# Tagged Questions

Questions on the (continuous or discrete) convolution of two functions. It can also be used for questions about convolution of distributions (in the Schartz's sense) or measures.

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### Contracting a sum of convoluted delta functions using plus/minus symbol?

So I'm currently reading a journal article (I must say, I'm not too savvy with math notation). However, I'm coming across the following relationship: http://i.stack.imgur.com/omwcN.png The equation ...
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### Convolution can smooth an input function, is there an operation which bunches it up?

An easy to remember description of what the convolution of two functions is, is to say that one is a weight function and the result is a weighted average of the other function. The canonical example ...
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### Clarification on Wolfram Mathworld's explanation of the connection between Gelfand Transform and Fourier Transform

http://mathworld.wolfram.com/GelfandTransform.html In the definition, what does $x$, $\hat x(\phi)$, and $\phi$ represent exactly if we were to consider definition of the Fourier transform? Can ...
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### Compute the convolution of two compactly supported functions

I'm looking for a concrete example to understand the computation procedure for convolution: Let $f, g \in C_{0}^{\infty}(\mathbb R)$ be defined as follows: $$f(x) := e^{-\frac{1}{1-x^2}}1_{(-1,1)}$$ ...
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### Is the convolution pointwise bounded?

A problem from an old exam: Prove or disprove: if $p,q \in [1,\infty)$ such that $p^{-1}+q^{-1}=1$ and $f\in L^p, g\in L^q$, then the convolution $f*g$ is pointwise bounded. First of all: what ...
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### How to take derivatives of a convolution when the kernel's derivative is in the distribution sense?

I came need to take the derivative of the following convolution: $$\int_{-\infty}^\infty \operatorname{sgn}(x-y)e^{-|x-y|}f(y) \, dy$$ However, the derivative of the kernel only exists in the sense ...
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### On a problem of weak convergence for a particular convolution of pr. measures

Assume that $\{P_n: n \in N\}$ and $\{Q_n: n \in N\}$ are sequences of probability measures. Assume that $P_n \stackrel{w}{\to} P.$ Also, assume that $Q_n = \delta_{b_n},$ the Dirac measure and ...
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### Convolution of probabilities

It is a well known fact that for a random variable $Z=Y_1+Y_2+...+Y_n$ where $Y_i$ are independently distributed then the probability density function of $Z$ is the convolution of the density ...
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### Convolution of a gabor function and gaussian noise?

I am convolving the same image with a 2D Gabor over different gaussian noise masks that are generated in every trial. The convolution naturally takes time, is there any way to speed up the process by ...
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### What plays the role of the identity for the generalized convolution associated to the Fourier-Bessel transform?

In traditional Fourier theory, the Dirac delta plays the role of an "identity" for the $L^1$ algebra with respect to the usual convolution. The convolution is traditionally built out of group ...
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Given $p\in\mathbb{Q}[x]$, we define the weigth of $p$ as: $$W(p) = \#\{n\in\mathbb{N}: [x^n]\,p(x)\neq 0\}$$ i.e. as the number of non-zero terms. By playing a bit with the Taylor series of ...
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### Integrability of Maximal Convolution Operator

Let $f\in L^{\infty}(\mathbb{R}^{n})$ be supported in the unit ball and have mean zero. Let $\phi\in L^{1}(\mathbb{R}^{n})\cap C^{\alpha}(\mathbb{R}^{n})$ be a Holder continuous function with exponent ...
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### Reconciling two intuitions about convolution

There are two intuitive things convolution does. In the time domain, it represents the distribution of the sum of two independent random variables. In the frequency domain, it's just multiplication. ...
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### Translations AND dilations of infinite series

Sometimes, when working with infinite series, it's useful to add "dilated" or "translated" versions of the infinite series, term by term, back to the original. There are ways of making this rigorous ...
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### how to express separable 2D convolution using simple properties?

I have two 2D sequence $x(n_1,n_2)=f(n_1)g(n_2)$ and $y(n_1,n_2)$. How can I express this convolution x(n_1,n_2)*y(n_1,n_2)=(f(n_1)g(n_2))*y(n_1,n_2)=(f(n_1)*y(n_1,n_2))*(g(n_2)) ...
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### Convolution with one of the variables is mixed and the other continuous

Suppose $X$ and $Y$ are independent random variables with CDF $F$ and $G$ and nonnegative support. If $X$ has a point mass $p$ at $0$ and otherwise some "density" $f$ (that is, ...
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### Integration of convolution

I'm trying to solve the following equation $$\int\limits_{-\infty}^t \,(f\ast g)(t')dt'.$$ $f$ could be a kind of $\delta$-function: $f(t) = \delta(t)$ but should not be limited to be one. $g$ is ...
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### Simple case of Young's inequality

I have a question concerning Young's inequality stated as follows: $||a∗b||_{ℓ_q}≤||a||_{\ell_1}||b||_{ℓ_q},~~~~ 1≤q≤∞$. Here you can find something on $\ell_q\big(\mathbb{Z}\big)$: Young's ...
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### When is convolution associative?

Convolution is associative on e.g. integrable function on $\mathbb{R},$ but not on distributions. What about the convolution of measures on an unimodular group $G$?
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### A convolution equation with two unknowns

I consider the following convolution-type equation with two unknowns $f_1$, $f_2$: $$a_1 * f_1 + a_2 * f_2 = 0,$$ where $a_1$, $a_2 \in L^1(\mathbb R)$ and $*$ is the ordinary convolution. This ...
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### Reversing results for sums of independent variables

Please let me use a specific example to illustrate the general title above. (1) It is well known that if $X$ and $Y$ are independent and $X,Y\sim N(0,1)$ then $$Z\equiv X^2+Y^2\sim\chi_2^2$$ where ...
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### On the convolution of $f(x)=\sin x/x$ and $g(x)=1-|x|$

I am having trouble with computing the convolution of $f(x)=\sin x/x$ and: $$g(x)=\begin{cases} 1-|x|,& -1 \leq x \leq 1 \\ 0, & x \notin [-1,1] \end{cases}$$ I ...