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
51 views

The convolution is in $L^1$

According to my notes: The convolution is in $L^1$. Indeed $$\left| \int_{\mathbb{R}^n} \left( \int_{\mathbb{R}^n} f(y) g(x-y) dy\right) dx\right| \leq \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} |f(y) ...
0
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1answer
29 views

Convolution of Gaussian and error function

I am trying to evaluate the following integral: $$ \int_{-\infty}^{\infty}e^{-\frac{x^2}{2}}\Phi(x-t)dx $$ where $$ \Phi(y) = \frac{1}{2} + \frac{1}{2}erf\left(\frac{y}{\sqrt{2}}\right) $$ I have ...
4
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0answers
73 views
+50

Convolution: How to construct it for a given function?

While working on my thesis my advisor handed me an unfinished paper which states the following: First, define the operators \begin{align*} A_i &:= -\operatorname{div}(\sigma_i\nabla) \\ A_e ...
-1
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0answers
7 views

How would I draw (sketch) the result of DT convolution sum

This picture shows the result of a convolution sum. PICTURE IS HERE! The question is how can I draw $y[n]$. I would appreciate any hints to start
0
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2answers
20 views

Alternative integration limits in a Laplace transform

The unilateral Laplace transform of $f(t)$ is $\int_0^\infty e^{st} f(t) \mathrm{d}t$. If we define the transform as $\int_{a}^\infty e^{st} f(t) \mathrm{d}t$, would it conserve all the nice ...
0
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2answers
23 views

DFT and windows

I am using DFT with windows. The way I understand how a window makes the DFT "look" better, is that multiplication in time domain is convolution in frequency domain. Therefore a window with following ...
0
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1answer
34 views

Continuity of characteristic function

Problem: Let $G$ be an open subset of $\mathbb{R}$. Show that $\chi_G$ is continuous on $G\cup(\mathbb{R}\backslash\overline{G})$. Consequently, $\chi_G$ is continuous a.e. on $\mathbb{R}$. My ...
0
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0answers
11 views

Can 2d convolution been represented as matrix multiplication?

Discr. convolution on a discrete periodic signal can be represented as multiplication of input with matrix M. Where M is presented a special case of Toeplitz matrices - circulant matrices. The ...
0
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0answers
15 views

Verify the Green's function for Helmholtz equations

It is well known that $$ G(x)=\frac{1}{4\pi}\frac{\exp(ik|x|)}{|x|} $$ is the Green's function for Helmholtz equation $$ (\Delta+k^2)f=0 $$ in $\mathbb{R}^3$. My question is, given $v\in ...
17
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4answers
3k views

Why convolution regularize functions?

There is a tool in mathematics that I have used a lot of times and I'm still not confortable with. In fact I can't figure out (by this I mean that I cannot understand it geometrically) why does ...
0
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0answers
26 views

Laplace transform of a convolution-like function

Is there a way to calculate the Laplace transform of the following function? $$ \sum_{k=1}^{+\infty}f(t-(g(t)-\theta_k))h(g(t)-\theta_k), \qquad t>0. $$ Thanks in advance.
2
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1answer
42 views

Convolution - Hölder inequality

I wonder if you guys can help me out with a question(not homework). I have $\phi(x)=\int_\mathbb{R} |f(t)g(x-t)|dt$ where $f \in L^1(\mathbb{R}) $ and $g \in L^p(\mathbb{R})$ and p and p' are ...
1
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1answer
489 views

Distribution of the sum of many lognormal random numbers from same distribution

In my application I have to sum up a lot (between 1000 and 2000) lognormally distributed random numbers and use their sum. All random numbers that I sum up follow the same distribution. The current ...
3
votes
1answer
53 views

Why is a convolution of an $L^1$ and an $L^p$ functions well-defined?

I was reading the wikipedia article about convolution and I found this one: If $f \in L^1(\mathbb{R}^n)$ and $g\in L^p(\mathbb{R}^n)$, then $||f\ast g||_p \leq ||f||_1||g||_p$. But to $f\ast g$ ...
0
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1answer
21 views

Calculating convolution of binomial distribution using moment generating function

I have two independent random variables $X_{1}, X_{2}$ on the same probability space. $X_{1}$ is bin bin(n, p) and $X_{2}$ ís bin (m, p) with n, m natural numbers and p in the interval [0,1]. I need ...
1
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0answers
13 views

Can't integrate when solving this convolution question

Suppose we have $f(t) = 1$ and $g(t) = 100\cos(20t)$. Find $p(t) = f(t)*g(t)$. Solution $$(f*g)(t) = \int f(t-x)g(x)\,dx$$ $$f(t-x)=1$, $g(x)=100\cos(20x)$$ $$=100 \int \cos(20x)\,dx$$ $$p(t) = ...
1
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0answers
22 views

Calculation of renewal function $R(t) = \sum{F^n(t)}$?

