We define the following spaces for future reference:
Where the support of f, denoted by , is defined to be: $$supp\ f = {x\in \mathbb{R}^{n}: f(x) \neq 0}$$
Orthogonal Basis
In finite dimension, an euclidean space is a vector space equipped with an inner product. The space of polynomials of degree less than or equal is of dimension . We can identify it with , and on we can define a scalar product :
Note that the standard basis are a pairwise for the scalar product but not for the other two products. Orthogonal polynomials for are for example the Legendre polynomials which satisfy the following recurrence relation: