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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-1
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0answers
18 views

$2D$ convolution associative property with dot product [on hold]

If I have three square matrices $a,b,$ and $c$ of equal size, say each of them are $3 \times 3$ matrices. Then practically it is possible that $$(a \cdot b) \ast c = a \ast (b \cdot c),$$ that is the ...
1
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1answer
24 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 = ...
0
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0answers
10 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 ...
0
votes
0answers
49 views

Region of Convergence

I know to find the Region of Convergence you find the poles of the denominator, but I'm unsure what to do for the case where there is no denominator (for a and d) and what to do if the poles are ...
0
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0answers
17 views

Why is multiplication in frequency domain equals convolution in time domain? [on hold]

The title says it all in my mind. I do not know what else to add. Instead of simply putting the question on hold, could you be so kind as to actually point out what's missing?
0
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0answers
20 views

Approximation of Sobolev function with convolution

As a homework exercise in Sobolev spaces course we have the following: Suppose $u \in W^{1,p}(\mathbb{R}^n_+)$ and $u_\varepsilon(x)=u(x+2\varepsilon e_n)$, where $e_n=(0,\dots, 0,1)$ and ...
0
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0answers
28 views

Use convolution theorem to evaluate $\int_0^\infty e^{-((|a+su|)/c)^b}e^{-(u/k)^p}du$

$$\int_0^\infty e^{-((|a+su|)/c)^b}e^{-(u/k)^p}du$$ I cannot figure out what to do to solve a case like this, where the variable $u$ is only supported from $0$ to $\infty$. Some further information: ...
1
vote
1answer
57 views

Convolution $f * g$

Assume that $f$ is in $L^1 (\mathbb{R})$ and $g(x)= e^{2iπx}$. Compute $f * g$ I just need a hint and not the entire answer. How can I compute the convolution when I don't know what $f$ is?
1
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1answer
40 views

Graphical Convolution

For the first part of the above problem, I copied an example from my book and I got the answer to be $$t(t-1)+t(t-2)=t^2-3t$$ considering that the integral is the sum of the area of the rectangles ...
0
votes
0answers
26 views

Can you define a Cauchy product from this identity?

I can write vague expessions for $\zeta(3)$, the Apéry's constant, for example when I multipliy by $\frac{1}{n^4}$ the recursion relation (2) in page 2 here, and after I take the sum ...
0
votes
1answer
12 views

Why is the inf-convolution of lower semicontinuous functions continuous?

I'm confusing now about the continuity of inf-convolution. I understand that the inf-convolution of lower semicontinuous functions is semiconcave and so it's locally Lipschitz continuous (in ...
0
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0answers
18 views

convolution properties of distributions

Let $f,g,h \in D'(R^n)$. How we define the convolution of these functions? I'm trying to show some properties of convolutions such as $\delta\ast f=f$ $(f\ast g)' = f'\ast g=f\ast g'$ $(f\ast g) ...
0
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0answers
17 views

Meaning of standard integral convolution

In this paper, in the proof of Lemma $13$, there is this sentence: Now, we find a $1$ Lipschitz $\bar{g} \in C^1(\mathbb{R}^n)$ with $\| f - \bar{g}\|_{|\infty} < \epsilon / 2K$, using the ...
1
vote
2answers
28 views

Fourier series and convolution

Let $f$ and $g$ be $2\pi$-periodic, piece-wise smooth functions having Fourier series $f(x)=\sum_n\alpha_ne^{inx}$ and $g(x)=\sum_n\beta_ne^{inx}$, and define the convolution of $f$ and $g$ to be ...
0
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0answers
20 views

Sum of two random variables - uniform distributions [duplicate]

I have two continuous uniform random variables I need to add. I read that to get the sum of two pdfs you convolve them. I'm getting a bit confused on the limits of integration though. If both RVs are ...
1
vote
1answer
16 views

Convolution for two random variables

In the textbook i'm currently reading it is said that for two independent random variables $X$ and $Y$ density function of variable $Z=X+Y$ can be found from the equation: $$ g(z) = ...
1
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0answers
22 views

Convolution of two $L^\infty$ function with compact support.

I have the following lemma without proof: Lemma. Let $f, g \in L^\infty(\mathbb R^n)$ with compact supports. Then $f \ast g \in C(\mathbb R^n)$. Is this even true? I get this: $$ \begin{align*} ...
3
votes
2answers
47 views

Convolution Integral to Evaluate Fourier Transform

According to Mathematica with Fourier transform convention $$\widehat{f}(\xi)=(2\pi)^{-1/2}\int_{-\infty}^{\infty}f(x)e^{i\pi x}dx$$ The Fourier transform of the function $f(x):=|x|^{-1/2}e^{-|x|}$ ...
2
votes
0answers
35 views

Conditions under which an Convolution operator is normal.

I have a possibly complex valued convolution operator given by $ \int_{\mathbb{R}}K(x-y)f(y)dy$ I know that the operator is self-adjoint if $K(x)=\overline{K(-x)}$ holds. But are there softer ...
1
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0answers
14 views

Convolution of basis functions is a member of the same set of basis functions?