My textbook defines the renewal function $R(t) = E[N_t] = \sum_{n=0}^\infty F^n(t)$, where $F^n(t)$ appears to be the n-fold convolution of $F$ with itself. $F$ is the distribution of the interrenewal ...
1
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1answer
25 views

Convolution with Uniform and Exponential Random Variables

If $X$ Unif~$[2, 5]$ and $Y$ Exp~$(4)$ are independent, what is the probability density function of $X + Y$ ? I'm a bit confused about what the limits of integration should be to find the ...
0
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1answer
39 views

Fourier transform of square function

I am solving a problem about calculating the Fourier transform of the following quadratic function: $$f(x) = \frac{x^2 + 6x + 9}{16}$$ I tried to solve it directly by taking the transform of each ...
0
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1answer
416 views

Relation between Correlation and Convolution

We have two functions of time $f(t)$ and $g(t)$, for which convolution and correlation are defined as following: Convolution: $(f(t)\ast g(t))(\tau) = \int_{-\infty}^\infty{f(t)g(\tau-t)dt}$ ...
3
votes
2answers
248 views

Fourier transform of convolution for $L^2$ functions

If $f,g\in L^1(\mathbb{R})$, it is not hard to show by definition that $$(\hat{f\ast g)}(t)=\hat{f}(t)\hat{g}(t).$$ But what about if $f,g\in L^2(\mathbb{R})$? The Fourier transform on ...
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2answers
42 views

Extension of the convolution theorem

From the convolution theorem, we know that the multiplication in the frequency domain is equivalent to convolution in the time domain, and vice-versa. I am wondering if there is some kind of ...
3
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1answer
381 views

Convolution with itself equals itself times a function

Consider the case that $f \in L^1(\mathbb{R})$ and $g \in L^1_{loc}(\mathbb{R})$. Then look at the equation $$ f*f=g\cdot f. $$ I know that if $g$ is constant, then $f=0$. But what about other $g$'s? ...
2
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2answers
310 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 $f\ast g \in \mathcal{S(\mathbb R)}$ ? ...
7
votes
2answers
178 views

Extend a function by convolution

Let $f \in \mathcal{C}^{\infty}(\mathbb{R})$ be a compactly supported function ($supp(f)\Subset\mathbb{R})$. I am wondering about the existence of a $g \in L^p(\mathbb{R})$, for some $p$, such that ...
4
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2answers
1k views

What will be the support of the convolution of two test functions.

If $g\in C^{\infty}_c$ defined on $\Bbb R^n$ and K is the support of function $g$. I want to find the support of $g_\epsilon$. Where $g_\epsilon$ is regularization of $g$. Regularization of $g$ is ...
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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2answers
299 views

Function as a convolution product of other two

I need help with this: I have to prove that a function $f\in L_{2}(T)$ can be expressed as $f=g*h$ (convolution product) for some functions $g,h\in L_{2}(T)$ if and only if $(\hat{f}(n))_{n}\in ...
1
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0answers
18 views

Negative Binomial convolution

I've seen a couple of questions where some users provide some help on how to calculate the convolution of two independent variables $X\sim NB(r,p)$ and $Y\sim NB(s,p)$ link 1, link 2. However they ...
0
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3answers
24 views

Convolution of $te^{2t}$ and $\delta_1-\delta_2$?

I seek to find $f*g$ where $f=te^{2t}$ and $g=\delta_1-\delta_2$ and $\delta_a(t)= \displaystyle \lim_{\epsilon \to 0^+}d_{a,\epsilon}(t)$; i.e. $\delta$ is the Dirac Delta function. We have learned ...
1
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1answer
36 views

A Function of a Convolution (Laplace)

A paper I am reading makes the following claim: Assume that $a_n$ is a series of of positive, distinct, real numbers. Assume that $\epsilon_n$ are independent random standard exponential variables. ...
1
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1answer
426 views

Convolution: $ f (-)*g = g(-)* f$ does this mean both $f$ and $g$ have to be even functions?

Assuming $f$ and $g$ are different functions, does $ f (-)*g = g(-)* f$ mean both $f$ and $g$ have to be even functions? In fact, this is equivalent to $f\star g = g \star f$ (i.e., cross-correlation ...
5
votes
2answers
260 views

Number Theoretic Transform (NTT) to speed up multiplications

I recently heard that the Number Theoretic Transform (NTT), which is a specialization of Fast Fourier Transformation (FFT) over the ring modulo an integer, can be used to speed up certain ...
0
votes
1answer
11 views

How does multiple integral change into terms multiplying each other in convolution theorem of Laplace?

In the steps of the proofs highlighted below, how does a multiple integral changes in to multiplication of two integral. This is only possible if V is independent of u, but as it turns out V = t - ...
0
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0answers
9 views

How to compute this sum over values of the derivative of the sinc function?

