# Questions tagged [convolution]

Questions on the (continuous or discrete) convolution of two functions. It can also be used for questions about convolution of distributions (in the Schwartz's sense) or measures.

2,608 questions
Filter by
Sorted by
Tagged with
1 vote
11 views

In the below there is a proposition i have read in a scientific paper , i understood it but not completely just i have missed how the researcher goes here \ \begin{equation} \begin{aligned} \parallel ...
1 vote
26 views

### Find a function $g$ such that $(1*g)(t)=t^{-1/2}$.

Is there some function $g: (0,+\infty) \to \mathbb{R}$ satisfying $$(1*g)(t)=t^{-1/2}$$ for all $t>0$? Here the sign $*$ represents the convolution. I tried apply the Laplace transform both site of ...
10 views

### Solving the convolution equation $U*g=\sin{2x}$ where $g(x)=e^{-|x|}$.

The problem is as stated in the title but in more detail: find all tempered distributions $U\in\mathcal{S'(\mathbb{R})}$ that solve the convolution equation given in the title. My approach uses the ...
19 views

### Is the convolution of $L^2$ functions continuous? [duplicate]

Is the answer to the following question positive? The convolution of two $L^2(\mathbb R)$ functions is continuous I briefly recall it here: Take $f$ and $g$ $\in L^2(\mathbb R)$, then I want to show ...
9 views

### Smooth approximation of characteristic functions using convolution

Given a subset $A\subseteq \mathbb{R}^n$, define $A_{\varepsilon}:=\{x\in\mathbb{R}^n: d(x,A)<\varepsilon\}$. Assume that $A$ is Lebesgue integrable. Show that, given any $\varepsilon>0$, there ...
1 vote
53 views

1 vote
24 views

### Is convolution integrable in $L_1(\mathbb{R}^n)$?

Let $|h(y)|\in L_1(\mathbb{R}^n)$, i.e. $\int\limits_{\mathbb{R}^n}|h(y)|\,dy<+\infty$. Consider the function $F(x)=\int\limits_{\mathbb{R}^n}|h(x-y)|\,dy$. It is known that $F(x)$ be bounded and ...
1 vote
27 views

36 views

### How general is this property about correlation and the sum of two normal RVs?

Edited to make this more concrete: Given a random vector $(X_1,X_2)$ that is jointly normal with means / sd's $\mu_1,\mu_2, \sigma_1,\sigma_2$ and correlation $\rho$, the sum of $S=X_1+X_2$ is ...
1 vote
50 views

### Solving Backward Heat Equation with a Backward Heat Kernel?

Let $D>0$ be a constant. Imagine we have the following forward heat conduction problem: \begin{align*} \begin{cases} \partial_t u = D \partial_x^2u &, \quad (x,t) \in \mathbb{R} \times (0, \...
36 views

### Fourier Transform of Product of 2 Function (inverse convolution)

Having the Fourier Transform defined as $$\mathcal{F}(f(x)) = \hat{f}(k) = \frac{1}{\sqrt{2\,\pi}} \int_{-\infty}^{+\infty} f(x)\,e^-i\,k\,x}\,\mathrm{dx$$ I am asked to form the FT of ...
1 vote
32 views

### Generalization of convolution theorem [closed]

I am for years interested in things related to convolution, and different groups, and fast transformations If the convolution is defined as $$\sum_{j+k \equiv i \operatorname{mod}N} u_j v_k$$ Then we ...
1 vote
32 views

16 views

### Confusion regarding the derivation of graph convolution

I am currently studying Spectral Graph Convolutions, and I am following this document: https://atcold.github.io/pytorch-Deep-Learning/en/week13/13-1/. They have derived the convolution as follows: The ...
19 views

### Show that $f(t-i) \star g(t-k)=m(t-i-k)$

Let $f,g\in L^1$, and let $m=f \star g$ the convolution product of $f$ and $g$, $i,k\in \mathbb Z.$ Show that $$f(t-i) \star g(t-k)=m(t-i-k)$$ \begin{align*} f(t-i)\star g(t-i)&=f(t-i)\star g(t+i-...
1 vote
49 views

1 vote
20 views

47 views

135 views

### Folland theorem 6.19: $p=\infty$

Consider the following theorem from Folland's book "Real analysis: Modern techniques and their applications": In the book, it can be found on p194 (second edition). I understand the proof ...
24 views

1 vote
40 views

### Convolution of distributions is commutative

Let $u,v\in \mathcal{E}'(\mathbb{R}^n)$ be two compactly supported distributions. Define $u*v$ to be the distribution $u*v(\phi) = u(Rv*\phi)$, where $v*\phi =v(\tau_xR\phi)$ for $\phi\in C^\infty$, ...
18 views

### How to simplify this expression including integral, modulus and convolution?

I encountered this expression during simplification: $$\int \lvert exp(ixy)g(y)\star f(x)\lvert^2dy$$ $i$ is the imaginary unit, $g(y)$ is a real function while $f(x)$ is a complex function. $\star$ ...
20 views

### How to prove that probability density function of the sum of 2 independant variable is equal to their convolution?

The probability density function of the sum of two independent random variables is the convolution of their individual probability density functions. What is the simplest demonstration that proves ...
65 views

### Integral with respect to convolution of measure

Let $E$ be a $\mathbb R$-Banach space, $$\theta_n:E^n\to E\;,\;\;\;x\mapsto\sum_{i=1}^nx_i$$ for $n\in\mathbb N$ and $\lambda$ be a measure on $\mathcal B(E)$. Remember that the $n$-fold convolution ...
50 views

43 views

27 views

### Mollification of a product of two functions in $\mathbb{R}^n$

Consider $f$ and $g$ to be two functions such that $f$ is supported on the unit ball and $g$ is a function that vanishes in the unit ball (for points when $|x|<1$), and is non-zero for $|x|\geq 1$ ...
1 vote
For a time-invariant convolution, given the convolution theorem, we know that $\mathcal{F}\{(h*x)(t)\}=\hat{h}(\omega).\hat{x}(\omega)$. My question is what if the convolution is time-variant. Let \$(h*...