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.

Filter by
Sorted by
Tagged with
1 vote
0 answers
11 views

Asking about proposition related to Sobolev and Bergman spaces

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 ...
user avatar
  • 41
1 vote
0 answers
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 ...
user avatar
0 votes
0 answers
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 ...
user avatar
  • 133
0 votes
0 answers
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 ...
user avatar
0 votes
0 answers
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 ...
user avatar
  • 2,180
1 vote
0 answers
53 views

Is this integral convolution continuous?

The following question is related to the one in Is this convolution continuous? Let $0 \leq \xi \leq T$, $Y_0 \in L^2([-T, T])$, $a_1 \in L^2([0,T])$ and consider the following function: $$ Y_{1}(\xi)...
user avatar
3 votes
0 answers
45 views

$T*\varphi=\sin(x)$

I am looking for a distribution $T:\mathcal D(\mathbb R)\to\mathbb C$ and a test function $\varphi\in C_c^{\infty}(\mathbb R)$ with $$T*\varphi = sin(x),$$ but I can't think of any pair $(T,\varphi)$ ...
user avatar
  • 655
0 votes
0 answers
49 views

When $(f\cdot g)*h=f(g*h)$? [closed]

Inquiry. It is known that the convolution is commutative, that is, $f*(g*h)=(f*g)*h$ It is possible to obtain in some case that $(fg)*h=f(g*h)$? ($fg$ is the product of $f$ and $g$) Thanks
user avatar
  • 2,509
0 votes
0 answers
20 views

Convolution behaviour when applied to matrix product

Consider the discrete convolution operator, applied for example in convolutional neural networks. Let $W\in\mathbb{R}^{k\times k}$ be the convolutional filter and $X\in\mathbb{R}^{m\times m}$ be a ...
user avatar
  • 644
2 votes
0 answers
17 views

Expressing the convolution integral of scaled and translated arguments

I was reviewing the Fourier transform where I came across the convolution integral. If the convolution $x(t)*y(t) = \int_{-\infty}^{\infty}x(\tau)y(t-\tau)d\tau$ be defined, how the convolution of $x(...
user avatar
  • 2,817
0 votes
1 answer
74 views

How to compute the Fourier transform of a zero-centered Gaussian pulse over a specified range, say $[0,1]$?

Given an even Gaussian kernel $f : \Bbb R \to \Bbb R^+$ $$f(x) = \exp \left(- \frac{1}{2\sigma^2} x^2 \right)$$ and a probability density function (or, if you discretize $[0,1]$, then a probability ...
user avatar
0 votes
1 answer
25 views

Relationship between convolution for functions and for measures?

Is the definition of the convolution of two measures in any way analogous to the definition of convolution of two functions? I know almost nothing about measure theory and have trouble making sense of ...
user avatar
  • 167
1 vote
1 answer
38 views

Cyclic convolution of a periodic function with itself is a constant?

Let $f(x)$ be a periodic function of period $T$. Now let us define $c(x)$ the cyclic convolution (on the same period T) of $f$ with itself: $$c(x)=\int_t^{t+T}f(\tau)f(x-\tau)d\tau$$ I have the ...
user avatar
1 vote
0 answers
22 views

Convolution of harmonic function $h$ constant implies that $h$ is constant?

Let $C\subseteq \mathbb{R}^d$ be a compact set and $\rho\in L^1(\mathbb{R}^d,[0,+\infty))$ a function whose essential support is $C$. For some $R\in (0,+\infty)$, let $U:=C+B(0,R)$ and $h:U\to \mathbb{...
user avatar
0 votes
1 answer
23 views

Converence of the derivative of the convolution

Let $z \colon \mathbb R^2 \to \mathbb{R}$ be $\mathcal C_B^{1,1}(\mathbb R^2)$. Fix $\epsilon>0$, choose $\rho \in \mathcal C^\infty $ a mollifier and consider the convolution of $z$: $$z_\epsilon(...
user avatar
1 vote
1 answer
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 ...
user avatar
1 vote
0 answers
27 views

How to solve integral equation of convolution using Fourier transform

I'm having trouble solving the following exercise: Use the Fourier transform to solve the integral equation $$f(x) = \int_{-\infty}^{\infty} e^{-|x-\xi|}u(\xi)\,d\xi$$ Then verify your solution when $...
user avatar
  • 83
0 votes
2 answers
47 views

