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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0answers
23 views

Young's inequality for convolutions for functions of bounded support

If $$f\in L^P(\mathbb{R}^d), g\in L^q(\mathbb{R^d}), \; \frac{1}{p}+\frac{1}{q}=1+\frac{1}{r},$$ then Young's inequality for convolutions states $$\|f*g\|_{L^r}\leq\|f\|_{L^p} \|g\|_{L^q}.$$ In ...
0
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0answers
30 views

Branch points of functions defined as convolution integrals

I am studying sets of equations containing convolution integrals of the following type: $$ u\mapsto \int_D dz g(z) f(z-u), $$ where $g$ is analytic, but $f$ has a pole at the origin (so colloquially ...
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1answer
1k 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 ...
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1answer
26 views

what is the Convolution the expression [on hold]

what is the convolution of the expression, $\ x(t)*y(-t)\ $ I want to apply it to be written in integral form.
4
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1answer
97 views

A convolution involving binomials

Given $$f(i)\gt0,\:g(i)>0,\:i =0,1,2,3,...\:$$and$$\sum_{i=0}^{\infty}f(i) = 1,\sum_{i=0}^{\infty}g(i) = 1$$Prove that, if$$\frac{g(l-k)f(k)}{\sum_{i=0}^{l}f(i)g(l-i)}=\binom{l}{k}p^k(1-p)^{l-k}\: ...
1
vote
1answer
16 views

Fourier transform of a product of two rect functions

I am trying to evaluate the following expression $$\mathcal{F}\{\mathrm{rect}_{L_{x}}(x)\mathrm{rect}_{L_{y}}(y)\}$$ which denotes the 2-dimensional Fourier transform (reciprocal variables $k_x$, ...
1
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1answer
17 views

integral over domain $1_{(x+y≥100)}$

The problem is: Compute: $\frac {1}{40^2}\int_{40}^{80}\int_{40}^{80} (100-x)1_{(x+y≥100)}dxdy$ my attempt: $$=\frac {1}{40^2}\int_{40}^{80}\int_{100-x}^{80} (100-x) dydx = \frac ...
1
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0answers
18 views

Multiple convolutions

Let $\phi(x)=1$ on [0,1] and 0 anywhere else. Is there a was to say what the support of the n-times convolution of phi with itself, that I wann to denote by $B_n$, is? In especially is it possible to ...
0
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0answers
13 views

Derivative of convoluted 2D image w.r.t. to its coefficients

I am creating an image with the following variables with the following dimensions: $A: (1,i)\\ X_a: (i,x,y)\\ B: (1,j)\\ X_b: (j,x,y)\\ Image=A\cdot X_a\odot B\cdot X_b $ Where $\odot$ stands for ...
1
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1answer
39 views

PDF & CDF of a Sum of Weighted Independent Random Variables $Z=aX+bY$

From this question here, I learned that the Cumulative Distribution Function (CDF) of $Z=X+Y$ is: \begin{eqnarray*} F_Z \left( z \right) & = & \int F_X \left( z - y \right) dF_Y \left( y ...
5
votes
2answers
328 views

The condition for $Y$ to make $\mathbb{E}[\max\{X_1+Y,X_2\}] > \mathbb{E}[\max\{X_1, X_2\}]$

I would like to know the condition for a random variable $Y$ in order to make $\mathbb{E}[\max\{X_1+Y,X_2\}] > \mathbb{E}[\max\{X_1, X_2\}]$, where $X_1$ and $X_2$ are iid. Any help would be ...
0
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0answers
31 views

Fourier transform and recursion

Starting with the first derivative of a continuous general function $u(x)$, say $\frac {du}{dx}$ and I take the Fourier Transform of it, I know the solution is $i\cdot k\cdot U$, where $ U$ is the ...
2
votes
2answers
369 views

Is the convolution an invertible operation?

If I have a signal $f(x,y)$ (discrete) and I convolve this signal with a kernal $h(x,y)$: $y(x,y) = f(x,y) \star h(x,y)$ (where $\star$ is the convolution operator) Can I obtain $f(x,y)$ given only ...
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4answers
39 views

Laplace of $\int_0^t \frac{sinx}{x}dx$

What is the Laplace transform of $\int_0^t \frac{\sin x}{x}dx$ I'm thinking about approaching it as a convolution but I am not sure how. Could I define it as the convolution of $1$ and $\frac{\sin ...
-3
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1answer
68 views

Fourier Analysis question on convolution [closed]

Let $f(x) = \operatorname{sinc}(x)^2$. Find $(f*f)(x)$? This is what I tried $f(x)=\mathrm{sinc}(x)^2$ $$ \begin{align} f\ast f(x) &=\int f(u)\cdot f(x-u)\,\mathrm{d}u\\ ...
1
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1answer
22 views

Conditional distribution of the sum of k, independent ordered draws

Suppose I make k independent draws from the same distribution (say uniform). The distribution of the sum of these order statistics should equal the distribution of the sum of k independent random ...
1
vote
1answer
20 views

Convolution, g(t)=sin(t)

I have two functions: $f(t)=(t+\pi)\theta(t+\pi)-2t\theta(t)+(t+\pi)\theta(t-\pi)$, (looks like $-|x|+1, -\pi < x < \pi$ ) and $g(t) = \sin{(t)}$ Could someone please point me in the right ...
1
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2answers
31 views

Sums of independent random variables (more than two) [closed]

I read that the convolution of two iid random variables is $$(f * g) (z) = \int f(z-y) g(y) dy$$ What is the general formula for more than two RVs? For example, for three RVs.
7
votes
4answers
5k views

Can someone intuitively explain what the convolution integral is?

