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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0
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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 ...
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1answer
10 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 - ...
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0answers
15 views

Limit of a convolution product

For $\gamma \in \mathbb{R} \setminus \{ 0 \} $ let $k : \mathbb{R}^n \to \mathbb{R}$ be defined by $k(x) = |x|^{-n + i \gamma}$; I've been asked to prove that $\text{p.v. } k * f = \lim_{\epsilon \to ...
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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}^{} ...
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0answers
19 views

convolution of h[n]=a^(-n) u[-n] and x[n] =u[n] [closed]

I wanted to know how the shifted h[n-k] graph would look like for h[n] = a^-nu[-n]. I knew its on left side with a decreasing function, but when we need to take its convolution do i need to flip it ...
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0answers
24 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. ...
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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 ...
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0answers
20 views

How do solve the integral $\int_{-\infty}^{\infty} exp(-a|t|)exp(-a|t-\tau|)dt$?

How do you determine the autocorrelation of $g(t) = exp (-a|t|)?$ Plugging it into the equation $ \int_{-\infty}^{\infty} g(t)g(t-\tau) dt$ would result into something like $ \int_{-\infty}^{\infty} ...
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0answers
22 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 & ...
-1
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0answers
15 views

Convolution - Trigonemtry Manipulation

Please see image, I'm just wondering how you get from 1st to the 2nd line? Thanks enter image description here
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0answers
6 views

How can I write a gradient of sobel filter in continuous formula?

Let $*$ denote a convolution operation, $G$ denote a kernel, and $I$ is a given image. The gradient of the image $I$ is equivalent: $\nabla (G*I) = (\nabla G) * I$ The Sobel filter approximtes two ...
3
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1answer
49 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 ...
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0answers
14 views

$\|h\|_{L^{p}} \leq C \|f\|_{L^{p}} \implies \|g\ast h \|_{L^{p}} \leq C_1 \|g\ast f\|_{L^{p}}$?

Suppose that $f, h \in L^{p}(\mathbb R) (1\leq p \leq \infty)$ so that $\|h\|_{L^{p}} \leq C \|f\|_{L^{p}}$ for some constant $C$. Take $g\in \mathcal{S}(\mathbb R^{d})$ (Schwartz Space). We note ...
1
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1answer
19 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 ...
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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 ...
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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
25 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 ...
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3answers
50 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 ...
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0answers
26 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 ...
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0answers
12 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) = ...
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0answers
12 views

Iterated convolutions w.r.t. different variables of a function

I do not understand a claim from a paper: Let $b:[0,T] \times \mathbb{R}^n \rightarrow \mathbb{R}$ be a bounded function and let $$b^{n} (t,x) = b(t,x) \ast \psi_n(t) \ast\phi_n(x), $$ where ...
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0answers
51 views

Square root of dirac delta function

Is there a measurable function $ f:\mathbb{R}\to \mathbb{R}^+ $ so that $ f*f(x)=1 $ for all $ x\in \mathbb{R} $, i.e $$\int\limits_{-\infty}^{\infty} f(t)f(x-t) dt=1 $$ for all $ x\in \mathbb{R} $.
2
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1answer
15 views

Uniqueness result for convolution

I have seen the convolution operator in different settings, and I was wondering about the following: Suppose $h=f\ast g$ for an unordered pair of functions $(f,g)$. Does there exist a pair of ...
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0answers
35 views

Convolution of two distributions

Consider the convolution product: $$H(x)\ast\operatorname{Pf}\dfrac{H(x)}{x},$$ where $\operatorname{Pf}$ denotes pseudo function. This means, that $\operatorname{Pf}\dfrac{H(x)}{x}$ is, as defined ...
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1answer
20 views

The product of a uniform probability density function and -1

What happens when you multiply a uniform probability density function between -1 and 1 by -1? Does the new uniform distribution become -1/2 between -1 and 1? I am asking because I am trying to find ...
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0answers
34 views

