Questions tagged [convolution]

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51 views

partial differential inequality [duplicate]

I want to prove that if $f\leq l$ and $\lim_{a\to \infty}\frac{f(a)}{l(a)}=1$ (we can also suppose that $\lim_{a\to \infty}f(a)=\lim_{a\to \infty}l(a)=1$) where $f$ is a positive continuous bounded ...
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1answer
152 views

$\frac{\partial}{\partial x}\int_{\mathbb{R}}\frac{1}{\sqrt{2 \pi \varepsilon}}e^{-\frac{(x-y)^2}{2\varepsilon}}l(y)dy\leq C\frac{1}{x}$

Let $l$ be a continuous bounded function ($l$ is not differentiable). I want to prove for $x$ large enough that $$\frac{\partial}{\partial x}\int_{\mathbb{R}}\frac{1}{\sqrt{2 \pi \varepsilon}}e^{-\...
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0answers
21 views

Unable to do question 3 in 7.3 from Folland's Fourier Analysis and its Application [migrated]

I'm unable to answer this question, where we were given $f(x)$: $$f(x)=\begin{cases} 1, & \text{if }-1<x<1 \\ 0, & \text{otherwise}\end{cases}$$ The questions asks me to compute $f*f$ ...
3
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1answer
159 views

If the convolution of two functions $f\star g$ is equal to $g$, $f$ is even with compact support and $g$ is bounded, implies that $g$ is constant?

Let $f$ be an even continuous function with compact support such that $$ \int f(t)\,\mathrm{d}t=1, $$ and let $g$ be a bounded continuous function such that the convolution $f\star g$ satisfies the ...
3
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1answer
131 views

Prove or disprove the linearity of expectiles

For expectation (mean), there are many useful properties such as Linearity of Expectation: $\mathbb{E}[X+Y]=\mathbb{E}[X]+\mathbb{E}[Y]$ $\mathbb{E}[\alpha X]=\alpha\mathbb{E}[X]$ (The 2 equations ...
4
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2answers
380 views

Earliest use of deconvolution by Fourier transforms

From a previous discussion here Origin of the convolution theorem, it was shown that the property of convolution $y(t)$=$a$*$b$ becoming a multiplication after Fourier transform: $F$$(y(t))$= $F(a)F(b)...
2
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1answer
68 views

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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0answers
29 views

Deconvolution and mean-preserving spreads

Context I have been working on proving the existence of a mathematical object. After trying several things, I think that if I can show the following, an important step towards proving existence will ...
1
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1answer
145 views

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 ...
1
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1answer
119 views

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): \...
2
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1answer
101 views

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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0answers
20 views

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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39 views

Derivative of a convolution integral of the following type?

I'm looking to find the derivative of a convolution integral of the following form: \begin{equation} \frac{d}{dr}((G(r,t)*f(t)) = \frac{d}{dr} (\int_{-\infty}^{\infty} G(r,t-\tau)f(\tau) d\tau) \end{...
3
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3answers
198 views

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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71 views

Is it possible to define a product of two divergent integrals that would have the following properties?

Here I introduce an algebra of divergent integrals and series. But the theory is currently lacking an important element: there is no algorithm of construction of a divergent integral that would be ...
2
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1answer
194 views

Is there (fast) fourier transform for vector convolution?

Given a list of variables $u_1,\dots,u_m\in\mathbb R$ and $v_1,\dots,v_n\in\mathbb R$ the standard convolution is defined $$U*V(t)={\sum_{i}} u_iv_{t-i}.$$ Given a list of vectors $u_1,\dots,u_m\in\...
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1answer
322 views

Convolution in K-Theory via an Example (From StackExchange)

I've spent lots of time in Chriss and Ginzburg's "Complex Geometry and Representation Theory" and despite convolution (in Borel-Moore homology or K-theory) being very central, I feel like I'm still ...
2
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1answer
385 views

Norm of convolution operator

By Young's inequality for any $f\in L^p(\mathbf{R})$ the map $T_f:g\mapsto f\star g$ is a continuous operator from $L^q(\mathbf{R})$ to $L^r(\mathbf{R})$ where $1\leq p,q,r\leq \infty$ satisfy $1+\...
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0answers
129 views

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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3answers
363 views

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 ...
3
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1answer
223 views

Exponential deconvolution using the first derivative

There is an interesting observation using the first derivative to deconvolve an exponentially modified Gaussian: The animation is here at terpconnect.umd.edu. The main idea is that if we have an ...
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1answer
540 views

Origin of the convolution theorem

I am a chemist, with some interest in signal processing. Sometimes, we use the deconvolution process to remove the instruments response from the desired signals. I am looking for the earliest ...
10
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1answer
549 views

Gelfand's trick (Gelfand's lemma) in positive characteristic?

