How do you choose the interval? Does it matter conceptually that the Legendre polynomials, for instance, are orthonormalised over (-1,1)? Does that mean that even though they are not orthonormal over all space even though you can obviously graph them over all space? Specifically, I want to use Gramm Schmidt on the functions sin(x)/x and sin(2x)/(2x) and looking at their behavior at the origin but also comparing their long range oscillations. Therefore, I'm thinking 0 to 2pi might not be a good choice, but negative pi to pi or negative infinity to infinity are better choices. How do I choose the appropriate interval?