Finding a function given as a part of a convolution integral

I am trying to solve the following equation for the function $f$. $$t^{-\alpha} \exp{ \left(- \beta x^2 t^{-2 \alpha} \right)} = \int_0^t \frac{f\left(x, s\right)}{t - s}ds$$ where $\alpha$ and ...
2
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1answer
43 views

Why is a convolution 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 $f\ast g$ ...
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3answers
46 views

what is the convolution of $sin(bx)$ and $e^{-a|x|}$?

my textbook says that $F[f*g] = F[f] \cdot F[g]$ but what is $F[sinbx]$? It doesn't exist right? So how should I solve it? $F$ is fourier transform
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2answers
37 views

Show that these two differential equations have the same solution

Question: Show that the problems $ax'' + bx' + cx = f(t); x(0) = 0, x'(0) = v_0$ and $ax'' + bx' + cx = f(t) + av_0 \delta(t); x(0) = x'(0) = 0$ have the same solution for $t \gt 0$. Thus the effect ...
2
votes
2answers
57 views

Solving convolution problem with $\delta(x)$ function

Suppose we had the functions: $$g(t)=\theta(t)(e^{-t}+2e^{-2t})+2\delta(t)$$ and $$u(t)=2(\theta(t)-\theta(t-2))$$ Then we have ...
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0answers
24 views

Convolution of a Pareto and a Uniform distribuion

I would like to convolve a Pareto and a Uniform Distribution. Pareto's PDF: $f_X(x)=\begin{cases} 0 & x < c \\ b\frac{c^b}{x^{b+1}} & x\geq c \end{cases}$ The $[0,a]$ ...
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votes
1answer
35 views

How to show $ |f*g|_{1} \le |f|_{1}|g|_{1}$ [closed]

Well, as it is stated in the titel. I have to show $ \|f*g\|_1 \le \|f\|_1\|g\|_1$. thank you already now for your help.
0
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1answer
48 views

Proof of Rudin's Theorem 8.14, RCA

In Rudin's proof of Theorem 8.14, which states that convolutions of Lebesgue integrable functions over the real line are Lebesgue integrable, he first proves the result for Borel measurable functions, ...
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0answers
28 views

How to solve the laplace transform of $f_m(t_m)$ = $f_1(t)$ $\int_{0}^{\alpha} f_2(\tau) d\tau$ + $f_2(t)$ $\int_{0}^{\alpha} f_1(\tau) d\tau$.

Could you please help me to solve the following : if $t_m$ = min($t_1$,$t_2$) The probability density function $t_1$ is $f_1(t_1)$ and $t_2$ is$f_2(t_2)$ then $f_m(t_m)$ is the probability density ...
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0answers
24 views

How to calculate the volume of revolution around $x=3$ for the region bounded by x, y axis and $\sqrt{x}+\sqrt{y}=1$?

So I intend to do it in both shell and disk ways. Let's first use shell method (formula: $\int 2\pi x(f(x)-g(x))dx$): Since $x=3$ is the major axis, and $y=(1-\sqrt{x})^2$, we have $V=\int_0^1 2\pi ...
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0answers
34 views

What does closed under convolution mean in Probability Theory?

I understand what does it mean for a set to be closed under addition or multiplication, i.e. the sum/product of elements in a set, is still in a set. Now, I am a little bit confuse when it says the ...
1
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1answer
22 views

Laplace and unit step- multiplication vs convolution

Please be gentle if the question is stupid. When using the laplace transform, you often multiply the function of interest by a shifted unit step function to operate on the positive portion of the ...
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1answer
17 views

Approximations of $L^p$ functions, convolutions, mollifiers, etc. (resource needed)

What is a good resource in which I can read about mollifiers, basic theorems regarding convolutions, smooth approximations of $L^p$ functions and the like? (the presence of exercises would be great, ...
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1answer
19 views