I came across this preprint that claims in Lemma 1.1 that Gelfand's trick (also known as Gelfand's lemma) only works in characteristic zero: Let $H < G$ be finite groups. Suppose we have an anti-...
2
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1answer
79 views

De-Convolution of Distributions

Under what conditions a continuous unimodal distribution G(x) can be represented as a convolution of N of the same F(x) distributions? I.e. G(x)= F(x) * F(x) * F(x) *...... Also does F(x) also ...
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1answer
141 views

Is this operator invertible?

Let $T(t)$ be a strongly continuous semi-group on a Banach space $X$, and let $A(\cdot)\in C(0,\tau; \mathcal{L}(X))$ for some $\tau>0$. The operator $G:C(0,\tau;X)\to C(0,\tau;X)$ maps every $h\in ...
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2answers
147 views

L1 distance after Convolution

Given two discrete distributions $P$ and $Q$ with the same support $x_1,\cdots,x_n$. Assume $K \in L^1(\mathbb{R})$ is a nonnegative function with $\int_\mathbb{R} K(x)dx = 1$, and let $K_h(x) = \frac{...
2
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1answer
95 views

Product of independent random variables and tail deconvolution

Suppose $X, Y$ are two independent non-negative random variables. The conditions (i) $\mathbb{P}(X > t) = \frac{C}{t^p} + o(t^{-p})$ (ii) $\mathbb{P}(Y > t) = o(t^{-q})$ for any $q > ...
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0answers
37 views

Show a convolution of distributions ε-close to min-entropy k is ε^t-close to min-entropy k

Assume $X_1,...,X_t$ are independent distributions on $\mathbb{Z}_2^n$ s.t. each $X_i$ is $\epsilon$-close to min-entropy $k$; i.e. there exist distributions $Y_1,...,Y_t$ on $\mathbb{Z}_2^n$ s.t: $$ \...
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2answers
2k views

When can a function be made positive by averaging?

Let $f: {\bf Z} \to {\bf R}$ be a finitely supported function on the integers ${\bf Z}$. I am interested in knowing when there exists a finitely supported non-negative function $g: {\bf Z} \to [0,+\...
6
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1answer
236 views

Dualizable presheaves with respect to Day convolution

This question was posted on MSE and got very little attention, so I'm also posting it here. Let $\mathcal{C}$ be a closed symmetric monoidal category and let $PSh(\mathcal{C}):=Fun(\mathcal{C}^{op}, \...
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1answer
121 views

A question about the convolution theorem

I have the following "argument" about Fourier series, which I know is wrong because it yields a ridiculous conclusion. However, I don't know where the mistake is, and need to know which step is the ...
12
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1answer
483 views

Fast convolution of sparse functions

Let $F:\mathbb{R}\to \mathbb{Z}$ be a step function with at most $k$ discontinuities, at given rationals $a_1<a_2<\dotsc<a_k$. Let $g:\mathbb{R}\to \mathbb{Z}$ be given as a linear ...
3
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1answer
75 views

Level sums, displacements: how to determine them efficiently?

Let $R =\mathbb{Z}/N \mathbb{Z}$. Let $f:R\to \mathbb{R}$, $\rho:R\to \lbrack 0,1\rbrack$. We assume that it takes trivial time to compute any given value $f(m)$ or $\rho(m)$. Define $$S(\delta,m) = ...
1
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1answer
187 views

Variance of convolution between filter $A$ and Ornstein-Uhlenbeck process $x_t$

If we consider $x_t$ an Ornstein-Uhlenbeck process (with $W_t$ the Wiener process), does anyone know what would be the variance of the convolution of $x_t$ with a given filter $A$ i.e. $V(x_t \star A)$...
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0answers
216 views

Positive Convolution Root

I try to compute the convolution root of a symmetric, positive definite, nonnegative, one dimensional function $f: \mathbb R\to \mathbb R^+_0$. Furthermore I assume $f$ is bounded and $\int_{\mathbb R}...
3
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2answers
199 views

The disintegration of the convolution of two probability measures

Let $G$ be a topological group with all the topological conditions in order that some form of the disintegration theorem be applicable (for instance, take $G$ metrizable). Let $N$ be normal and closed,...
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1answer
226 views

How to prove the combinatorial equality? [closed]

Please, help me to understand following convolution (or give a reference): $$ \sum_{R=0}^N \binom{R}{r} \binom{N-R}{n-r} = \binom{N+1}{n+1} $$ Why is it true? Thank you!
3
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1answer
122 views

When are all the convolution roots of an infinitely divisible probability measure infinitely divisible?