I'm having a hard time understanding how the convolution integral works (for Laplace transforms of two functions multiplied together) and was hoping someone could clear the topic up or link to sources ...
1
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1answer
14 views

Convolution of derivatives

When transforming nonlinear PDE to its Fourier space, I encounter the following problem: Consider the equation $u_t=(u^3-u)_{xx}$. Then, when transforming to Fourier space we get \begin{equation*} ...
1
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1answer
41 views

Convolution of Two Random Variables

I have been working a few hours on this particular problem. Please excuse my lack of formatting. This is the question: Let $X$ and $Y$ be random variables with density function $f(x) = 2x$ on $[0, ...
1
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1answer
25 views

What is this Toeplitz like matrix called and how do I represent it as a convolution?

I have a matrix that is used to represent the Green's function in a popular class of fast E & M solvers (CG-FFT). The matrix represents distances, that are later filled in using the appropriate ...
0
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0answers
24 views

Convolution of two equations f*g(t)

I am trying to calculate the convolution of: $$ f*g(t) $$ where $f(t) = \delta''(t) -\delta(t) $ and $ g(t)=e^{-t^2}+e^{-t} \theta (t) $ Maybe someone can give me a solution or point me in the ...
0
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0answers
33 views

How to convert infinite intergral to sum

How to convert Wiener filter formulas from integral to sum? They are for images therefore it must be possible to convert them to sums. Any help will be appreciated: I could not find much info on ...
2
votes
1answer
17 views

Is convolution a coercive bilinear form in $L^2$ -space?

This is one of the problems in functional analysis course I'm having. Suppose $f,g \in L^2(0,10)$. Then define a bilinear form $$ B:(f,g)\mapsto \int_0^{10} f(x)g(10-x) dx. $$ Now I have to find out ...
5
votes
2answers
97 views

Proving the sum of two independent Cauchy Random Variables is Cauchy

Is there any method to show that the sum of two independent Cauchy random variables is Cauchy? I know that it can be derived using Characteristic Functions, but the point is, I have not yet learnt ...
0
votes
1answer
9 views

Graphs of functions defined by convolution

A sequence of functions on the real line is defined as $$f_0=\chi_{[-1,1]},\qquad f_{n+1}=f_n*f_0, n=0,1,2,\dots $$ Here * means convolution. I tried to draw the graphs of the functions and see what ...
0
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1answer
34 views

The maximum value (peak) of multiple self-convolution of rectangular function

In Multiple self-convolution of rectangular function - integral evaluation, formula for self-rectangular function of rectangular function seems to have been derived. How do we prove that this formula ...
0
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1answer
37 views

Sum of two uniform independent random variables

I would like to find the cdf of $Z=X_1+X_2$, with $X_1\sim U(0,1) $, $X_2\sim U(0,2)$ I always prefer to find the cdf instead of the pdf with convolution, and this time I am having troubles with the ...
0
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1answer
28 views

Convolution of an integrable function an $L^\infty$ function [duplicate]

Let $f$ be an integrable function on $\mathbb{R}$, and $g$ be an $L^\infty$ function on $\mathbb{R}$. Then, the convolution $f*g$ is said to be continuous and bounded on R. I managed to show that it ...
0
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0answers
14 views

How to compute this convolution in matlab?

The equation is as above, where $f_x$ and $f_y$ refer to axises in frequency domain, $x$ and $y$ refer to axises in space domain and $F$ refers to Fourier transform. My main problems lie on 1) how ...
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0answers
15 views

linear convolution and circular convolution

Can anyone explain for me when circular convolution is equal to linear convolution. I get this example but i need more expanation esp on the underlined
0
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0answers
8 views

Convolution in linear phase FIR filter

$G(z)$ is a linear phase FIR filter. is it possible to design another a realizable filter $H(z)$ such that it will undo the effect of the filter. That is $y(n)=g(n)*x(n)$ and the one going to be ...
0
votes
2answers
110 views

Proper convolution notation

What would be the correct way to write down the convolution in "star" notation for these two functions; $h(t)$ and $\delta(t-x)$. $\delta$ is the Dirac delta function. The integral notation should ...
0
votes
1answer
17 views

Discrete convolution equation

Let $x_1 = (x_1^k)_{k =-\infty}^{+\infty}$, $x_2 = (x_2^k)_{k=-\infty}^{+\infty}$, $x_3 = (x_3^k)_{k=-\infty}^{+\infty}$ be three sequences of real numbers such that $x_j^k = 0$ for $k < -m_j < ...
0
votes
2answers
713 views

How to calculate a 1D convolution summation?

