[constantcrying]

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]

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]

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.

[creata]

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.

[sfpotter]

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.

[efavdb]

Could you guys recommend an easy book on this topic?

[sfpotter]

Which topic?

[constantcrying]

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.

For polynomials it is this: https://en.m.wikipedia.org/wiki/Polynomial_chaos

[sfpotter]

What you're saying isn't wrong, per se, but...

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...

[constantcrying]

I think you are very confused. My post is about approximation. Obviously other areas use other polynomials or the same polynomials for other reasons.

In this case it is about the principle of approximation by orthogonal projection, which is quite common in different fields of mathematics. Here you create an approximation of a target by projecting it onto an orthogonal subspace. This is what the Fourier series is about, an orthogonal projection. Choosing e.g. the Chebychev Polynomials instead of the complex exponential gives you an Approximation onto the orthogonal space of e.g. Chebychev polynomials.

The same principle applies e.g. when you are computing an SVD for a low rank approximation. That is another case of orthogonal projection.

>Instead, you would normally use Chebyshev interpolation

What you do not understand is that this is the same thing. The distinction you describe does not exist, these are the same things, just different perspectives. That they are the same easily follows from the uniqueness of polynomials, which are fully determined by their interpolation points. These aren't distinct ideas, there is a greater principle behind them and that you are using some other algorithm to compute the Approximation does not matter at all.

>I am also not totally sure what polynomial chaos has to do with any of this.

It is the exact same thing. Projection onto an orthogonal subspace of polynomials. Just that you choose the polynomials with regard to a random variable. So you get an approximation with good statistical properties.

[sfpotter]

No, I am not confused. :-) I am just trying to help you understand some basics of numerical analysis.

> What you do not understand is that this is the same thing.

It is not the same thing.

You can express an analytic function f(x) in a convergent (on [-1, 1]) Chebyshev series: f(x) = \sum_{n=0}^\infty a_n T_n(x). You can then truncate it keeping N+1 terms, giving a degree N polynomial. Call it f_N.

Alternatively, you can interpolate f at at N+1 Chebyshev nodes and use a DCT to compute the corresponding Chebyshev series coefficients. Call the resulting polynomial p_N.

In general, f_N and p_N are not the same polynomial.

Furthermore, computing the coefficients of f_N is much more expensive than computing the coefficients of p_N. For f_N, you need to evaluate N+1 integral which may be quite expensive indeed if you want to get digits. For p_N, you simply evaluate f at N+1 nodes, compute a DCT in O(N log N) time, and the result is the coefficients of p_N up to rounding error.

In practice, people do not compute the coefficients of f_N, they compute the coefficients of p_N. Nevertheless, f_N and p_N are essentially as good as each other when it comes to approximation.

[constantcrying]

At this point I really hate to ask. Do you know what "orthogonal subspace" means and what a projection is?