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0 pointssfpotter1y ago0 comments

Chebyshev polynomials are useful in approximation theory because they're the minimax polynomials. The remainder of polynomial interpolation can be given in terms of the nodal polynomial, which is the polynomial with the interpolation nodes as zeros. Minimizing the maximum error then leads to the Chebyshev polynomials. This is a basic fact in numerical analysis that has tons of derivations online and in books.

The weight function shows the Chebyshev polynomials' relation to the Fourier series . But they are not what you would usually think of as a good candidate for L2 approximation on the interval. Normally you'd use Legendre polynomials, since they have w = 1, but they are a much less convenient basis than Chebyshev for numerics.

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3 comments · 1 top-level

creata1y ago· 2 in thread

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.

sfpotterOP1y ago

If there is, I'm not aware of it. Orthogonal polynomials come up in random matrix theory. Maybe there's something there?

But your last paragraph is exactly it... it is a "basic" fact but the consequences are profound.

efavdb1y ago

Could you guys recommend an easy book on this topic?

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