Convolution product and zero-product property

The convolution product for continuous integrable functions on $[0,+\infty)$ is defined as $$ (f * g)(t) = \int_0^t f(s) g(t-s) ds. $$ Does it has a zero-product property? The paper Variational ...
user avatar
  • 9,592
0 votes
1 answer
28 views

Calculation of $\chi_1\ast \chi_n$

Let $(f\ast g)(x)=\int f(y)g(x-y)dy$, $\chi_1=\chi_{[-1,1]}$ and $\chi_n=\chi_{[-n,n]}$ for $n\in \mathbb N$. I am trying to calculate $f_n=\chi_1\ast \chi_n$ and show $\|f_n\|_ \infty=2$. Since $\...
user avatar
0 votes
0 answers
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 ...
user avatar
  • 9
1 vote
0 answers
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, \...
user avatar
3 votes
1 answer
36 views

Fourier Transform of Product of 2 Function (inverse convolution)

Having the Fourier Transform defined as $$\mathcal{F}(f(x)) = \hat{f}(k) = \frac{1}{\sqrt{2\,\pi}} \int_{-\infty}^{+\infty} f(x)\,e^{\displaystyle -i\,k\,x}\,\mathrm{dx}$$ I am asked to form the FT of ...
user avatar
  • 985
1 vote
0 answers
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 ...
user avatar
  • 163
1 vote
0 answers
32 views

Can the $\log(z)$ function be evaluated as a sum over exponential terms?

I've noticed the exponential function $e^{-z}$ can be evaluated as a sum over $\log$ terms as follows where the evaluation limits $N$ and $f$ are both assumed to be positive integers. $$e^{-z}=\...
user avatar
  • 4,553
2 votes
1 answer
32 views

Does the Bayes-Filter perform a convolution in the prediction step?

I am watching the (fantastic) SLAM lectures of Claus Brenner, where he introduces the Bayes-Filter (Kalman-Filter, Particle-Filter, Histogram-Filter). He says, that the prediction step involves the ...
user avatar
0 votes
0 answers
23 views

U substitution to turn a double integral into a convolution?

I'm trying to work through the results of a classic paper in super-resolution microscopy (C. Sheppard, Optik, 1988). The main result sort of hinges on writing the double integral: $$ \int dx \int dx^\...
user avatar
0 votes
0 answers
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 ...
user avatar
0 votes
1 answer
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-...
user avatar
  • 31
1 vote
1 answer
49 views

When is the mollification of an $L^p$ function in $L^p\cap L^{\infty}$?

Let $f\in L^p(\mathbb{R}^3)$ and $f_{\varepsilon}=f\ast \varphi_{\varepsilon}$ its convolution with the standard mollifier $\varphi_{\varepsilon}$. Then it is well-known that $f_{\varepsilon}\in L^p(\...
user avatar
  • 582
1 vote
1 answer
22 views

When performing a convolution of probability density functions, how does one determine the intervals?

I am having trouble applying the convolution formula for probability density functions when the pdfs of the random variables in question are defined over different intervals. Here is an example: Say $...
user avatar
  • 11
1 vote
0 answers
20 views

Is this mollification smooth?

Let $n\geq 2$. Let $f\in C^{\infty}(\mathbb{R}^n)$ be some mollified function, i.e. $f=f_{\varepsilon}=\tilde{f}\ast \varphi_{\varepsilon}$ for some $\tilde{f}\in L^p(\mathbb{R}^n)$ and $g=\frac{1}{|x|...
user avatar
  • 582
0 votes
1 answer
29 views

Convolution of 2D functions - Matlab

I need to compute the convolution of two functions in $\mathbb{R}^2$ numerically. Basically, I have two functions, $f(x,y)$ and $g(x,y)$, and I want to compute $f\cdot g$. I have a rectangular domain $...
user avatar
  • 1
2 votes
0 answers
47 views

Convex conjugate of sum of convex functions — infimal convolution

I have a function $f: \mathbb{R} \to \mathbb{R}_+$ defined by $$ f(x) = f_1(x) + f_2(x) + f_3(x) - f_4(x) $$ where every $f_i$ is a proper, closed convex function defined over some interval $[a,b] \...
user avatar
0 votes
0 answers
24 views

