# Questions tagged [convolution]

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56 questions
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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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### 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 ...
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 ...