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
73 views

What is the solution of this differential equation? / How to solve it?

I have the following problem : $$m\ddot{x} + c\dot{x} + kx = f_f\delta(t-t_0) + f_c \sin(\omega t) + f_h \theta (2t_0-t)$$ where $x(t)$ is a function of time, $t>0$ and $t_0>0$ and where ...
1
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0answers
89 views

Convolution of a product with focal kernel

Consider the following convolution of a product of two functions $f(x)$ and $g(x)$: $\int f(x')g(x')K_n(x-x') dx'$ where the kernel $K_n$ is a sequence of functions that approach a Dirac delta ...
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 ...
3
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2answers
585 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 ...
3
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1answer
312 views

Differentiating a convolution integral

I'm trying to turn the integro-differential equation $\phi'(t) + \phi(t) = \int_0^t \sin{(t - \xi)} \, \phi(\xi) \, \mathrm{d} {\xi}$ into the differential equation $\phi'''(t) + \phi''(t) + ...
1
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1answer
79 views

Sums of normal CDF's

This is my following problem: $$ CDF_A:F_A(x)=\Phi(x)^2 $$ $$ CDF_B:F_B(x)=1-\Phi(-x)^3 $$ $$ Defining: X=A+(-B) $$ I have those two CDF's and I want to calculate the probability that X is smaller ...
1
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0answers
39 views

Asymptotics of a convolution

For $r>1$ define the functions $$f(x)=|x|^{-1/2}\chi_{[-1,1]\setminus\{0\}}\quad\text{and}\quad g(x)=|x|^{-1/2r}(-\chi_{[-1,0)}+\chi_{(0,1]}).$$ I am interested in the asymptotic behavior of ...
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2answers
42 views

Is convolution associative with regards to the complex unity?

Setup: I need to do a convolution with the function $\cfrac{i}{x}$, and I would like to get rid of the $i$. My functions to be convolved are all real valued. According to the ever-failable wikipedia, ...
1
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1answer
45 views

Sums of random variables expressed as convolution

If $X_1, \dots, X_n$ are a sequence of IID random variables, and $S_1, \dots, S_n$ is the sum of the first n random variables, i.e. $S_1 = X_1$ and $S_n = \sum_{i = 1}^{n} X_i$. Apparently, we can go ...
1
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1answer
104 views

Hölder's inequality. Understanding proof?!

I know how most standard textbooks show that $||f*g||_r \le ||f||_p||g||_q$ with $\frac{1}{r}+1=\frac{1}{p}+ \frac{1}{q}$, but I found a book where the hint $|f(x-y)g(y)|\le ...
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2answers
60 views

Third Moment of a Sum of Normal and Gamma

I just ran into the next problem: The random variables $X$ and $Y$ are independent, where $X \sim Normal(1,1)$ and $Y \sim Gamma(\lambda,p)$ with $E(Y) = 1$ and $Var(Y) = 1/2$ How do we find ...
1
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2answers
405 views

Evaluating the convolution integral of two sine functions

This is a homework problem, so I'm not looking for a worked out solution, merely to be pointed in the right direction. Convolve x(t) with h(t) where: $$ x(t) = sin(t) \\ h(t) = e^{-.1t}sin(2t)u(t) ...
1
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1answer
211 views

$f$ is bounded and continious $\Rightarrow$ the convolution integral $\int f(\tau)g(x-\tau)\text{ d}\tau$ is bounded and continuous

Let $g\in L^1(\mathbb{R}^n)$ and $f:\mathbb{R}^n\to\mathbb{R}$ be bounded and continuous. Why is the convolution integral $$f*g:\mathbb{R}^n\to\mathbb{R}\;,\;\;\;\int f(\tau)g(x-\tau)\text{ d}\tau$$ ...
1
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3answers
79 views

Proof of convolution inequality

I have to prove that if $f$, $g$ $\in L^1(\mathbb{R^n})$ then $\operatorname{dom}\left(f*g\right)$ is a set of full measure and: $\left\|f*g\right\|_{L^{1}} \le \left\|f\right\|_{L^1} ...
3
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1answer
127 views

Show that for any $f\in L^1$ and $g \in L^p(\mathbb R)$, $\lVert f ∗ g\rVert_p \leqslant \lVert f\rVert_1\lVert g\rVert_p$.

