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.
2,906
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Convolution Integral with Same Direction in Integrand
I am working on the convolution below, however I have gotten stuck. I am not sure how to think about changing the bounds of the integrals to give me an answer. Here is the problem, where $u(t)$ is the ...
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Convolution with the "wrong" sign
Let $u\in L^\infty(\mathbb R^n)$, $\eta\in C([-1, 1]^n, [0, 1])$ and, given $\epsilon>0$ consider the mollifier $\eta_\epsilon(x) = \epsilon^{-n} \eta(x/\epsilon)$.
Define $u_\epsilon = u\ast \eta_\...
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Convolution and fractional operator
Let $h\in C_c^\infty(\mathbb R^n)$ and consider the equation
$$(-\Delta)^s u = h,$$
where $(-\Delta)^s u$ denotes the fractional Laplacian of $u$.
Let $\Gamma$ be its fundamental solution, i.e. $(-\...
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Fourier Transform Method for Solving Fredholm Translation-Invariant Covariance-Kernel Integral Equations
I'm examining a problem involving the Fredholm integral equations of the second kind and trying to apply Fourier transform techniques. Particularly, the challenge arises when seeking to express the ...
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Do mollifiers exist in dimensions higher than 1?
A mollifier is defined as a function $\varphi: \mathbb{R}^n \rightarrow \mathbb{R}$ such that
$$\int_{\mathbb{R}^n} \varphi(x) dx = 1$$
$\varphi$ has compact support
$$\lim_{\epsilon \rightarrow 0} \...
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Understanding convolution formula for density function
Let $X$ and $Y$ be uniformly distributed, independent random variables on $[0,1]$. Put $Z = X-Y$. How can I obtain the density function of Z?
I understand the following solution by geometric ...
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Representation formula for the $\Delta^2 u =f$
I already know the representation formula of $\Delta u =f$ is the that $u(x)$ equal to the $\phi * f(x)$, where $\phi$ is fandamental solution.
Now I have wonder konw the representation formula of $\...
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Convolution of two exponential functions where x > 0
Assuming $X$ and $Y$ are i.i.d. random variables let $Z = X + Y$
$$
f_X(x) =
\begin{cases}
\lambda e^{- \lambda x} & x \gt 0 \\
0 & \text{else}
\end{cases}
$$
$$
f_Y(y) =
\...
- 51
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Do there exist mathematical transforms other than the Fourier Transform for which there exists some sort of a fast convolution theorem?
One nice property of the Fourier transform is it's famous convolution theorem :
$$f*g = \mathcal{F}^{-1} \left\{ \mathcal{F}\left\{ f \right\} \cdot \mathcal{F}\left\{g\right\} \right\}$$
If we want ...
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Convolution of two uniform probability densities (two square waves)
Assuming $X$ and $Y$ are i.i.d. random variables let $Z = X + Y$
$$
f_X(x) = f_Y(y) =
\begin{cases}
1/2 & -1 \le x \le 1 \\
0 & \text{else}
\end{cases}
$$
Find the density ...
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Is the convolution of a tempered distribution and a Schwartz function also a function?
Let $T \in \mathscr{S}'(\mathbb{R}^n)$ and $f \in \mathscr{S}(\mathbb{R}^n)$. We may define their convolution as
$$(T * f)(\varphi) = T(\tilde{f}*\varphi)$$
where $\tilde{f}(x) = f(-x)$. The above ...
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Trouble visualizing a graph of the sum of infinite Dirac functions
I'm currently trying to solve a problem on my "Signals and Systems" class and I'm stumped.The question has 2 parts:
1-) The function T.I can be defined as:
$$
T\cdot I(t) = \sum_{n=-\infty}^{...
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Convolution of Sets or Geometric Objects
I'm trying to find the term for something I thought of but I assume someone else has done it before.
Let us say I have two stamps, a square one and a circle one. I first stamp down the square one, ...
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Is there a function whose autoconvolution is its square? $g^2(x) = g*g (x)$
I am looking for a function over the real line, $g$, with $g*g = g^2$ (or a proof that such a function doesn't exist on some space like $L_1 \cap L_2$ or $L_1 \cap L_\infty$). This relation can't hold ...
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Riemann-Lebesgue Lemma and trigonometric polynomial
from my understanding Riemann-Lebesgue Lemma says that every continuous and 2π periodic function, the Fourier coefficient go to zero as n approach to infinity
$\lim_{|n|\to+\infty}\hat{f}(n)=0,$
but i ...
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Is $(df(t)/dx) g(t)\star f(t)$ equal to $f(t) g(t)\star(df(t)/dx) $?
i tried to prove：
$$
\dfrac{df(t)}{dx} \cdot g(t)\star f(t) = f(t)\cdot g(t)\star \dfrac{df(t)}{dx}
$$
where $\star$ is convolution, $f$ is both a function of $x$ and a function of $t$, and $g$ is ...
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Positivity of the gaussian convolution at the root
I am currently struggling on the following problem. Let $\sigma>0$ a real number and $g_\sigma$ the gaussian function with mean 0 and deviation $\sigma$. Let also $f$ be a smooth real function such ...
