The operator (9.33) is a projection in the sense that
In order to obtain an orthonormal basis of , a Gram–Schmidt procedure could be applied to the standard basis . However, exploiting properties of polynomials, a more efficient approach can be used, where the first two polynomials are constructed and then a three-term recurrence formula is used.
In the following construction, each orthonormal polynomial is chosen to be monic, meaning that its leading coefficient is one.
The conditions that has degree zero and that it is monic only leaves the choice
Writing the condition yields
Theorem 19 (Three-term recurrence formula for orthogonal polynomials). For monic polynomials , which are orthogonal with respect to the scalar product , where each is of degree , the following relation holdsfor .
Proof. Let . Since is a polynomial of degree , it can be expanded as9.40). □
Notice that , as defined in Eq. (9.40), remains monic and can therefore be automatically used for constructing , without any rescaling.
Eqs. (9.38, 9.39, 9.40) allow one to compute orthogonal polynomials for any weight function , without the expense of a Gram–Schmidt procedure. For specific weight cases, there are even more explicit recurrence formulae, such as those in Eqs. (9.43, 9.44) and (9.48, 9.49, 9.50) below for Legendre and Chebyshev polynomials, respectively.
Living Rev. Relativity 15, (2012), 9
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