Comment by creata

True, and there are plenty of other reasons Chebyshev polynomials are convenient, too.

But I guess what I was asking was: is there some kind of abstract argument why the semicircle distribution would be appropriate in this context?

For example, you have abstract arguments like the central limit theorem that explain (in some loose sense) why the normal distribution is everywhere.

I guess the semicircle might more-or-less be the only way to get something where interpolation uses the DFT (by projecting points evenly spaced on the complex unit circle onto [-1, 1]), but I dunno, that motivation feels too many steps removed.