9.2 Some properties of orthogonal polynomials

Given any weighted scalar product ⟨⋅,⋅⟩ω as in Eq. (9.17View Equation), the expansion of a function into polynomials is optimal in the associated norm if orthogonally projected to the space of polynomials PN of degree at most N. More precisely, given a function 2 f ∈ L ω(a,b) on some interval (a,b) and an orthonormal basis PN of polynomials N {pj}j=0, where pj has degree j, its orthogonal projection,
∑N 𝒫N [f ] = ⟨pj,f ⟩ωpj (9.33 ) j=0
onto the subspace spanned by PN satisfies
𝒫N [f ] = {fN ∈ PN : fN minimizes ∥f − fN ∥ω}, (9.34 )
∥𝒫N [f] − f ∥ω ≤ ∥fN − f∥ω (9.35 )
for all fN in the span of PN, that is, it minimizes the error for such fN.

The operator (9.33View Equation) is a projection in the sense that

2 𝒫N = 𝒫N , (9.36 )
and it is orthogonal with respect to ⟨⋅,⋅⟩ω: the residual r := f − 𝒫N [f ] satisfies
⟨r,𝒫N [f]⟩ω = 0. (9.37 )
Notice that, unlike the interpolation problem (discussed below) here, the solution of the above least-squares problem (9.34View Equation) is not required to agree with f at any prescribed set of points.

In order to obtain an orthonormal basis of PN, a Gram–Schmidt procedure could be applied to the standard basis {xj}Nj=0. However, exploiting properties of polynomials, a more efficient approach can be used, where the first two polynomials p0,p1 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.

Eqs. (9.38View Equation, 9.39View Equation, 9.40View Equation) allow one to compute orthogonal polynomials for any weight function ω, without the expense of a Gram–Schmidt procedure. For specific weight cases, there are even more explicit recurrence formulae, such as those in Eqs. (9.43View Equation, 9.44View Equation) and (9.48View Equation, 9.49View Equation, 9.50View Equation) below for Legendre and Chebyshev polynomials, respectively.

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