# Questions tagged [convolution]

The convolution tag has no usage guidance.

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### 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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### 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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### 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,+\...

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### 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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### 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 ...

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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 ...

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### 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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### 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 ...

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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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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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### 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\...

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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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### 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!

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### 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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### 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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### 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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### 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.
...

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### 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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### 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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### 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 $G$ ...

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### 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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### 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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### 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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### 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 ...

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### 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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### 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(\...

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### 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 ...

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### 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) ...

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### 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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### 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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### 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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### 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 ...

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### 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_{...

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### 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 $...

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### 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 $$...

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### 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} $.

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### 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 $x$ ...

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### 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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### 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 ...

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### 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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### 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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### 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 ...

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### 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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### 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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### 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 ...

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### 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 ...

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### 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 = ...

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### 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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### 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}...

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### 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_{-\...