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.
3,091 questions
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Is there a deconvolution of two Gaussians, which conserves area?
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 ...
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On the convolution identity of a sub arc of circle and the open set which is thickened epsilon amount of another subarc in circle.
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 ...
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A singular identity involving the arithmetic derivative and the divisor counting function
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 ...
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If all convolutions of a tempered distribution T with any compactly supported function are compactly supported, is T compactly supported?
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{...
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Is there a formula for calculating the convolution of a polynomial and an exponential? [closed]
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, ...
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Can Young's inequality give a pointwise bound for a convolution?
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 ...
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Closed form of $\operatorname{Ei}(-t) \theta(t) \star \operatorname{Ei}(t) \theta(-t)$
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:
$$\operatorname{Ei}(x) = \...
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Evaluation of a Broken-up Convolution Integral
I would like a comment on the correctness of my derivation below:
I have the following Duhamel Convolution integral
$$
U(t)=\int_0^t e^{-p(t-y)}f(y)\text{d}y
$$
For reasons that have to do with the ...
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Hecke algebras over fields other than $\mathbb{C}$
I am trying to prove some statements about specific Gelfand pairs, when considering representations of some finite groups over field $k$ which can be of positive characteristic. For this I study some ...
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Difficult Convolution Problem -- I Am Stuck with the Integration
I have the following function which is a probability density function on the $-1 < t < 1$ interval: $$f(t) = \frac{2\sqrt {1-t^2}}{\pi}$$
I wish to convolve this function with itself to get a ...
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On consequences of Titchmarsh theorem: can the analytical extension of the complex refractive index cross the negative real axis?
I will begin with the mathematical question at hand, and then describe technical details that were the background of the question, and then some possible approaches, although I clearly have not solved ...
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Circular convolution modulo $3$
I am working with a convolution sum of the form
$$ h(j) = \sum_{k=0}^2 f\!\big((j-k) \bmod 3\big)\, g(k), $$
where $f, g : \{0,1,2\} \to \mathbb{C}$. Because of the modulo $3$ structure in the index ...
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Convolution theorem applied on integral function
Consider $f\in L^2(\mathbb{T})$ where $\mathbb{T}=[0,2\pi]$ and further denote by $\mathcal{F}$ the Fourier transform. Define
\begin{align*}
g(t):=\int^t_0f(s)ds.
\end{align*}
Can we find a function $...
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How to recover differentiator approximation from RC circuit solution?
I'm reading The Art of Electronics, in 1.4.3, the section on passive differentiators. The math of the situation is given by
$$\frac{d}{dt}(V_{in}(t)-V_{out}(t))=\frac{1}{RC}V_{out}(t)\tag{1}$$
In the ...
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Pointwise Positivity of a convolution
I am trying to prove the point wise positivity of a convolution
$(r * u)(x,y) = \int r(x-\xi,y-\eta) \, u(\xi,\eta)\, d\xi d\eta,$
where $u(x,y)$ is pointwise positive (PP), and $r(x,y)$ is the ...
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