ABSTRACT
In order that polynomials belong to the space, we shall also require that
the moments xnd[i, n = 0 ,1, • • •, exist. This is no problem if (a, b) is
a finite interval but i t may be if i t is infinite. There generally exist many orthogonal polynomial systems associated
with /x, but we shall usually consider systems where Pn(x) is of degree n. This may be ensured by orthogonalizing the sequence
That is, we let Po(x) — 1 and then subtract from x its projection on the space spanned by Po(x). This gives us P\(x). We then subtract from x2 its projection on the space spanned by Po(x) and P\(x) to get ^ 2 ( ^ ) 7 etc. This wi l l give us a sequence {Pn} of polynomials that are of exact degree n and orthogonal wi th respect to d\i. They may be made into an orthonormal system by normalization
pn(x) = Pn(x)/\\Pn\\^.
