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Gaussian Integration Is Cool
58 beansbeansbeans 9 6/8/2025, 8:35:54 AM rohangautam.github.io ↗
What is also worth pointing out and which was somewhat glanced over is the close connection between the weight function and the polynomials. For different weight functions you get different classes of orthogonal polynomials. Orthogonal has to be understood in relation to the scalar product given by integrating with respect to the weight function as well.
Interestingly Gauss-Hermite integrates on the entire real line, so from -infinity to infinity. So the choice of weight function also influences the choice of integration domain.
Like, is it possible to infer that Chebyshev polynomials would be useful in approximation theory using only the fact that they're orthogonal wrt the Wigner semicircle (U_n) or arcsine (T_n) distribution?
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.
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
That is, the integral from - to + infinity of e^(-x^2) dx = sqrt(pi).
I remember being given this as an exercise and just being totally shocked by how beautiful it was as a result (when I eventually managed to work out how to evaluate it).
[1] https://mathworld.wolfram.com/GaussianIntegral.html
It's the gateway drug to Laplace's method (Laplace approximation), mean field theory, perturbation theory, ... QFT.
https://en.m.wikipedia.org/wiki/Laplace%27s_method