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Convolving an image with mean filter infinite times
I was taught in my image processing class that when a mean filter is applied infinite times on a given image, the intensity of each pixel reaches the same value. I understood this that time entirely ...
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Prove $\underbrace{\frac{1}{2a} \chi_{[-a,a]} \ast...\ast \frac{1}{2a} \chi_{[-a,a]}}_{k-\text{times}} \ast \chi_{[-B-ka,B+ka]}=1$ if $|\xi|\leq B$.
My Question is regarding the answer in the following question Deriving the Oversampling formula
To be more specific
let $g(\xi) = \underbrace{\frac{1}{2a} \chi_{[-a,a]} \ast...\ast \frac{1}{2a} \chi_{[...
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Spatial convolution of Taylor Expanded function which depends on isostable coordinates
I have been looking at reducing a neural field model into isostable equations and eventually into a Kuramoto-Sivashinksy type form to study patterns. I am now however confused on a particular part of ...
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1
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Convolution calculation
I'm trying to calculate the convolution between $x(t)=e^{-t}u(t)$ and $y(t)=e^{-(t-2)}u(t-2)$.
$\int_{-\infty}^{\infty} e^{-τ}u(τ)e^{t+τ-2}u(t+τ-2)dτ=\int_{0}^{t-2} e^{t-2}dτ=e^{t-2}\int_{0}^{t-2} dτ= ...
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0
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Fast convolution with "small" values
Say we have two sequences of integers $a = \{a_1 \dots a_n\}$ and $b = \{b_1 \dots b_n\}$, where $a_i, b_i \in \mathbb{Z}_q$, but we know some value $p<q$ such that $0 \leq a_i < p$.
We want to ...
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Ratio of Gaussian convolution is bounded
I am struggling with the following problem. Let $f$ be a smooth real function with only a finite number of zero. Let $\sigma>0$ a real number. Is it possible to show that the function $$\...
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Calculate the limit of $\langle \partial_x f,\rho_\varepsilon \ast \chi_B \rangle $
Let $f(x_1,x_2)=x_2^2\chi_E$ where $E=\{(x_1,x_2)\in \mathbb{R}^2:x_1\geq 0 \}$ and let $B$ be the unit ball in $\mathbb{R}^2$. Since $f$ is locally integrable, it can be interpreted as a distribution ...
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Equation for Gaussian kernel's effect on a frequency's amplitude
Applying a Gaussian blur/kernel with a sigma of $\sigma_{gau}$ to a sine/cosine wave of frequency $f_{sin}$ will cause what percent reduction in the amplitude $p_{amp}$ (not power) of the sine wave?
...
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votes
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A nested double sum(to do with e?)
I’ve come across this nested double sum while doing an investigation but cannot seem to find a closed form for it.
$$1+\sum_{i=1}^\infty{\frac{1}{i!} \sum_{j=0}^i}{\frac{1}{j!}}$$
This is about the ...
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Is the integral $\int_{-\infty}^\infty dx \int_{-\infty}^\infty dy \frac{e^{i a x}}{\sqrt{x^2+y^2}}$ convergent?
Is the integral
$$I = \int_{-\infty}^\infty dx \int_{-\infty}^\infty dy \frac{e^{i a x}}{\sqrt{x^2+y^2}}$$
convergent for real $a$?
I have an idea to calculate it but I am not sure if it is correct:
...
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What is the admissible function in the convolution integral?
I know this is a dumb question, but I'm having some trouble with the notation.
In 1-D, the solution to a BVP with some linear differential operator $$Lu(x) = f(x)$$ is of the form: $$u(x)=\int_{-\...
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About the notation of $2$D Laplace operator
I'm reading a paper on $2$D discrete Laplace operator, and perhaps because it's an old paper, the notation in it really bothers me a lot. So can someone please explain it to me? For example, the ...
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Computing $I=\int_{-\infty}^\infty \frac{e^{-\frac{1}{2}\left(\frac{x-\mu)}{\sigma} \right)^2}}{\sigma \sqrt{2 \pi}}\frac{1}{1+e^{-x}}\, \mathrm{d}x$
I had posted this on Stats Stackexchange as I initially thought a probabilistic approach would be best suited. But posting it here for as it has more traffic.
Problem
Evaluate $$I=\int_{-\infty}^\...
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Exact Successor State Distribution for a Pendulum
I want to solve the following problem. Suppose we have a simple pendulum, which follows the differential equation
\begin{equation}
\dot{x} = f(x) = [x_2, -\sin(x_1)]^T, \text{with } x=[x_1, x_2]^T.
\...
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Convolution of Double Coset Indicator Function in Hecke Algebra of Locally Profinite Group
Let $G$ be a locally profinite group (i.e. a topological group that is locally compact Hausdorff and totally disconnected or, equivalently, a Hausdorff topological group s.t. $1 \in G$ has a ...
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Method to solve a first order non linear ODE with a convolution inside
Working on thermodynamics, I arrived to an equation like this in spatial and frequency domain:
$$ \partial_z G(z,\omega) + \left( G(z,\omega) F(\omega) \right) \ast G(z,\omega) = 0 $$
with $0 < ...
