What is Fig. 1 showing? Is it the value of the integral compared with two approximations? Would it not be more interesting to show the error of the approximations instead? Asking for a friend who isn’t computing a lot of integrals.
constantcrying · 5h ago
A good introduction to the basics.
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.
creata · 2h ago
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?
sfpotter · 1h ago
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.
constantcrying · 1h ago
Yes. Precisely that they are orthogonal means that they are suitable.
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.
There are polynomials that aren't orthogonal that are suitable for numerics: both the Bernstein basis and the monomial basis are used very often and neither are orthogonal. (Well, you could pick a weight function that makes them orthogonal, but...!)
The fact of their orthogonality is crucial, but when you work with Chebyshev polynomials, it is very unlikely you are doing an orthogonal (L2) projection! Instead, you would normally use Chebyshev interpolation: 1) interpolate at either the Type-I or Type-II Chebyshev nodes, 2) use the DCT to compute the Chebyshev series coefficients. The fact that you can do this is related to the weight function, but it isn't an L2 procedure. Like I mentioned in my other post, the Chebyshev weight function is maybe more of an artifact of the Chebyshev polynomials' intimate relation to the Fourier series.
I am also not totally sure what polynomial chaos has to do with any of this. PC is a term of art in uncertainty quantification, and this is all just basic numerical analysis. If you have a series in orthgonal polynomials, if you want to call it something fancy, you might call it a Fourier series, but usually there is no fancy term...
seanhunter · 3h ago
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.
srean · 1h ago
Indeed.
It's the gateway drug to Laplace's method (Laplace approximation), mean field theory, perturbation theory, ... QFT.
There is a relationship here, in the case of Gauß-Hermite Integration, where the weight function is exactly e^(-x^2) the weights have to add up sqrt(pi), because the integral is exact for the constant 1 polynomial.
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
There are polynomials that aren't orthogonal that are suitable for numerics: both the Bernstein basis and the monomial basis are used very often and neither are orthogonal. (Well, you could pick a weight function that makes them orthogonal, but...!)
The fact of their orthogonality is crucial, but when you work with Chebyshev polynomials, it is very unlikely you are doing an orthogonal (L2) projection! Instead, you would normally use Chebyshev interpolation: 1) interpolate at either the Type-I or Type-II Chebyshev nodes, 2) use the DCT to compute the Chebyshev series coefficients. The fact that you can do this is related to the weight function, but it isn't an L2 procedure. Like I mentioned in my other post, the Chebyshev weight function is maybe more of an artifact of the Chebyshev polynomials' intimate relation to the Fourier series.
I am also not totally sure what polynomial chaos has to do with any of this. PC is a term of art in uncertainty quantification, and this is all just basic numerical analysis. If you have a series in orthgonal polynomials, if you want to call it something fancy, you might call it a Fourier series, but usually there is no fancy term...
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