Suppose $\left\lbrace \Phi_i\right\rbrace_{i=0}^{\infty}$ is a complete basis of $\ell_1$. So if $M\in\ell_1$ we can write it as a linear combination of the basis functions $M=\sum_{i=0}^\infty ...
0
votes
0answers
13 views

Convergence of convolution with an even summability kernel

Suppose that $f(\theta) : [0, 2\pi] \rightarrow \mathbb{R}$ is a monotone increasing real valued function and $\{k_n\}$ is an even summability kernel. I want to show that $f \star k_n$ converges ...
1
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0answers
16 views

Applying the convolution theorem in the presence of a twiddle factor

The convolution theorem says that a 2-d cyclic convolution like $C = U \ast V$ can be evaluated more quickly than doing the raw sum $C_{i,j} = \sum_{a,b}^n U_{a,b} V_{i-a,j-b}$ for each point (assume ...
1
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0answers
47 views

Convolution of measures, why is the notation like this?

In both my book, and on Wikipedia they define convulution of two measures like this: $(\mu_1*\mu_2)(B)=\int_{\mathbb{R}^d}\mathcal{X}_B(x+y)d\mu_1(x)d\mu_2(y)$ It doesn't seem like a typo, but ...
0
votes
0answers
13 views

Is there an alternate name for the symplectic convolution?

Looking into the Wigner-Weyl transformation mapping Hilbert space operators to functions on phase-space, I've run up against the need for a symplectic convolution $$[F\star G](x,p) = \int \!dy\,dk\, ...
0
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0answers
16 views

Is the convolution of two waves again a wave?

Call a function $f:\mathbb{R}^{1+n}\to \mathbb{R}$ a wave, if it satisfies the wave equation $$\frac{1}{c^2} \frac{\partial^2 f}{dt^2} = \sum_{i=1}^n \frac{\partial^2 f}{dx_i^2} = div(grad\,f)$$ ...
2
votes
2answers
26 views

Why does convolution of delta function commute (test distribution perspective)?

If I understand correctly, for test functions $f$ we define $$ \langle\delta, f\rangle = f(0)$$ and convolution is defined as $$ \langle g * T, f\rangle = \langle T, g^- * f\rangle,$$ where $f$ ...
1
vote
0answers
23 views

Limit of sequence with discrete convolution $a_n = 1 - b * a$

I have a sequence \begin{equation} $a_n = 1 - \sum\limits_{i=1}^{n-1}b_na_{n-i}$ where $b_n\leq{c}^n$, and $c<1$ Based on multiple simulations with varying parameters, I think that the sequence ...
0
votes
0answers
12 views

Convolution of cosine and shifted unit step

I'm trying to understand the basics of a convolution and have troubles with the following task: $$ x_1 = \cos(2 \pi t ) \cdot u(t) $$ $$ x_2 = u(t-0,5)$$ The task is to compute the convolution $$x_1 ...
1
vote
1answer
15 views

What is the purpose of the requirement on mollification radius?

On page 66 of "Sobolev Spaces (Adams ed2)" in the proof of Lemma 3.16 (Mollification in $W^{m,p}(\Omega)$), it is mentioned that $\varepsilon < {\rm dist}(\Omega', \partial\Omega)$. However, I ...
1
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0answers
17 views

analyze convolution in spatial domain against multiplication in frequency domain

Lets say I have a image of $NxN$ and a separable filter that I want to apply on it. there are 2 ways to do that: 1. By convolution in spatial domain. 2. By multiplication in frequency domain. I need ...
0
votes
0answers
22 views

Convolution of delta-ish functions

I would like to compute the convolution of a function with itself, where the function is $f(x) = \frac{\delta(x)}{x}$. When there is a shift in the delta function it is easy to compute, but this one ...
0
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0answers
23 views

The Deconvolution Integral

The standard 1D continuous convolution integral is defined as: $$y(t) = h(t)*x(t) = \int^{+\infty}_{-\infty}h(\tau)\cdot x(t-\tau)\ d\tau$$ Using fourier transform, $$Y(j\omega) = X(j\omega)\cdot ...
0
votes
1answer
18 views

Conditional Expectation of Poisson Distribution

So I am not sure how to go about this, Say that $X_j\sim$Pois$(\theta)$, and are iid. Find the following: $$ E[X_1+2X_2+3X_3|\sum_{j=1}^nX_j] $$ I am aware that I am suppose to somehow make use of ...
0
votes
0answers
14 views

Multivariate probit gaussian convolution

For univariate normal distribution, we know the following formula exists $\int\Phi(a)\mathcal{N}(a|\mu,\sigma^2)da=\Phi\left(\frac{\mu}{\sqrt{1+\sigma^2}}\right)$ Is there a similar formula for ...
0
votes
0answers
21 views

existence of a function $f\in C_c^{\infty}(A_2)$ s.t. $f_{A_1}$ is constant 1

Let $A_1,A_2\subseteq \mathbb{R}^d$ two domains such that $A_1\subset \subset A_2$. Why exists a function $f\in C_c^{\infty}(A_2)$ such that $f_{|A_1}$ is constant 1? My idea is to define $f=\rho ...
1
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1answer
20 views

discrete convolution $f*g$ belongs to $\ell_\infty$, i.e. the sup norm is finite

Definitions. Fix any $\phi\in(0,1)$ and $\theta\in(0,1)$, and let us define functions \begin{equation}f(n)=\left\{\begin{array}{ll}n^{-\phi},&\text{ if }n\geq 1\\0,&\text{ ...
0
votes
1answer
36 views

Convolution of two functions.

