Orthogonal polynomials

Factual Questions
1

I’ve been playing around with orthogonal polynomials as part of my research recently, and a question occurred to me. It seems like a simple enough question to pose, but I’m not sure how to go about addressing it.

Background: the Legendre polynomials {1, x, 3x[sup]2[/sup] - 1, …} are orthogonal if you integrate them over the domain [-1,1] with no weighting. The Hermite polynomials {1, 2x, 4x[sup]2[/sup] - 2, …} are orthogonal if you integrate them over the domain (-∞,+∞) with a weighting of e[sup]-x[sup]2[/sup]/2[/sup]. The Chebyshev polynomials of the first kind {1, x, 2x[sup]2[/sup]-1, …} are orthogonal over the domain [-1,1] with a weighting function of 1/√(1-x[sup]2[/sup]). My questions are:
[ul][]Does there exist a weighting/measure and domain of integration such that the monomials {1, x, x[sup]2[/sup], x[sup]3[/sup], …} are orthogonal? If so, can an explicit expression for this weighting be given?[]More generally, if you hand me a countably infinite set of polynomials of degrees 0, 1, 2, 3, …, can I always find an inner product (i.e., domain of integration and weighting) such that these polynomials are orthogonal? Does the answer change if I let this inner product be indefinite?[/ul]I tried playing around with Sturm-Liouville theory for a bit, but it’s been a while since I dealt with such things. I have a niggling suspicion that I asked this question of one of my professors once, but I can’t recall what the answer was (or if he had one.)

2

If the weighting is to be non-negative, you wouldn’t be able to, since 1, x^2, x^4 are all non-negative as well, so they couldn’t be made orthogonal.

Even with arbitrary weighting, they couldn’t be made orthonormal. The self term of x would give the same result as the cross-term between 1 and x^2, for example.

That leaves you with Integral{w(x) * x^n dx} = 0 for all n > 0. I guess W(x) = delta(x) works, but that’s boring. Maybe something oscillating infinitely many times as x -> 1, like W(x) = cos(1/(1-x))*, could work over the range +/- 1.

  • I didn’t work out those integrals, so no complaining if they aren’t zero.

ETA: For your more general question, I think any set polynomials where more than one is non-negative would have the same sort of problems.

3

While Zen Beam is obviously correct, a better place to ask such technical questions is a site called Math Overflow where I have found useful answers to questions.