Past exam of the approximation theory and finite element methods course of the University of Oxford / 2007 / A / 3 / b

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- Solution b

Solution

For the weighted inner product,

(x^{j}, x^{k}) = 1/ (j + k + 2)

. Start Gram-Schmidt process with the monomials and choose monic normalization:

ϕ_{0} = 1, ϕ_{1} = x - \frac{( x , 1 )}{( 1 , 1 )} = x - \frac{2}{3} .

Its squared norm is

1/4 - 4/9 + 2/9 = 1/36

. Next

(x^{2}, 1) / (1, 1) = 1/2

and

(x^{2}, ϕ_{1}) = \frac{1}{5} - \frac{1}{6} = \frac{1}{30}, \frac{( x ^{2} , ϕ _{1} )}{∥ ϕ _{1} ∥ _{2}^{2}} = \frac{6}{5} .

Therefore

ϕ_{0} = 1, ϕ_{1} = x - \frac{2}{3}, ϕ_{2} = x^{2} - \frac{6}{5} x + \frac{3}{10} .

Subtracting these orthogonal projections gives orthogonality by construction; independent nonzero rescalings would give equivalent unnormalized orthogonal polynomials.

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