Questions tagged [convolution]

Question 1

I'm searching for the solution $\bar{a}$ to the system of equations $\bar{e}_1 = B\bar{a}$ given by \begin{equation} \left[\begin{array} & 1 \\ 0 \\ \vdots \\ 0 \\ 0 \end{...

Question 2

Is there a way to conserve area of two Gausians, such that $\int f(x)dx + \int g(x)dx = \int\left[ \int f(\tau)*g(x-\tau)d\tau\right]$? For context, I have stumbled upon a math problem in my lab and I ...

Question 3

Let $\sigma_I=\{e^{2i\pi t}:t\in I\}$ and $\sigma_J=\{e^{2\pi i t}:t\in J\}$ be two disjoint subarcs on the the first quadrant of unit circle of arc length $\theta$, where $I,J\subseteq [0,1/4]$ of ...

Question 4

For integers $n$, the arithmetic derivative $D(n)$ is defined as follows: $D(p) = 1$, for any prime $p$. $D(mn) = D(m)n + mD(n)$, for any $m,n\in\mathbb{N}$ (Leibniz rule). The Leibniz rule implies ...

Question 5

What I mean is that we have $T \in \mathcal{S}'(\mathbb{R}^n)$ such that for all $f \in C^\infty_c (\mathbb{R}^n)$, we have that $\mathrm{supp}(T \ast f)$ is compact, and we're asking whether $\mathrm{...

Question 6

Let us assume that we have a $n$-th order polynomial $f=y(t)$ and an exponential function $g=e^{-\alpha t}$. Is there any formula or property that can be used to compute the convolution $f*g$? Or, ...

Question 7

I’m trying to understand a step in Appendix A.1 of Bejenaru and Herr, The cubic Dirac equation: small initial data in $H^1(\mathbb{R}^3)$. The paper proves global well-posedness for the cubic Dirac ...

Question 8

Question: Is there a (simpler) closed form of $\color{blue}{C(t) = \operatorname{Ei}(-t) \theta(t) \star \operatorname{Ei}(t) \theta(-t)}$? Definitions: Exponential Integral: $$\operatorname{Ei}(x) = \...

Question 9

I would like a comment on the correctness of my derivation below: I have the following Duhamel Convolution integral $$ U(t)=\int_0^t e^{-p(t-y)}f(y)\text{d}y $$ For reasons that have to do with the ...

Question 10

I am trying to prove some statements about specific Gelfand pairs, when considering representations of some finite groups over field $k$ which can be of positive characteristic. For this I study some ...

Question 11

I have the following function which is a probability density function on the $-1 < t < 1$ interval: $$f(t) = \frac{2\sqrt {1-t^2}}{\pi}$$ I wish to convolve this function with itself to get a ...

Question 12

I will begin with the mathematical question at hand, and then describe technical details that were the background of the question, and then some possible approaches, although I clearly have not solved ...

Question 13

I am working with a convolution sum of the form $$ h(j) = \sum_{k=0}^2 f\!\big((j-k) \bmod 3\big)\, g(k), $$ where $f, g : \{0,1,2\} \to \mathbb{C}$. Because of the modulo $3$ structure in the index ...

Question 14

Consider $f\in L^2(\mathbb{T})$ where $\mathbb{T}=[0,2\pi]$ and further denote by $\mathcal{F}$ the Fourier transform. Define \begin{align*} g(t):=\int^t_0f(s)ds. \end{align*} Can we find a function $...

Question 15

I'm reading The Art of Electronics, in 1.4.3, the section on passive differentiators. The math of the situation is given by $$\frac{d}{dt}(V_{in}(t)-V_{out}(t))=\frac{1}{RC}V_{out}(t)\tag{1}$$ In the ...

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Questions tagged [convolution]

Solving linear system with triangular Toeplitz matrix

Is there a deconvolution of two Gaussians, which conserves area?

On the convolution identity of a sub arc of circle and the open set which is thickened epsilon amount of another subarc in circle.

A singular identity involving the arithmetic derivative and the divisor counting function

If all convolutions of a tempered distribution T with any compactly supported function are compactly supported, is T compactly supported?

Is there a formula for calculating the convolution of a polynomial and an exponential? [closed]

Can Young's inequality give a pointwise bound for a convolution?

Closed form of $\operatorname{Ei}(-t) \theta(t) \star \operatorname{Ei}(t) \theta(-t)$

Evaluation of a Broken-up Convolution Integral

Hecke algebras over fields other than $\mathbb{C}$

Difficult Convolution Problem -- I Am Stuck with the Integration

On consequences of Titchmarsh theorem: can the analytical extension of the complex refractive index cross the negative real axis?

Circular convolution modulo $3$

Convolution theorem applied on integral function

How to recover differentiator approximation from RC circuit solution?

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