CSE 401/CS 450/MATH 450/ECE 491: Message 234

Date: Mon, 06 Nov 2006 08:58:46
From: S Bond
Subject: Re: Gram-Schmidt orthogonalization on polynomials?

Jonathan William Ray wrote:
> This definitely wasn't taught in math 415, but the lectures assume we 
> can do Gram-Schmidt orthogonalization on polynomials.
> How is there any meaningful concept of orthogonal projection in the 
> space of polynomials? 

We define the (L2) inner product between two functions as the integral
of their product.  Two functions are said to be orthogonal if the
integral of their product is zero.  The integration is typically
restricted to an interval, say [-1,1].

< p , q > = Integral from -1 to 1 of the product of p and q.

On unbounded intervals, there is usually a weight function used in
the definition of the inner product.  The Legendre polynomials are
orthogonal on the interval [-1,1] without including a weight
function.  Chebyshev, Jacobi, Laguerre, and Hermite polynomials all
use weight functions.

Page 321 in the book defines the inner product.  You won't need
to know how to generate any set of orthogonal polynomials.  We
won't actually use them at all midway through Chapter 8, and for
a little bit in Chapter 10.