We define what it means for polynomials to be orthogonal with respect to an inner product. As an example, we discuss the Legendre polynomials and their construction via the Gram-Schimdt process. Then we turn to a definition of the Legendre polynomials via a differential equation, and use it to prove their orthogonality. This approach generalizes to all classical orthogonal polynomials.
Loading...
Reload document 
Open in new tab
Hi Nick, do you have plans for future exploration using these orthogonal polynomials and their differential equations?
Hi Lisa, I’m glad you asked. You can see more about some ideas involving this approach here: http://arxiv.org/abs/2010.10690
Taking the differential equation as a starting point is apparently not so common for people who study orthogonal polynomials. There is something called the Askey scheme which tries to characterize all orthogonal polynomials in terms of hypergeometric functions (see here http://homepage.tudelft.nl/11r49/askey/) Proving orthogonality straight from the diff. eq. lets us look at some known results in a new light (such as the forward shift operator for differentiating) in addition to other possibilities still to be explored.
I can say a bit more to elaborate here: on the second to last line of slide 4, there is an expression for an integral of Legendre polynomials over the interval [x,1]. Just after this, we plug in x = -1 to get orthogonality; but, this is not the only possible use for that formula!
If -1 < x < 1, we can write x = \cos \theta for some 0 < \theta < \pi. If we do this, then P_n(x) for this fixed x behaves like a damped sine wave, whose argument involves \theta and n, as n becomes very large. This sort of effect also extends to integrals of products of Legendre polynomials, i.e. those shown in the slides, if m is fixed and x is fixed (and more generally, Jacobi polynomials, which include the former as a special case).
I hope this sparks some curiosity, and please reach out to me at nicholas.juricic@uconn.edu if you have more questions! (Lisa, or anyone who is also reading 🙂 )