Past exam of the approximation theory and finite element methods course of the University of Oxford / 2008 / A / 2 / b

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- Solution b

Solution

The displayed recurrence requires monic normalization, which is not explicitly stated in the preamble. Arbitrary independent rescaling of orthogonal polynomials preserves orthogonality but changes the recurrence coefficients. Choose their monic representatives for this and the following calculations.

Expand

x ϕ_{n}

in the orthogonal polynomial basis:

x ϕ_{n} = ϕ_{n + 1} + \sum_{j = 0}^{n} a_{j} ϕ_{j} .

The leading coefficient of

ϕ_{n + 1}

is one. For

j \leq n - 2

,

(x ϕ_{n}, ϕ_{j}) = (ϕ_{n}, x ϕ_{j}) = 0

, because the second factor has degree below

n

. Thus only

j = n, n - 1

remain. Inner products give

a_{n} = \frac{( ϕ _{n} , x ϕ _{n} )}{∥ ϕ _{n} ∥ ^{2}}, a_{n - 1} = \frac{( ϕ _{n} , x ϕ _{n - 1} )}{∥ ϕ _{n - 1} ∥ ^{2}} = \frac{∥ ϕ _{n} ∥ ^{2}}{∥ ϕ _{n - 1} ∥ ^{2}} .

The last equality uses

x ϕ_{n - 1} - ϕ_{n} \in P_{n - 1}

. Rearranging yields exactly the three-term recurrence for monic orthogonal polynomials

ϕ_{n + 1} = (x - a_{n}) ϕ_{n} - \frac{∥ ϕ _{n} ∥ ^{2}}{∥ ϕ _{n - 1} ∥ ^{2}} ϕ_{n - 1} .

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