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.
Sorry if this is a stupid question, but is there a direct link between the choice of weight function and the applications of the polynomial?
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?
I thought when I first saw the title that it was going to be about the Gaussian integral[1] which has to be one of the coolest results in all of maths.
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).
Gaussian integrals are also pretty much the basis of quantum field theory in the path integral formalism, where Isserlis's theorem is the analog to Wick's theorem in the operator formalism.
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.
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