Computing convolutions of measures

What's new

If $latex {f: {\bf R}^n \rightarrow {\bf C}}&fg=000000$ and $latex {g: {\bf R}^n \rightarrow {\bf C}}&fg=000000$ are two absolutely integrable functions on a Euclidean space $latex {{\bf R}^n}&fg=000000$, then the convolution $latex {f*g: {\bf R}^n \rightarrow {\bf C}}&fg=000000$ of the two functions is defined by the formula

$latex \displaystyle f*g(x) := \int_{{\bf R}^n} f(y) g(x-y)\ dy = \int_{{\bf R}^n} f(x-z) g(z)\ dz.&fg=000000$

A simple application of the Fubini-Tonelli theorem shows that the convolution $latex {f*g}&fg=000000$ is well-defined almost everywhere, and yields another absolutely integrable function. In the case that $latex {f=1_F}&fg=000000$, $latex {g=1_G}&fg=000000$ are indicator functions, the convolution simplifies to

$latex \displaystyle 1_F*1_G(x) = m( F \cap (x-G) ) = m( (x-F) \cap G ) \ \ \ \ \ (1)&fg=000000$

where $latex {m}&fg=000000$ denotes Lebesgue measure. One can also define convolution on more general locally compact groups than $latex {{\bf R}^n}&fg=000000$, but we will restrict attention to the Euclidean…

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An improved Type I estimate

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As in all previous posts in this series, we adopt the following asymptotic notation: $latex {x}&fg=000000$ is a parameter going off to infinity, and all quantities may depend on $latex {x}&fg=000000$ unless explicitly declared to be “fixed”. The asymptotic notation $latex {O(), o(), \ll}&fg=000000$ is then defined relative to this parameter. A quantity $latex {q}&fg=000000$ is said to be of polynomial size if one has $latex {q = O(x^{O(1)})}&fg=000000$, and bounded if $latex {q=O(1)}&fg=000000$. We also write $latex {X \lessapprox Y}&fg=000000$ for $latex {X \ll x^{o(1)} Y}&fg=000000$, and $latex {X \sim Y}&fg=000000$ for $latex {X \ll Y \ll X}&fg=000000$.

The purpose of this (rather technical) post is both to roll over the polymath8 research thread from this previous post, and also to record the details of the latest improvement to the Type I estimates (based on exploiting additional averaging and using Deligne’s proof of the Weil conjectures) which lead…

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Spacetime, Relativity, and Quantum Physics

Spacetime, Relativity, and Quantum Physics

http://www.ws5.com/spacetime/

Spaun brain simulation

Spaun brain simulation

http://www.nature.com/news/simulated-brain-scores-top-test-marks-1.11914

AI research at Google

AI research at Google

http://www.artificialbrains.com/google

Tetrapod Quantum Dots Light the Way to Stronger Polymers

Tetrapod Quantum Dots Light the Way to Stronger Polymers

http://newscenter.lbl.gov/feature-stories/2013/07/29/tetrapod-quantum-dots-light-the-way-to-stronger-polymers/

 

The Riemann hypothesis in various settings

What's new

[Note: the content of this post is standard number theoretic material that can be found in many textbooks (I am relying principally here on Iwaniec and Kowalski); I am not claiming any new progress on any version of the Riemann hypothesis here, but am simply arranging existing facts together.]

The Riemann hypothesis is arguably the most important and famous unsolved problem in number theory. It is usually phrased in terms of the Riemann zeta function $latex {\zeta}&fg=000000$, defined by

$latex \displaystyle \zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}&fg=000000$

for $latex {\hbox{Re}(s)>1}&fg=000000$ and extended meromorphically to other values of $latex {s}&fg=000000$, and asserts that the only zeroes of $latex {\zeta}&fg=000000$ in the critical strip $latex {\{ s: 0 \leq \hbox{Re}(s) \leq 1 \}}&fg=000000$ lie on the critical line $latex {\{ s: \hbox{Re}(s)=\frac{1}{2} \}}&fg=000000$.

One of the main reasons that the Riemann hypothesis is so important to number theory is that the zeroes of…

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