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Questions tagged [convolution]

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1
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1answer
75 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
29 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: $$ \...
35
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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,+\...
5
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1answer
194 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
100 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 ...
9
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1answer
345 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
74 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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0answers
18 views

Variance of filtered OU process

I derived the variance of an Ornstein-Uhlenbeck process filtered with a function A (I never came cross such derivation). My problem is that when I define A, I can't get an acceptable answer. I wonder ...
1
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1answer
109 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
108 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}...
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0answers
173 views

Does $\int_0^\infty f(x+\theta)g(x) \, dx=0\, \forall \theta \in \mathbb{R}$ imply $f=0$ almost everywhere, if $g$ is smooth and strictly positive?

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be an integrable function, and $g:(0,\infty)\rightarrow(0,\infty)$ a smooth, strictly positive function. If $$\int_0^\infty f(x+\theta)g(x)\,dx=0\qquad\forall\...
3
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2answers
173 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
172 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
103 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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0answers
64 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
140 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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0answers
109 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. ...
3
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1answer
256 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} ...
6
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2answers
247 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
118 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
137 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
119 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},\,...
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0answers
105 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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0answers
95 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
70 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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0answers
43 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(\...
3
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1answer
269 views

Norm inequality for convolution operators on groups

Let $G$ be a discrete, finitely generated group. Let $f\in \mathbb{C} G$ be given. Consider $g\in G\setminus \operatorname{supp} f$ and let $\delta_g$ denote the Dirac delta at $g$. Is it true ...
4
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4answers
406 views

Self convolutions of singular continuous measure

Let $\mu$ be a finite measure on $\mathbb{R}$. Define the measures $(\mu_n)_{n\geq 1}$ by $\mu_{n+1}=\mu\ast \mu_n$ and $\mu_1=\mu$ Is there a singular (with respect to the Lebesgue measure) ...
3
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0answers
52 views

Is there a name for the general type of operation that sweeps a kernel over a function (e.g. like convolution, morph. dilation, registration, etc)

There is a certain family of 'sweeping' operators / functions $S(y; f,k,g)$, where: $f$ is a function $f : x \mapsto \mathbb{R}^N$ $k$ is a 'kernel' function $k : x \mapsto \mathbb{R}^N$ $y$ ...
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0answers
35 views

Envelope of a parametrized family of convolutions

For a certain application I need to compute a pointwise supremum of this family of gaussian convolutions: $$\sup_s f(x)\otimes e^{-\frac{x^2}{s^2}}$$ where $f(x),x\in \mathbb{R}^2$ is known and $\...
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0answers
71 views

Convolution in Hardy spaces

Question Are there non-trivial restrictions on the coefficients of functions in Hardy spaces ($H_p(\mathbb{D})$, $p<1$) that make a subspace that is closed under convolution? Definition The Hardy ...
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0answers
181 views

Some detail in Fefferman's thesis

Recently, I am reading "Inequalities for strongly singular convolution operator" written by Fefferman. I have some question on the detail of proof of Theorem 2'. Let $\theta \in (0,1)$. Let $f \in ...
2
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1answer
323 views

Relation between Cox-deBoor recursion and Convolution (b-spline basis)

Consider the Cox-deBoor recursion formula for producing b-spline basis functions given a knot vector: $N_{i,0}(u)=1 $ if $u_i\leq u < u_{i+1}$ otherwise, $=0$ $N_{i,p}(u)=\frac{u-u_{i}}{u_{...
2
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2answers
230 views

Approximate identities and pointwise convergence

I'm studying Fourier analysis and have a question about approximate identities. Let $k_{\epsilon}$ be an approximate identity on $L^{1}(\mathbf{T})$. We know that $k_{\epsilon}*f\to f$ in $L^{1}$ as $...
2
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0answers
177 views

Link between standard convolution and Day convolution

There is a notion of convolution product between two functors called "Day convolution". (See here nlab for instance) I know that the definition of this notion is inspired by the discrete convolution $$...
52
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1answer
3k views

Square root of dirac delta function

Is there a measurable function $ f:\mathbb{R}\to \mathbb{R}^+ $ so that $ f*f(x)=1 $ for all $ x\in \mathbb{R} $, i.e $$\int\limits_{-\infty}^{\infty} f(t)f(x-t) dt=1 $$ for all $ x\in \mathbb{R} $.
1
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1answer
67 views

Subquadratic multiplication of probability mass functions (with log-convolution?)

