# Mathematics Colloquia and Seminars

### Interpolation of data by nonnegative $$C^2$$ functions

PDE and Applied Math Seminar

 Speaker: Black Jiang, UC Davis Location: 2112 MSB Start time: Fri, Feb 8 2019, 4:10PM

In this talk, we consider the following extension problem of Whitney-type: given a finite set on the plane with prescribed nonnegative values, how do we decide the minimal $$C^2$$ norm of all the globally nonnegative functions that interpolate the data?

There is a twofold solution to the problem, namely, an algorithm which allows us to compute the minimal norm, and a bounded extension operator which recovers an interpolant witnessing the minimal norm.

The idea of the proof relies on the observation that any finite set on the plane lies on a curve on a sufficiently small scale, so the local problem is essentially one-dimensional and readily solvable. We will look at some key objects that will allow us to control these scales so that the local solutions are compatible with each other.

This is joint work with Kevin Luli.