If $g(t)=\frac{sin(\pi t)/T}{\pi t/T}$ and $g'(t) = \frac{\partial}{\partial t}g(t)$, then how to compute this sum? $$SUM = \sum_i a_i \sum_m h_m g'(kT - iT - \tau_k -mT),$$ where $\{a_i\} \in \{\pm ...
0
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0answers
14 views

Convolution to establish Gaussian process

A Gaussian process $z(s)$ can be established by convolving a gaussian white noise process $x(s)$ with a smoothing kernel $k(s)$ http://ftp.stat.duke.edu/WorkingPapers/01-03.pdf $$\\z(s)=\int_{S}^{} ...
1
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0answers
118 views

Using the Kuramoto-Sivashinsky operator applied on the Korteweg–de Vries Soliton as a filter for image processing

When the Kuramoto-Sivashinsky operator (Kuramoto-Si) is applied to the Korteweg–de Vries Soliton (Soliton) we obtain a very interesting filter which is able to process an image via convolution. An ...
0
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0answers
29 views

Convolution Finite vs Infinite Support

It is known that the convolution of two Gaussian function is also a scaled Gaussian function. This convolution is taken from $–\infty$ to $\infty$ since the Gaussian function has infinite support. ...
0
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0answers
11 views

Upper bound for ratios of “nearly negative binomial” probabilities

Let $\lambda\in(1/2,1)$, and define an iid sequence of nonnegative random variables $\{X_i\}$ which are "nearly" geometric, in that their distribution behaves similarly to the geometric distributions ...
0
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0answers
23 views

Discrete convolution of function with itself

Can anyone confirm if I am getting this right. I need to compute discrete convolution of a function with itself. $$f_i = \begin{cases} 1/2 & \mbox{if } i \in \{0,1\} \\ 0 & ...
3
votes
1answer
50 views

Help in finding a function in $L^p$ such that $f*g=||g||_1 f$ with $g\in L^1(\mathbb{R})$ non negative and fixed

I consider a non negative function $g\in L^1(\mathbb{R})$. I want to find a function $f\in L^p(\mathbb{R})$ such that $$ f*g=||g||_1 \:f ,$$ where $*$ is the convolution. I would be very thankful if ...
0
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2answers
2k views

Undecimated Wavelet Transform (a trous algorithm) - how to determine 'anchor'/'center' of convolution filter

i am currently implementing the 'Undecimated Wavelet Transform' with the 'a trous' algorithm. See e.g. http://www.znu.ac.ir/data/members/fazli_saeid/DIP/Paper/ISSUE2/04060954_2.pdf, section II-A. As ...
1
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1answer
20 views

Bounds on derivatives of harmonic functions on unit ball

Let $u$ be a harmonic function on the unit ball in $\mathbb{R}^n$. Show that $$\sup_{B_{1/2}} \lvert \nabla u \rvert \leq C(n) \sup_{\partial B_1} \lvert u \rvert$$ More generally, show that ...
0
votes
1answer
18 views

Calculate basic convolution

I'm not totally sure I understand the concept, maybe an easy example will help me understand it. Let f be $ f(x) = 1 $ if $ 0 \le x \le 1 $ and $f(x) =0$ elsewhere. So the convolution is defined ...
0
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0answers
17 views

Convolution of two gaussian functions

I want to calculate the convolution $F * G$ of two Gaussian functions without resorting to Fouritertransforms: $F(t) := \exp(-at^2), G(t) := \exp(-bt^2) \qquad a,b>0$ But intuitively I expected ...
1
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1answer
26 views

If $f \in L^p(\Omega)$, then $(\rho_n *f) \to f $ in $L^p(\Omega)$, for a sequence $(\rho_n)$ of mollifiers.

I am reading Functional Analysis, Sobolev Spaces and Partial Differential Equations, by Haim Brezis. There it is shown that: If we have $f \in L^p(\mathbb{R}^n)$, then $(\rho_n *f) \to f $ in ...
0
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3answers
52 views

Convolution of sine and cosine.

So I came across this question while studying for the GRE Subject Exam, and I am not really sure how I am supposed to handle it. Let $$ f(x) = \int _0 ^{\pi} \sin t \cos (x+t) dt $$ I am to find where ...
0
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0answers
33 views

Convolution of Gaussian and parabolic function.

What is convolution of $\exp(-x^2)$ i.e Gaussian function and $2x^2$? I don't have any idea,as I have found that Laplace transform of Gaussian function involves complementary error function,so inverse ...
0
votes
0answers
13 views

Convert a landau function to a gauus function

Assume the Landau Distribution $$p(x) = \dfrac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}\left(x+e^{-x}\right)}$$ What I would like to do is "convert" it to a gauss function $$g(x) = ...
0
votes
1answer
407 views

Find convolution of u[n]-u[n-2] and u[n]-u[n-2]

Question: Find convolution of $u[n]-u[n-2]$ and $u[n]-u[n-2]$ I have found that $u[n]\cdot u[n]=n$, $u[n]\cdot u[n-2]=n-2$, $u[n-2]\cdot u[n-2]=n-4$ Use linear property, my answer is: ...