Let $G$ be a topological group. Let us say that a probability measure $\mu$ on $G$ is strongly infinitely divisible (SID) if $\mu$ is infinitely divisible and any probability measure $\nu$ on $G$ ...
2
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0answers
68 views

Are the roots of an infinitely divisible probability infinitely divisible themselves?

Let $\mu$ be an infinitely divisible probability on a topological group $G$. If $\nu ^{* n} = \mu$ for some $n$, is $\nu$ an infinitely divisible probability too? A sufficient criterion would be to ...
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0answers
192 views

When convolution with exponential kernel is bounded

Let $g(t)=e^{-\omega t}$, $\omega>0$. What is, in terms of well-known function spaces, the space $X$, $L_{loc}^2(0,\infty)\subset X$, of all functions $f:\mathbb{R}^+\to \mathbb{R}^+$, satisfying $...
3
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0answers
139 views

Is there a closed form for the discrete convolution of $\sigma_1$ and $\sigma_2$?

I am trying to find a closed form for the following sum: $$\sum_{k = 1}^{n-1}\sigma_1(k) \sigma_2(n-k)$$ where $\sigma_i$ is the sum of divisors and $\sigma_2$ is the sum of squares of divisors. ...
4
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1answer
502 views

Existence of solutions to first-order PDE involving convolution

Let $f(x,\alpha)$ be a smooth function of compact support in $x$. Now, let its $\alpha$-dependence be determined by the following first-order equation, \begin{align} \frac{\partial}{\partial \alpha} ...
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2answers
287 views

What is $\int_{0}^{z} e^{-a^{2} x^{2}} {\rm erf}(bx)\, dx$?

The integral $$\int_{0}^{z} e^{-a^{2} x^{2}} {\rm erf}(bx)\, dx$$ is related to the convolution of two half-normal distributions. This can be inferred from this question on MSE. The following ...
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0answers
131 views

Convolution of $\ell$-adic sheaves and group homomorphisms

This question follows this one , where I defined convolution of $\ell$-adic/perverse sheaves. Here I am working with a perfect field $k$ ($char(k)\neq l$) and with a smooth separated groupscheme $G$ ...
3
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1answer
158 views

Convolution of $\ell$-adic sheaves is commutative if the group is commutative

[This is a duplicate of this question on Stackexchange] I am trying to figure out how to prove a very basic statement about convolution of $\ell$-adic/perverse sheaves in Katz's "Rigid local systems" ...
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0answers
124 views

Convolution integral of series involving the non-trivial zeros of $\zeta(s)$

Let us consider the convolution $$f\left(x\right):=\int_{2}^{x-2}\sum_{\rho_{1}}\frac{u^{\rho_{1}}}{\rho_{1}}\sum_{\rho_{2}}\frac{\left(x-u\right)^{\rho_{2}}}{\rho_{2}}du,\,x>4$$ where $\rho_{i},\,...
3
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0answers
113 views

Any chance to get the moments of this exotic distribution?

Let us define the following cumulative distribution: \begin{align} \Pr (Y(t)\geq a)=\sum_{n=0}^\infty \frac{e^{-kt}(kt)^n }{n!}\int_a^\infty[\circledast_{f}\circ H_a ]^n\delta (x) dx \end{align} where ...
1
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0answers
101 views

Existence theorems Volterra Equation of second kind on unbounded domains

The general Volterra Equation of the second kind in convolution form can be described by: $$ \phi(x) = \int_a^x K(x-t)\phi(t)\, \mathrm{d}t + f(x), \text{ for } x\geq a $$ Suppose we wish to ...
3
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0answers
73 views

Computing distribution of non-identical coin flips

Suppose I have $N$ coins, where coin $i$ has probability $p_i$ of coming up heads. I flip all $N$ coins and let $S_N$ be the number of heads. How can I compute the distribution of $S_N$ efficiently? ...
1
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0answers
49 views

Classifying Algebras of Convolution

$L^1(\mathbb R^n)$, $L^1(\mathbb R_+)$, $C^0_c(\mathbb R_+)$, $C^\infty_c(\mathbb R_+)$ are algebras of convolution. Question 1: is there a classification of subalgebras of convolution of $L^1(\...