We are currently looking for a fast, i.e. subquadratic, algorithm for the following equation: $z_m = \sum_{i,j :\, (i \cdot j) = m} x_i \cdot y_j$. That is, we are given two finite input vectors $x$ ...
6
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1answer
442 views

Is there an alternate name for the symplectic convolution?

Looking into the Wigner-Weyl transformation mapping Hilbert space operators to functions on phase-space, I've run up against the need for a symplectic convolution $$[F\star G](x,p) = \int \!dy\,dk\, ...
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2answers
452 views

Are there multiplicative functions which are not rational?

Vaidyanathaswamy calls an arithmetic function rational if it is the convolution of some finite collection of functions which are either completely multiplicative or inverse to a completely ...
4
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0answers
126 views

Shifted convolution problem for Coefficients of automorphic forms

The shifted convolution problem for coefficients of modular forms is well studied and many estimates were established for the shifted convolution sums of Hecke eigenvalues. So, one may ask about the ...
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2answers
422 views

Is there a Gelfand-Naimark-like characterization of group algebras $L_1(G)$?

The Gelfand-Naimark theorem establishes that a complex commutative Banach algebra $A$ with an identity and an involution $x\to x^*$ satisfying $\|x x^*\|=\|x\|^2$ is (isometrically isomorphic to) a $C(...
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2answers
205 views

Is there a version of the Titchmarsh Convolution theorem to find singular support?

Okay, some terminology, correct me if I'm wrong. Singular support - the set on which a distribution fails to be smooth. In this case a piecewise function. Is there a name for $f*f*f$? The ...
2
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1answer
195 views

Convolution vanishes on an interval

Fix a "test" function $f(x)=x\exp(-x^2)$, which is nonzero except $x=0$. Suppose that $g$ is a function with some necessary regularity. Consider the convolution. $$ (f\ast g )(x)=\int_{-\infty}^{+\...
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0answers
186 views

Which classes of functions are “convolution ideals”?

If $g$ is continuous then $f*g$ is continuous. If $g$ is smooth then $f*g$ is smooth. If $g$ is a polynomial then $f*g$ is a polynomial. If just one of the two functions belongs to the class of well-...
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1answer
592 views

Is the set of the convolutions of two-point measures dense in the set of all measures?

A measure supported in two points is a measure of the form $$ \mu=\alpha\delta_a+(1-\alpha)\delta_b, $$ where $a<b$ and $\alpha\in (0,1)$. The question is: Given a finite non-negative measure ...
3
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1answer
433 views

Convolution of measures - entropy growth

Imagine you have two shift-invariant measures $\mu, \nu$ in the Bernoulli space $\{0,1\}^{\mathbb{N}}$ with positive entropy and both are not the Bernoulli measure $(\frac{1}{2},\frac{1}{2})$. I know ...
2
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0answers
893 views

convolution integral involving modified Bessel functions of the first kind

I'm stuck with this convolution integral ($z \geq 0$)... \begin{equation} f_{Z}(z)=\int^{\infty}_{-\infty}f_{1}(x)f_{2}(z-x)dx = \mbox{ } ??? \end{equation} which represents the pdf of the sum $Z = ...
3
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1answer
254 views

Complete solution set of a Convolutional Equation?

Here is a problem that am I stuck and I appreciate any help. In essence, I am trying to show that the only solutions for the described problem are the ones provided below. Best.. Setup: In what ...
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0answers
92 views

Do they have the same limit?

Suppose $a(\cdot)\in L^p$ and is symmetric and $b(\cdot)\in L^q$, where $1/p+2/q=2$, $p,q\ge 1$. Consider the quantity $Q_T=$ $$ \frac{1}{T}\int_{\mathbb{R}}dx\int_{[-T,T]^2}d\mathbf{v}\int_{[-T,T]^2}...
3
votes
1answer
530 views

Maximum of a mollified/convolution function

I have a function $f:{\mathbb R}\rightarrow {\mathbb R}_+$ which has a unique maximum at $x=0$. $f$ can be symmetric or asymmetric. I am interested on the mollified-f function $$\tilde{f}(x)=\int_{-\...