Compute $1_{[0,n]} * 1_{[0,n]}$

$n$ is a natural number. I want to find the convolution of $f = 1_{[0,n]}$ with itself ($1$ is for indicator). Is my work correct $$(f *f)(x) = \int_0^n 1_{[0,n]}(x-y)dy = 1_{[0,2n]}(x)$$ thanks
3
votes
3answers
64 views

Compute $e^{-x^2} * e^{-x^2}$

How to compute the convolution of $e^{-x^2}$ with itself? $$e^{-x^2} * e^{-x^2} = \int_{\mathbb R} e^{-(x-y)^2} e^{-y^2}dy = e^{-x^2}\int_{-\infty}^{\infty} e^{2xy - 2y^2} dy$$ I can't solve it. I ...
0
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1answer
34 views

Fourier Transform Proof $ \mathcal F(f(x)g(x))=(\frac{1}{2\pi})F(s)*G(s)$

I need to prove this: $$ \mathcal F(f(x)g(x))=(\frac{1}{2\pi})F(s)*G(s)$$ So far, I believe I have to use the Fourier transform standard equation $$ \mathcal ...
3
votes
1answer
29 views

convolution: how can I show that $(y*f)'(t) = (y'*f)(t) + y(0)f(t)$

I have the following math problem from my intro to dif. eq. class: (so don't just give an answer) If the convolution $$ (y*f)(t) = \int_0^t y(t-v)f(v)\,dv$$ then show that $$ (y*f)'(t) = ...
1
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1answer
38 views

Laplace Transform: $g(x)=a\sin(x)+\int_0^x \sin(x-u)g(u) du$ [closed]

$$g(x)=a\sin(x)+\int_0^x \sin(x-u)g(u)du$$ I need to find $g(x)$ I believe I need to use Laplace Transform with this in mind (Convolution Thm): $$(f*g)(x)= \int_0^x f(x-t)g(t)dt$$ However I don't ...
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0answers
36 views

Convolutional codes in matlab

I'm trying to construct a convolutional code in Matlab and encode some random data. However the length of the codeword are not as expected. This is the problem information: ...
0
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0answers
9 views

Simple convolution between 2 signals

I have these two signals: $x_1(t) = 2 rect(\frac{t}{4})$ $x_2(t) = 2 u(t-1)$, where u is the $u(t)$ is the Heaviside function So, $x_1(t) (conv) x_2(t) = ...
0
votes
2answers
36 views

Help with integral $\int_0^y x^{-\alpha} (y-x)^{-\alpha} dx$

How should I proceed to work out following convolution integral: $\int_0^y x^{-\alpha} (y-x)^{-\alpha} dx$ for real $\alpha$ > 0. It is the convolution of a powerlaw decaying impulse response ...
3
votes
0answers
33 views

Equation of convolution of measures

Let $\mu_1,\mu_2$ be two locally finite complex regular Borel measures on $[0,+\infty)$ and $\delta_x$ be the Dirac measure at point $x\in[0,+\infty)$. Suppose that for all $x\in(0,+\infty)$ ...
1
vote
1answer
25 views

Prove that if $f\in L^p(\mathbb{R_d})$ and $\phi\in\mathbb{S^d}$ then $f*\phi\in\mathbb{C^\infty}$

Show that if $f\in L^p(\mathbb{R^d})$ and $\phi\in\ S(\mathbb{R^d})$ then $f*\phi\in\mathbb{C^\infty}$, where $S(\mathbb{R^d})$ is the Schwartz class. How does one prove this rigorously? I have ...
0
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0answers
13 views

Approximate 2D convolutions as a sum of separable convolutions

Just like this 3D question, but for 2D. I have a set of 2D convolution kernels, not separable. Is there a good methods to approximate them as a sum of a relatively small number of separable arrays? ...
1
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1answer
30 views

Asymptotic of a convolution integral

$f(x) \ge 0$, $g(x) \ge 0$ are defined on $[0,\infty)$ and $f(x) \sim x^{-a}, \ x \to \infty$, where $a>1$. The integrals $\int_0^\infty f(x)dx<\infty$ and $\int_0^\infty g(x) dx<\infty$. ...