# Questions tagged [convolution]

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### Young's convolution inequality for weighted norms

Young's convolution inequality states that, for $1/p+1/q=1/r+1$ ($1\leq p,\, q, r\leq \infty$), we have $$\lVert f * g \rVert_r \leq \lVert f\rVert_p \lVert g\rVert_q.$$ It is implicit here that the ...
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### Algebraic relation amongst an elliptic function and its convolution

NOTE: I edited this question, following the comments of Alexander Eremenko and Paul Garrett. I have a question concerning elliptic functions that maybe you can help me shed light on. I am a ...
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### Hopf "algebroid" structure of a groupoid convolution algebra?

This question is already posted in math.stackexchange, but didn't receive any answer. I'm not sure if this question fits in here, but surely someone in here can guide me to the correct answer. To make ...
108 views

### What is the computational complexity of the calculation of $\Psi(x)$?

What is the computational complexity of the calculation of $\Psi(x)$ described below: Let $\left\{ f_i : \{0,1,\dots,m\} \to \mathbb{R} \right\}_{i=1}^n$. For each $x \in \{0,1,\dots,m\}$ we ...
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### If $(\exp(\mu_n))_{n\in\mathbb N}$ is weakly convergent, is the normalized sequence convergent as well?

Let $E$ be a metric space and $\mathcal M(E)$ denoote the space of finite signed measures on $\mathcal B(E)$ equipped with the total variation norm $\left\|\;\cdot\;\right\|$. I would like to know ...
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### Which random variables can be written as the difference of two independent positive random variables?

Can we characterize random variables $X$ that satisfy $$X\sim Y - Z$$ for two independent positive random variables $Y$ and $Z$? Are $Y$ and $Z$ unique in some sense? Can (one possible choice of) $Y$...
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### Convolution, Fourier transforms, and area preservation [closed]

Consider the convolution of two functions, f * g. And let us assume, for practicality, some example case where an integral of f or g can be interpreted as the "area under the curve" (or the ...
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### Existence of unique convolution semigroups of probability measures on more general spaces then $\mathbb R^d$

Let $E$ be a $\mathbb R$-Banach space, $\mathcal M_1(E)$ (resp. $\mathcal M_1^\infty(E)$) denote the set of probability measures (resp. infinitely divisible probability measures) on $E$, $\varphi_\mu$ ...
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### Convolution of ball measures

It is well known that convolution of two ball measures (i.e. a uniform measure over a ball) in $\mathbb{R}^{n}$ is absolutely continuous with respect to the Lebesgue measure. My question is - how to ...
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### Does this formula correspond to a series representation of the Dirac delta function $\delta(x)$?

Consider the following formula which defines a piece-wise function which I believe corresponds to a series representation for the Dirac delta function $\delta(x)$. The parameter $f$ is the evaluation ...
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### Convergence of semi convex functions

Definition. Let $u:\Omega \rightarrow \mathbb{R}$. A function $u$ is called semiconvex if $u=v+w$ for some $v\in C^{1,1}(\Omega)$ and a convex function $w$. Note. Saying that $u$ is semiconvex is ...
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### The derivative of a filter with respect to a output singal [closed]

I have two signals, $d(t)$ and $p(t)$, respectively the input and the output of the matching filter $w(t)$, i.e. $$d(t)*w(t)=p(t)$$ where $*$ denotes convolution.The impulse response $w(t)$ may be ...
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### Wavelet momentum identity

I am reading an article on wavelet connection coefficients (G. Beylkin, "On the representation of operators in bases of compactly supported wavelets", 1992 (MSN)) and I came across Equation (3.31): \...
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### History- calculating convolution by tabular method

I often see a trick for calculating convolution of discrete data by a so-called Tabular method. There are a lot of Youtube videos and many Indian textbooks on Signal Processing [Books].1 Basically, ...
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### Convolve a 4D Gaussian function along a plane?

There is a 4D Gaussian function $G(u,s)=G(x|c,\mu,\Sigma )$ where $x=\begin{bmatrix}u\\ s\end{bmatrix}$,$u$ and $s$ is all 2D vector. Now I want to blur (convolve) it along with $u$ by another 2D ...
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### Derivative of a convolution integral of the following type?

I'm looking to find the derivative of a convolution integral of the following form: \frac{d}{dr}((G(r,t)*f(t)) = \frac{d}{dr} (\int_{-\infty}^{\infty} G(r,t-\tau)f(\tau) d\tau) \end{...
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### When does convolution erase non-monotonicities?

Suppose $\phi:\Bbb R\to[0,\beta]$ is a bounded continuous function such that $\phi(-\infty)=0$ and $\phi(\infty)=\beta$. Assume $\phi$ is non-decreasing except near zero, i.e. there exists $r>0$ ...
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### Bijections $\mathbb{Z}\times\mathbb{Z}\to\mathbb{Z}$ with vanishing local means

This is just a summer-time curiosity arisen after a recent question by Dominic van der Zypen. For a finite subset $S$ of $\mathbb{Z}\times\mathbb{Z}$ and a function $f$ on \$\mathbb{Z}\times\mathbb{...
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### Mathematical Techniques to Reduce the Width of a Gaussian Peak

In the chemical analysis by instruments, the signals of several molecules are overlapped which makes it difficult to determine the true area of each peak, such as those shown in red. I simulated this ...