I'm interested in uniform approximation with generalized polynomials -- these are linear combinations from families of parametrized continuous functions over some domain that satisfy some technical conditions, but its also fine to think of them as sums of regular monomials like 1, x, x^2, ..., x^N. This problem has been well understood for real intervals (classical case) for a long time, but I'm interested in this problem where we're approximating functions over complex domains.

There is a theoretically stable algorithm for the classical problem called the Remez exchange algorithm, and an extension to complex domains due to P.T.P. Tang in his 1987 PhD thesis at Berkeley. Theoretically Remez and its complex extension are very stable, but unfortunately implementations my advisor and I are aware of seem to struggle with large degree polynomials, where large is bigger than say n=45 -- errors begin to explode.

In any case, independently of this I've been learning more of the nitty gritty details of deep learning for a project at work (I'm a SWE in my day job, the math is more moonlighting), so to ground my efforts there I've been exploring deep learning approaches to this problem of complex uniform approximation, implementing results from various papers and tweaking things for my use case, and coming up with questions. That's much of what I'm thinking about this week!

Also, I'll be having a half-day long ADHD evaluation session on Friday -- so a bit apprehensive about that.