views

Proof of a corollary about associativity of (differential) convolution operater

I m working on the proof the following corollary for ages... I would appreciate any help!! Cor: Let $h : [0, \infty ) \rightarrow \mathbb{R}$. We define the convolution operator $*$ for the ...
0
votes
0answers
11 views

does convolution with the derivative of sinc give the derivative?

If we convolve a sampled waveform, which is sampled according to the Nyquist rate, with the sampled derivative of the sinc function do we get the sampled derivative of the original waveform? Is this ...
1
vote
0answers
36 views

Deconvolution vs convolution.

This is now a second time I am attempting to ask this very important but simple question here. What I want to know is can you do deconvolution by convolving a signal. It is often stated that, for ...
1
vote
1answer
115 views

convolution of function with itself 4 times

I have to compute the convolution of $ f(t) = \frac{1}{\pi}\frac{1}{t^2 + 1} $ with itself 4 times, i.e. $$ f \star f \star f \star f $$ I slightly doubt that doing it in steps, i.e. taking $f \star ...
4
votes
1answer
74 views

What is the pdf of $Z=X/\max(X,Y)$ with $X,Y$ exponentials of lambda parameter?

Given $X,Y$ 2 independent r.v.'s both distributed as $\exp(λ)$, what is the pdf of $Z=X/\max(X,Y)$?
0
votes
2answers
41 views

estimating $L^p$ norm of $\frac{x}{|x|^3} \ast (-\text{div}_x \ldots)$

I've been having problems with following an argument in a paper I'm reading, I hope someone can help me understand the point. Suppose $f(t,\cdot,\cdot)$ is a $C_0^1 (\mathbb{R}^6)$ function for all ...
0
votes
1answer
61 views

Convolution of ring Delta function

Assume $f(r)=\delta(r-R)$ where $\delta(\cdot)$ is a ring delta function. In other word, $f$ is a circular delta function on a circle with radius $R$. I want to do the convolution of $f$ with itself ...
0
votes
2answers
72 views

Derivative of a convolution

I need to find the derivative of the following equation, which I do think is a convolution: Could anybody give me a hint on how to find the derivative of V(x)? Many thanks in advance!
0
votes
2answers
83 views

Pdf of $Z=(XY)^{1/2}$. with X,Y independent r.v. with the same distribution (iid) [closed]

Let be $X,Y$ two independent random variables having the same distribution (the following is the density of this distribution) $$f(t)= \frac{1}{t^2} \,\,\, \text{for $t>1$}$$ Calculate the ...