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.
Some sort of numerical analysis book that covers these topics - minimax approx, quadrature etc. I’ve read on these separately but am curious what other sorts of things would be covered in courses including that.
I would check out "An Introduction to Numerical Analysis" by Suli and Mayers or "Approximation Theory and Approximation Practice" by Trefethen. The former covers all the major intro numerical analysis topics in a format that is suitable for someone with ~undergrad math or engineering backgrounds. The latter goes deep into Chebyshev approximation (and some related topics). It is also very accessible but is much more specialized.
I'd suggest: Trefethen, Lloyd N., Approximation theory and approximation practice (Extended edition), SIAM, Philadelphia, PA (2020), ISBN 978-1-611975-93-2.
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.