I hope I said that right. I'm trying to follow along with a convolution example but maybe I am in over my head. I don't understand how in this example they get the values on the right. For example, I ...
3
votes
2answers
581 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 ...
0
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0answers
19 views

Periodic convolution of functions

Define the periodic convolution of functions in L2([0; 1]). What theorem of convolution do I use to define this and how do I solve this?
3
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1answer
46 views

Given $f\in L^1(\mathbb{R})$ with $\|f\|_1<\infty$ and $g_n=\sqrt{n/2\pi}e^{-nx^2/2},f_n=g_n\ast f$, show that $\lim\|f_n-f\|_1=0$

Given $f$ a Lebesgue integrable function on $\mathbb{R}$ with finite $L^1$-norm, I am asked to show that $\lim_{n\to\infty} \|f_n - f\|_1 = 0$, where $f_n = f \ast g_n$ and $g_n = ...
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0answers
16 views

Finding A Convolution

Let $f(x)=e^{- \mid x \mid}, g(x)= e^{-x^2}$ What is $(f*g)(\xi)$? I have been trying to find it, but I am stuck on finding the integral of $e^{y^2+y+\xi}$ Thank you!
0
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0answers
18 views

How to represent a periodic function as the sum of sinc functions in fourier transform

Suppose function $f(t)$ is 1-periodic. This means that in fourier transform, $F(\omega)$ is sum of impulse signals (dirac delta function and its shifts) at the multiples of $1$. Now we can form $g(t)$ ...
3
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0answers
62 views

How do I apply this PDE as an image filter?

I'm trying to preprocess a height map image with a helmholtz-type equation as described in this paper. The equation is: $$ddx(h') + ddy(h') + y(h'-h) = 0$$ I solved for h and got: ...
3
votes
1answer
136 views

Haar measure, convolution and involutions

I have some problems to follow the proof of the anti commutativity property of the convolution and involution operations defined using a Haar measure as presented in Pedersen's book Analysis Now, ...
1
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0answers
35 views

How much does convolution with a compact C^m kernel increase the order of continuity.

Let $f \in C^n$ and $g \in C^m$, with $g$ compactly supported and integrable. How much does the convolution $f\star g$ of $f$ with $g$ increase the order of continuity? Statement: I think that, under ...
0
votes
1answer
7 views

Filter output of a signal

So I have a filter $$H(z) = 0.5 + 0.5z^3 = (1/2, 0, 0, 1/2)$$ and need to find the output of it on a cyclical signal $$x = (..., 3, -1, 2, 1, 5, 2, 3,-1, 2, 1, 5, 2, 3,...) $$ Would the output be ...
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0answers
27 views

Prove that the variance of a discrete random variable increases with a parameter

I have an infinite number of known probability density functions $f_1(x),f_2(x),f_3(x),...$. The PDFs $f_k(x)=\sum_{j=1}^k v(A+j-1)e^{-v(A+j-1)x}\binom{k}{j-1}q^{j-1}(1-q)^{k-j-1}$. Let ...
2
votes
2answers
79 views

Can we expect $g(f\ast h)= gf \ast gh$ for some $g\in C_{c}^{\infty}(\mathbb R)$?

Let $f,g:\mathbb R \to \mathbb C$ be nice functions so that their convolution make sense. My question: Is it possible to choose $0\neq g\in C_{c}^{\infty}(\mathbb R)$ (= the space of smooth ...
5
votes
2answers
76 views

Is the convolution of a function $f(x)$ and a polynomial $p(x)$ always a polynomial?

After reading the following question: How do I prove a convolution is a polynomial? I want to ask if that is always the expected result, that is to say, does the following holds? A convolution ...
1
vote
0answers
25 views

Can we choose $g$ so that $\|(g\widehat{(f^{3})})^{\vee}\|_{L^{p}} \leq C \|g_{1}f\|_{L^{2}}^{r} \|(g_{2}\hat{f})^{\vee}\|_{L^{s}}$?

Let $f, f^{2}, f^{3}\in L^{q}(\mathbb R)\cap C_{0}(\mathbb R)$ where $ q\geq p, \ \text{and}$ and $C_{0}(\mathbb R)$ is the class of continuous functions vanishing at infinity. My Questions: ...
-2
votes
1answer
31 views

A question related to convolution

Let $f\ge0$ such that $\int_{{R^n}} {f(x)dx} = A < 1$, and ${f_k} = f*...*f$ a convolution of $k$ times. My problem is how to prove $f_k$ is integrable and $f_k\rightarrow0$ in $L^1(R^n)$? ...