Approximation of $\|x\|_2$ via mollification

Let $f\colon \mathbb{R}^3\to \mathbb{R}$, $f(x)=\|x\|_2$. I want to approximate $f$ in $L^p$ by $C^1$ functions which are vanishing in a neighborhood of zero. Let $\varphi_{\varepsilon_n}=\frac{1}{\...
user avatar
  • 582
3 votes
1 answer
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 ...
user avatar
  • 985
0 votes
0 answers
24 views

Example of asymptotics for a discrete convolution

I am interested in the technique that this answer refers to: Consider the generating functions $F(z) = \sum_n u_n z^n$ and $G(z) = \sum_n v_n z^n$. The convolution has generating function $H(z) = F(z)...
user avatar
1 vote
1 answer
28 views

How to calculate the properties of the resultant 2D Gaussian from the deconvolution of one anisotropic, 2D Gaussian with another?

Anisotropic, two-dimensional Gaussians, are described by the equation (from here): $f(x,y)=A\exp \left( -(a(x-x_0)^2 + 2b(x-x_0)(y-y_0) + c(y-y_0)^2 )\right)$ where A is the amplitude of the peak, $...
user avatar
1 vote
1 answer
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 $\phi\in C^\infty$, ...
user avatar
  • 1,053
0 votes
1 answer
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. $\star$ ...
user avatar
  • 27
0 votes
0 answers
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 ...
user avatar
  • 137
4 votes
1 answer
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 ...
user avatar
  • 12.8k
0 votes
1 answer
50 views

Why the scale factor of the product of two gaussian functions is the convolution of the same gaussian functions?

The product of two Gaussian functions $$ f(x)=\frac{1}{\sqrt{2 \pi} \sigma_{f}} \exp\left(-\frac{x^{2}}{2 \sigma_{f}^{2}} \right) \quad \text { and } \quad g(x-y)=\frac{1}{\sqrt{2 \pi} \sigma_{g}} \...
user avatar
0 votes
0 answers
11 views

Numerically approximating a convolution integral with variable time steps

As part of a ODE system that I am trying to solve, one of the terms is calculated via a convolution integral. It's possible to approximate this integral numerically with a constant $\Delta t$ (using ...
user avatar
  • 101
0 votes
0 answers
7 views

Convolution theorem if one function depends on a 2 dimensional reciprocal space

I would like to write the following integral in terms of a convolution: \begin{equation} H(x_1,x_2) = \int dt e^{\mathrm{i} t (x_1+x_2)/2} f(t,x_1,x_2) g(t), \end{equation} where $f(t,x_1,x_2)$ ...
user avatar
  • 103
0 votes
0 answers
20 views

Cylindrical Coordinate Convolution Between Dirac Deltas for a Plane and Sphere

Assuming this the correct way to express the cylindrical coordinate convolution between Dirac deltas for a $\phi,\rho$-plane and a sphere of radius $r_0$ centered at the origin: $$\delta(\sqrt{\rho^2+...
user avatar
0 votes
1 answer
43 views

Convolution of $L^1 \cap L^q$ and $L^2 \cap L^{\infty}$

This is not from assignment, something on my notes. With $f \in L^1 \cap L^q$ for a $q > \frac{3}{2}$ and we take its convolution $u = \frac{1}{4 \pi|x|} * f$, we can show that $u$ is in $L^p \cap ...
user avatar
0 votes
0 answers
18 views

How can you convert a convolution stencil from a coarse to a fine mesh?

In the paper Learning Across Scales, the authors describe a multigrid method for transfering convolution operators to different meshes (pixel sizes in images); that is, given a 2d convolution kernel $...
user avatar
0 votes
1 answer
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 $|x|\geq 1$ ...
user avatar
  • 8,756
1 vote
1 answer
88 views

Fourier transform of a time-variant convolution

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*...
user avatar
  • 53
0 votes
1 answer
74 views

Convolution on a locally compact group is associative

Consider the following fragment from Folland's book "A course in abstract harmonic analysis" (question is below the image). Can someone explain why the boxed equality is true? Don't we need ...
user avatar
  • 985

1
2 3 4 5
53