# What is the "smoothest" average?

# What is the "smoothest" average?

In the real world, it is often desirable to obtain “smoothed” local averages of a function $f: \mathbb{Z} \to \mathbb{R}$ (for instance, new covid cases per day). But there are many possible ways to average, and it is not clear which one is best! Inspired by the axiomatic approach in game theory, we propose several criteria for “good” averaging schemes; these make it possible to show that some averages are better than others. In particular, we consider averaging $f$ by convolving it with a kernel on a fixed scale, and we measure the “smoothness” of the result by the $\ell^2$-norm of either the first or second discrete derivative. The determination of the optimal kernels involves the extremizing properties of Chebyshev polynomials and is related to new uncertainty principles for the Fourier transform. We also address a continuous analog of this problem where we work over $\mathbb{R}$ instead of $\mathbb{Z}$.

This talk is based on joint work with Stefan Steinerberger.