At this point I really hate to ask. Do you know what "orthogonal subspace" means and what a projection is?

I think you are very confused. My post is about approximation. Obviously other areas use other polynomials or the same polynomials for other reasons.

In this case it is about the principle of approximation by orthogonal projection, which is quite common in different fields of mathematics. Here you create an approximation of a target by projecting it onto an orthogonal subspace. This is what the Fourier series is about, an orthogonal projection. Choosing e.g. the Chebychev Polynomials instead of the complex exponential gives you an Approximation onto the orthogonal space of e.g. Chebychev polynomials.

The same principle applies e.g. when you are computing an SVD for a low rank approximation. That is another case of orthogonal projection.

>Instead, you would normally use Chebyshev interpolation

What you do not understand is that this is the same thing. The distinction you describe does not exist, these are the same things, just different perspectives. That they are the same easily follows from the uniqueness of polynomials, which are fully determined by their interpolation points. These aren't distinct ideas, there is a greater principle behind them and that you are using some other algorithm to compute the Approximation does not matter at all.

>I am also not totally sure what polynomial chaos has to do with any of this.

It is the exact same thing. Projection onto an orthogonal subspace of polynomials. Just that you choose the polynomials with regard to a random variable. So you get an approximation with good statistical properties.

Do you have a shred of evidence? How does a company function with managers only there 50% if the time.

Yes. Precisely that they are orthogonal means that they are suitable.

If you are familiar with the Fourier series, the same principle can be applied to approximating with polynomials.

In both cases the crucial point is that you can form an orthogonal subspace, onto which you can project the function to be approximated.

For polynomials it is this: https://en.m.wikipedia.org/wiki/Polynomial_chaos

There is a relationship here, in the case of Gauß-Hermite Integration, where the weight function is exactly e^(-x^2) the weights have to add up sqrt(pi), because the integral is exact for the constant 1 polynomial.