I write the exact statement of the problem: Show that for any $g \in L^1$ and $f ∈ L^p(\mathbb{R})$, p $\in (1, \infty)$, the integral for f ∗g converges absolutely almost everywhere and that $∥f ∗ ...
2
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1answer
192 views

Convolution of integrable function with bounded function

Let $H$ Lebesgue integrable. Let $f$ be measurable and bounded on $\mathbb{R}$ with $\lim_{\left|x\right|\rightarrow \pm \infty}f(x)=0$. Let $F(x)=K\ast f=\int_{\mathbb{R}} K(x-s)f(s) \, ds$ be the ...
1
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0answers
77 views

Pseudo-inverse of an underdetermined Toeplitz matrix

I have an undetermined Toeplitz matrix (more columns than rows). For example: \begin{equation*} T = \begin{pmatrix} 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 ...
1
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1answer
48 views

Interesting equation in L^1

Consider $L^{1}(T) = \{ f : R \rightarrow C \text{ with period 1 and } \int_{0}^{1} |f (x)| \ dx < \infty\}$. For $f,g \in L^{1}(T)$ the convolution is given by $(f * g)(x)= ...
3
votes
1answer
125 views

Is it possible to obtain the Uniform distribution as the difference of two independent random variables?

Is it possible to have two independent random variables X,Y with identical distribution, such that $X-Y \sim \text{Uniform}[a,b]$? I am almost certain that is not, but maybe I am overlooking ...
2
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2answers
344 views

How to show that the difference of two Gumbel distributed random variables follows a Logistic distribution?

How can you show that when you have two random variables $X,Y\sim\text{Gumbel}[0,1]$ , then $X-Y\sim\text{Logistic}[0,1]$ . I tried to use the convolution formula ...
1
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1answer
80 views

A question about stochastic ordering and convolution

Two probability density functions $f$ and $g$ are known to have distribution functions $F$ and $G$ respectively with $F(y)>G(y)$ for all $y$, say on $\mathbb{R}$. It is known that if we convolve ...
4
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1answer
89 views

Can FFT be adapted for deconvolution of non-periodic functions?

Can a non-periodic function be padded at the boundaries and deconvolved with inverse FFT? Since a Toeplitz matrix can be embedded in a circulant matrix to perform the deconvolution, is there an ...
0
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1answer
107 views

How to calculate the threefold convolution $f*f*f$

Somehow this convolution is driving me crazy. I am trying to calculate for the indicator function $f:=1_{[0,1]}$ the threefold convolution $$f*f*f$$ But honestly, it does not work somehow. ...
0
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1answer
59 views

Convolution of a function and Dirac delta - special case

Could anyone tell me where $f(n(a-b))$ came from? The thing is easy when there's $f(x)$ instead of $f(nx)$ - the result would be $f(a-b)$. Thanks in advance.
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0answers
106 views

Are any of those quotient rings isomorphic to other well known rings?

(1) Let $C_b(\mathbb{R})$ be the ring (with pointwise multiplication and addition) of bounded continuous functions. Let $I_0=\{f_{(x)} \in C_b(\mathbb{R}) \space | \space lim_{x \to \pm ...
0
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0answers
28 views

Convolution of an image with a kernel that is a product of two functions

Suppose that $G(i,j)$ is a Gaussian decay function on the distance between points $i$ and $j$ of an image. In addition, $D(i,j)$ is the difference between the VALUES of the image at those points. ...
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0answers
351 views

How to bring f(x) from denominator to numerator?

Is it possible to rewrite the following ratio in a way that $f(x)$, its powers or derivatives appears only as numerator. $$\frac{1}{\int_{0}^{c}(c-x)^{2}f(x)dx}$$ $c>0$ is a constant. $f(x)$ is ...
1
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0answers
158 views

Adding truncated normals: calculating convolutions

Problem: Suppose that $X$, $Y$, and $Z$ are independent standard normal random variables. What is the probability of: \begin{equation} P\{ X+Y+Z+\Delta>0 \, | \, Z+\Delta>0, \, ...
2
votes
1answer
63 views

Is there a function that replaces a product by convolution?

Consider two functions $f(x),g(x) = 0 \forall x<0$, I'd like to know if we can always find an $h(x)$ which satisfies the integral equation $$f(x)g(x) = h(x)*f(x)$$ where '$*$' is the convolution ...
0
votes
1answer
364 views

Green's Function vs. Fundamental Solution

From the texts I've used, the Green's function is of a problem is $G(x,y)$ such that $LG(x,y) = \delta(x-y)$. The fundamental solution is u(x) such that $Lu(x)=\delta(x)$. They seem to be used for the ...
0
votes
1answer
603 views

Application of Fubini's theorem to prove that convolution is integrable

I guess that this is an easy question, but I don't have a very solid math background. I'm trying to prove that if $f,g \in L^1(\mathbb{R})$, then $h = f \star g \in L^1(\mathbb{R})$. So, I have: $ ...
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0answers
80 views

Calculating convolutions of probability density functions

I have a PDE: $$\frac{\partial N (x,u)}{\partial x}=\int _0^uN(x,u)f(u-u')du'$$ $$N(0,u) = \delta (u)$$ Here $f(u)$ is a probability density function for $0 \le u \le u_{max}$, $\int _0 ^ {u_{max}} ...
1
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1answer
122 views

convergence of convolutions and approximation of unity

Let $\phi : \mathbb{R}\rightarrow \mathbb{R}$ be an integrable function with $\int \phi(x)dx = 1$. Let us define $\phi_\delta = \delta^{−1}\phi(\delta^{-1}x)$. Show that for every continuous ...
0
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1answer
82 views

Efficient polynomial evaluation using idea of fast fourier transform

Please would anyone suggest an efficient algorithm ($O(n \log n)$) to evaluate a polynomial at all the $n$th roots of unity, where $n$ is not a power of $2$?
2
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1answer
152 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 ...
1
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1answer
62 views

Confused with estimator for random variables.

