# Questions tagged [convolution]

The convolution tag has no usage guidance.

80
questions

**-4**

votes

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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 ...

**0**

votes

**1**answer

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^{-\...

**-1**

votes

**0**answers

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**

votes

**1**answer

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**

votes

**1**answer

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**

votes

**2**answers

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**

votes

**1**answer

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 ...

**0**

votes

**0**answers

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**

vote

**1**answer

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**

vote

**1**answer

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**

votes

**1**answer

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, ...

**1**

vote

**0**answers

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 ...

**0**

votes

**0**answers

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**

votes

**3**answers

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 $r>0$ ...

**0**

votes

**0**answers

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**

votes

**1**answer

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\...

**8**

votes

**1**answer

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**

votes

**1**answer

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+\...

**10**

votes

**0**answers

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{...

**5**

votes

**3**answers

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**

votes

**1**answer

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 ...

**7**

votes

**1**answer

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**

votes

**1**answer

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**

votes

**1**answer

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 ...

**1**

vote

**1**answer

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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vote

**2**answers

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**

votes

**1**answer

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 > ...

**1**

vote

**0**answers

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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votes

**2**answers

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**

votes

**1**answer

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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votes

**1**answer

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**

votes

**1**answer

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**

votes

**1**answer

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) = ...

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vote

**1**answer

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. $V(x_t \star A)$...

**0**

votes

**0**answers

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**

votes

**2**answers

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,...

**-3**

votes

**1**answer

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**

votes

**1**answer

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 $G$ ...

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votes

**0**answers

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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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
$...

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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**

votes

**1**answer

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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votes

**2**answers

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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vote

**0**answers

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**

votes

**1**answer

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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vote

**0**answers

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**

votes

**0**answers

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 ...

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vote

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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**

votes

**0**answers

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?
...

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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(\...