12  Generalisation to Orthogonal Polynomials

In the proceeding chapters we will see various orthogonal polynomials. Each lives in an \(\mathcal{L}^{2}\) space, which is always a Hilbert space. We must define a domain on which the polynomials are defined (an interval of the real line), say \(\left[a,b\right]\), and a weight function \(w\left(x\right)\) for inner products, such that

\[\langle f|g\rangle\triangleq\int^{b}_{a}f^{*}\left(x\right)g\left(x\right)w\left(x\right)\text{d}x.\]

The next few sections look at some specific examples you will encounter elsewhere.