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Convolution of an $L^{1}(\Bbb{R})$ function with $\frac{1}{\sqrt{|x|}}$ is well defined?
I think this should be true.
Let $f\in L^{1}(\Bbb{R})$ and consider $g(x)=\frac{1}{\sqrt{|x|}}$ , then is $(f*g)\,(y)$ well defined for $y$ almost everywhere?
I know that convolution of two $L^{1}$ ...
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A bit of confusion with the differentiation of convolution and support of functions
Let $f, g : \mathbb{R} \to \mathbb{R}$ be compactly supported smooth functions. Especially, let us suppose that their supports are both contained in $[0,\infty)$.
Then, I see that we may write their ...
- 6,808
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Convolution of f(-x) with delta
i'm struggling with the following convolution:
$\ f(-x,-y)* \delta(x,y-a)$
I don't know if it should be solved that way:
$\ f(-x,-(y-a))=f(-x,-y+a) $
or: (just the shifting)
$\ f(-x, -y-a) $
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Space-time convolution with the heat kernel - still continuous?
Let $K(x,t):=\frac{1}{(4\pi t)^{n/2}}e^{-\lvert x \rvert^2/(4t)}$ be the $n$-dimensional heat kernel.
Also, consider a locally-integrabl functon $f : \mathbb{R}^n \times [0,\infty) \to \mathbb{R}$. ...
- 6,808
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Definition of Convolution
I am currently studying calculus, but I am stuck with the definition of convolution in terms of constructing the mean of a function. Suppose we have 2 functions, f and g. We want to create the mean of ...
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How to approach the following differential equation
I have a differential equation of the form
$$
\frac{\mathrm{d}g}{\mathrm{d}x} = f(x) + \int_0^x g(y)f(x-y)\mathrm{d}y + \alpha g(x).
$$
$f$ is a monotonous decreasing function, satisfying $\int_0^\...
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Extension of single zero-crossing property
Let $f\in\mathscr{C}^2(\mathbb{R},\mathbb{R})$ a strictly increasing function, striclty convex on $(-\infty,0)$, strictly concave on $(0,\infty)$ and let $\sigma_1>\sigma_2>0$ be two real ...
- 302
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Zero-crossing for convolution
Suppose that you have a function $f\in\mathcal{C}^2(\mathbb{R},\mathbb{R})$. Can one show that if the function $g$ defined by:
$$g(x):=\int_\mathbb{R}f(s)e^{-(s-x)^2/2}ds$$
has to zeros $x_1$ and $x_2$...
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What is this operation?
Let $F[n]$ and $G[n]$ be arrays of length $N$. At first, $G=F$.
After initialisation, $G$ is calculated by the relation
...
- 1,060
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70
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Unique root for a simple convolution
I am struggling to show the following problem. Let $f \in \mathcal{C}^\infty(\mathbb{R},\mathbb{R})$ (satisfying the exponential growth condition) be a real function and $x^*\in \mathbb{R}$ such that $...
- 302
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Can the transpose of cropped block-Toeplitz matrix be represented as a cropped 2D convolution?
Suppose we have the following matrices defined over the field of complex numbers ($\Bbb C$):
a square input matrix $\mathbf{U}$ with dimensions $n \times n$
a symmetric convolution kernel $\mathbf{H}...
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Uniqueness of the root for a DoG function (Difference of Gaussian)
I am struggling with the following problem:
Let $f$ be a real function such that:
$f\in\mathcal{C}^\infty(\mathbb{R},\mathbb{R})$,
$f$ is strictly convex on $(-\infty,0)$, strictly concave on $(0,\...
- 302
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1
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Bus Station, Multiple buses arriving at the same time
Suppose we have a bus station. Each bus stops in this station for a pre-determined amount of time. due to traffic and other reasons, each bus can be late or early to arrive at the station, and we have ...
2
votes
1
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How to show a convolution $f \ast g$ vanishes at infinity, where $g(x)=\frac{1}{|x|}$?
Let $f$ be a smooth function defined on $\mathbb{R}^3$ with $f(x) = O(1/|x|^{2+\epsilon})$, and let $g(x)=\frac{1}{|x|}$. I want to show that $f\ast g (x) \to 0$ as $|x| \to \infty$.
(Maybe this is ...
- 983
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Conditions for independence of sum of i.i.d. random variables from remaining
Previously I asked the following question, that does not generally hold:
The following question is very similar to this reference.
Let $(X_{1},X_{2},X_{3})$ be a random vector with continuous ...
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0
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16
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How to do deconvolution but with exponential cross terms?
I have an equation
$$
g(x_0, y_0)=\int\int dxdy[h(x_0-x, y_0-y)f(x, y)e^{i\omega_0y(x_0-x)}]
$$
in the equation $g$ and $h$ are known complex functions, $\omega_0$ is a known real constant and $f$ is ...
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0
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Intuition for how linear superposition is related to the convolution theorem
This is my current understanding of convolution after having read through this blog post
The convolution operator can be thought of as an operation of linear superposition. If we have the response of ...
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