$f(x)=2x/3$, $0<x<3$, $f(x)=0$ otherwise $g(x)=1$, $-1<x<3$, $g(x)=0$ otherwise I am trying to work out the convolution $h=f*g= \int(f(y)g(x-y))dy$ I am able to show that: $x - 3 ...
0
votes
1answer
20 views

Convolution of function with itself

I'm trying to tackle the following question: Let $\displaystyle g_a(x)=\begin{cases}1-\frac{|x|}{a},&x<0\\0,|x|\ge0\end{cases}$. Find $g_a\ast g_a$. So, I tried to compute it by ...
1
vote
1answer
29 views

Fourier transform this convolution

So we have that $$ g(t) = \frac{1}{T}\int_{t-T}^{t}f(\tau) d\tau $$ for $T>0$ and I'm asked to show that $\left| \hat{g}(w) \right|≤\left| \hat{f}(w) \right|$. The hint I get from the question is ...
1
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0answers
38 views

What is the output $y(t)$ when you have input $x(t) = \cos(2 \pi t) $ and frequency response response $h(t) = u(t) - u(t - 1/2)$?

The output $y(t)$ is the convolution of input $x(t)$ with impulse response $h(t)$: $$ y(t) = h(t) * x(t) $$ This is a linear, time invariant system. What is the output $y(t)$ in real form when you ...
0
votes
1answer
37 views

Calculate $\frac{x}{(1+x^2)^2}\ast \frac{1}{1+x^2}$ using Fourier transformations

Calculate $\left(\frac{x}{(1+x^2)^2}\ast \frac{1}{1+x^2}\right)(y)$ using Fourier transformations. I have found a solution, but my method was very long. How could I shorten the solution? ...
2
votes
2answers
35 views

Calculating the convolution of a piecewise function

Let $$f(x) = \begin{cases} \frac{1}{2}, & \text{if $\rvert x\lvert \le 1$ } \\ 0, & \text{otherwise} \end{cases}$$ I want to calculate the convolution of $f$ with itself. I am ...
0
votes
1answer
18 views

A question on convolutions

Let $f$ be an $L^2$ function on the line. If $f*g$ is an $L^2$ function for every $g$ in $L^2$ does it follows that $f$ is in $L^1$?
3
votes
2answers
73 views

$E(X\mid X+Y)$, where $X$ and $Y$ are independent $U(0,1)$.

Given that $X,Y\sim U(0,1)$ calculate: (1) $E(X\mid X+Y)$ I am stuck at this point: $$E(X\mid X+Y)=\int_0^1 xf_{X\mid X+Y}(x\mid x+Y) \, dx=\int_0^1 xf_{X, Y}(x, Y) \, dx$$
1
vote
1answer
38 views

Construct a nonnegative nonzero Schwartz function whose Fourier transform is nonnegative and compactly supported.

I tried the exercise with the hint that $\phi(x)=|\varphi\star\hat{\varphi}|^2$ could be the solution with $\varphi$ compactly supported and odd. Thus, \begin{align*} ...
2
votes
0answers
15 views

Conditions under which a convolution transformation is injective in the 1-d Torus

Let $X=[0,1)$ the 1-d torus. Given a bounded positive function $w\colon X\to\mathbb{R}$ with unit integral (I mean $w\geq 0$, $w\in L^\infty(X)$ and $\int_X w\; dx=1$), define \begin{align*} T_{w} ...
6
votes
1answer
107 views

Inf-convolution, two basic questions?

Let $E$ be a normed vector space. Given two functions $\varphi$, $\psi : E \to (-\infty, +\infty]$, one defines the inf-convolution of $\varphi$ and $\psi$ as follows: for every $x \in E$, ...
2
votes
0answers
27 views

Convolution of two signals

I have a problem with the convolution of two signals: $$x_{1}(t) = e^{2t}*u(-t)$$ $$x_{2}(t) = u(t-3)$$ $$x_1 \mathbin{\mathrm{(conv)}} x_2 = \int_{-\infty}^{+\infty} x_2(\tau) * x_1(t-\tau) \, ...
0
votes
1answer
30 views

Finding a limit involving Fourier series and Dirichlet's kernel

Find the limit $$\lim_{n\to\infty} \int_0^{2\pi} (x+\frac{\pi}{2})^2 \frac{\sin((n+\frac{1}{2})x + x\cos nx}{\sin\frac{x}{2}}\ dx$$ So we may define $f = (x+\frac{\pi}{2})^2$ and then look at the ...
0
votes
1answer
31 views

Convolution domains probability theory

Problem 1.4 here: ...