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.

learn more… | top users | synonyms

0
votes
1answer
26 views

Use of convolutions to compute the distribution of the sample mean?

Let's consider N i.i.d continuous random variables from some arbitrary distribution. Why do we have to approximate the distribution of the sample mean using the CLT? Why can't we explicitly compute ...
1
vote
0answers
17 views

Convolution of $\mathbb{1}_{\mathbb{Q}}$

Is it possible to compute the convolution of $\mathbb{1}_{\mathbb{Q}}$ with $e^{-1/(1-t)^2}$?
4
votes
0answers
83 views

Symbolic math engines barf on this ostensibly tractable integral.

$$\frac14 \int_{-M\pi}^{N\pi - s} \cos(tu/M) \cos((t+s)u/M)(1-\cos(t/M))(1-\cos((t+s)/N))\space \mathrm d t$$ with integer $u$. Alpha runs out of time. Maxima gives a tremendous result that can ...
1
vote
1answer
54 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
vote
0answers
16 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 ...
0
votes
0answers
20 views

Convolution of two random variables, help with a proof

I know there are probably several ways to prove it, I'm interesting in this one in particular: Let $X,Y$ be two independent random variables. Then the probability distribution of $X+Y$ is: ...
4
votes
2answers
47 views

Does a convolution of a function $f(x)$ and a polynomial $p(x)$ will always result in 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 ...
2
votes
1answer
83 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 ...
1
vote
2answers
74 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) + ...
0
votes
0answers
20 views

Identities involving convolution and dot product?

In my work I commonly encounter things like $\mathbf A\ast (\mathbf B\cdot \mathbf C)$ and $\mathbf A\cdot (\mathbf B\ast \mathbf C)$, where the operator $\ast$ denotes convolution. Neither operation ...
1
vote
1answer
66 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
vote
0answers
29 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 ...
1
vote
2answers
29 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
vote
1answer
27 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
vote
1answer
71 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 ...
0
votes
0answers
24 views

similar to Plancherel theorem

I know that the Plancherel theorem is $$ \int |\hat{f}(x)\hat{g}(x)|^2 dx = \int |f*g(y)|^2 dy. $$ But I have a question why the below identity is holded $$ \int |\hat{f}(x)\hat{g}(x)|^2 dx = \int ...
0
votes
2answers
23 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
vote
2answers
116 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) ...
2
votes
1answer
87 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 $$f*g:\mathbb{R}^n\to\mathbb{R}\;,\;\;\;\int f(\tau)g(x-\tau)\text{ d}\tau$$ ...
0
votes
0answers
23 views

Thinking about a convolution on the circle as a convolution on the real line

I hope my question is not too vague. Let $f$ and $g$ be (smooth, say) real-valued functions on the circle. Let $f*g$ denote the standard $L^2$ convolution on the circle. Question: is it possible to ...
1
vote
3answers
58 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
votes
1answer
64 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
votes
1answer
71 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 ...
0
votes
0answers
30 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
vote
1answer
41 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
74 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
votes
1answer
75 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
vote
1answer
47 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
votes
1answer
31 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
votes
1answer
80 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
votes
0answers
34 views

Inverse Z transformation of 1/(z^2(z^2+1)^2)

I need to find the inverse Z transformation of $\frac{1}{z^2(z^2+1)^2}$ So far, I've tried using the convolution property, and so, inverting $\frac{1}{z(z^2+1)}$, gave me $-0.5i({{(-i)}^k-i^k})$ for ...
0
votes
1answer
38 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.
4
votes
0answers
98 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
votes
0answers
17 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. ...
0
votes
0answers
140 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
vote
0answers
58 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
49 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
82 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
114 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: $ ...
1
vote
0answers
49 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
vote
1answer
56 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
votes
1answer
62 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
votes
1answer
104 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
vote
0answers
48 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
vote
1answer
74 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} ...
0
votes
0answers
42 views

Probability that two bivariate distributions differ

My problem is similar to the unanswered Probability of collision (two bivariate normal distributions): I have two points in 2D space, let's say $A(\mu_x, \mu_y)$ and $B(r_x, r_y)$. The uncertainties ...
1
vote
0answers
69 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
vote
0answers
28 views

$f_{X^2}(x)$ VS $f_X(x^2)$ [duplicate]

Sorry, this time the format should be accurate. In probability, when we try to describe a pdf, we write it as $f_X(x)=1/x$, which means the random variable is X and the x is the specific variable in ...
1
vote
1answer
79 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? ...
2
votes
1answer
60 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", ...