I am working on a practice exercise in preparation for a final this week. I am really stuck on the following problem: Let $X_1, X_2$ be a random sample for a population with the probability density ...
1
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1answer
86 views

Any clue how to solve this convolution integral?

With other words: find a (closed) expression for $\;\overline{\mbox{sinc}}(x)$ . $$ \overline{\mbox{sinc}}(x) = \int_{-\infty}^{+\infty} \frac{\sin(\omega\xi)}{\omega\xi} ...
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0answers
203 views

Approximate convolution of independent Beta variates?

Is there a way to approximate the convolution of Beta variables? Specifically, I am trying to find an approximation to $g(x_0)$: $$g(x_0) = \int \delta(x_0-\sum_{i=1}^{n} a_i x_i) \prod_{i=1}^{n} ...
1
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1answer
307 views

Convolution of indicator functions is continuous

Suppose I have an indicator function on a set of measure $E$, which is a subset of $[0,1]$. Is the function of this indicator convoluted with itself a continuous function? How can I show that it is? ...
3
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1answer
137 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, ...
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0answers
39 views

Integral transforms with interesting pointwise multiplications?

The convolution theorem states that the Fourier transform of the convolution of functions equals the pointwise multiplication of Fourier-transformed functions, i.e.: $$\mathcal{F}\{f*g\} = ...
0
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1answer
110 views

convolution of measurable function with analytic function

Let $f$ be a bounded measurable function with support on the unit disk $\mathbb D \subset \mathbb R^2$ and let $g$ be an analytic function on $\mathbb R^2$. Is it true that the convolution $h = f ...
4
votes
1answer
64 views

Properties of the operator $T: f\to f*g$

Let g be the characteristic function of [-1/2,1/2]. $T: f\to f*g$ (convolution). I have managed to prove that T is a linear,bounded,self adjoint,injective operator and it's immage is inclused in ...
0
votes
2answers
310 views

What does triple convolution actually look like?

I have to prove associativity of the convolution of three functions. I'm having trouble picturing how the variables will look. For periodic functions $f$, $g$, and $h$, I have the definition $$ (f * ...
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0answers
227 views

Gamma random variables with fixed sum (different scale parameters)

Given a vector of independent random variables $\{X_i\}_{i=1..N}$, each of which is distributed according to a Gamma-distribution with pdf $Pr(X_i=x;\alpha_i,\beta_i) = \frac{1}{\Gamma ...
2
votes
1answer
45 views

Show $\int_{\mathbb{R}^n}\Delta_x \Phi(x-y)f(y)dy = \int_{\mathbb{R}^n}\Delta_y \Phi(x-y)f(y)dy.$

I read in an article about Laplace's equation that $$-\int_{\mathbb{R}^n}\Delta_x \Phi(x-y)f(y)dy = -\int_{\mathbb{R}^n}\Delta_y \Phi(x-y)f(y)dy.$$ Could someone explain to me why this is? I ...
2
votes
1answer
58 views

What is the easiest way to find the inverse Laplace of F(s)?

$$ F(s)= \frac{1}{(s-1)^2(1-1/s^2)} $$ Do I have to multiply by $s^2/s^2$ and then use partial fractions or is there a way to use the convolution theorem?
1
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1answer
145 views

A self-convolution formula that counts bracket expressions

Problem: Consider an alphabet of size $m+2$, consisting of the two bracket symbols $\ [ \ ] \ $ plus $m$ non-bracket symbols ($m \ge 0$). Define $f_m(n)$ to be the number of length-$n$ strings on this ...
1
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2answers
38 views

How do i find the lapalace transorm of this intergral using the convolution theorem?

$$\int_0^{t} e^{-x}\cos x \, dx$$ In the book, the $x$ is written as the greek letter "tau". Anyway, I'm confused about how to deal with this problem because the $f(t)$ is clearly $\cos t$, but ...
4
votes
2answers
288 views

Let $S$ be the Schwartz class. Show that if $f,g\in S$, then $fg\in S$ and $f*g\in S$, where $*$ denotes convolution.

Let $S$ be the Schwartz class. Show that if $f,g\in S$, then $fg\in S$ and $f*g\in S$, where $*$ denotes convolution. To differentiate $fg$, we may apply Leibniz's rule ( ...