This is joint work with Paul Carter.We completed this while on a cross-country drive as we moved the newly minted Dr. Carter from Brown to Arizona.
I've had a longtime fascination with the standard mean value theorem of calculus.
Mean Value Theorem Supposeis a differentiable function. Then there is some such that
The idea for this project started with a simple question: what happens when we interpret the mean value theorem as a differential equation and try to solve it? As stated, this is too broad. To narrow it down, we might specify some restriction on the
So I thought to try to find functions satisfying
This looks like a differential equation, which I only know some things about. But my friend and colleague Paul Carter knows a lot about them, so I thought it would be fun to ask him about it.
He very quickly told me that it's essentially impossible to solve this from the perspective of differential equations. But like a proper mathematician with applied math leanings, he thought we should explore some potential solutions in terms of their Taylor expansions. Proceeding naively in this way very quickly leads to the answer that those (assumed smooth) solutions are precisely quadratic polynomials.
It turns out that was too simple. It was later pointed out to us that verifying that quadratic polynomials satisfy the midpoint mean value property is a common exercise in calculus textbooks, including the one we use to teach from at Brown. Digging around a bit reveals that this was even known (in geometric terms) to Archimedes.
So I thought we might try to go one step higher, and see what's up with
That's a bit odd. It turns out that the midpoint itself is distinguished in this way. Why might that be the case?
It is beneficial to look at the mean value property as an integral property instead of a differential property,
An attentive eye might notice that the midpoint mean value theorem, written as the integral property
From this viewpoint, functions satisfying our original midpoint mean value property~
The weighted mean value property can also be written as an integral property. Trying to connect it similarly to harmonic functions led us to consider functions satisfying
Are there weighted harmonic functions corresponding to a weighted harmonic mean value property?
In one dimension, the answer is no, as seen above. But there are many more multivariable harmonic functions [in fact, I've never thought of harmonic functions on
This ends up being the focus of the latter half of our paper. Unexpectedly (to us), an analogous methodology to our approach in the one-dimensional case works, with only a few differences.
It turns out that no, there are no weighted harmonic functions on
Harmonic functions are very special, and even more special than we had thought. The paper is a fun read, and can be found on the arxiv now. It has been accepted and will appear in American Mathematical Monthly.
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