[Rayhem]

Obligatory callout to Nick Trefethen (et. al.)'s Chebfun[1] which extends this stuff in just about every direction you can think of. 'Chebfuns' are to functions as floats are to actual mathematical numbers. It's some seriously impressive software.

[1]: https://www.chebfun.org

[kxyvr]

I'll second this. Their methods are very powerful and very fast. For those out of the loop, the Chebyshev (and ultra-spherical) machinery allows a very accurate (machine precision) approximation to most functions to be computed very quickly. Then, this representation can be manipulated more easily. This enables a variety of methods such as finding the solution to differential algebraic equations to machine precision or finding the global min/max of a 1-D function.

I believe they use a different algorithm now, but the basic methodology that used to be used by Chebfun can be found in the book Spectral Methods in Matlab by Trefethen. Look at chapter 6. The newer methodology with ultraspherical functions can be found in a SIAM review paper titled, "A Fast and Well-Conditioned Spectral Method," by Olver and Townsend.