 Solution

ID: past-exam-of-the-approximation-theory-and-finite-element-methods-course-of-the-university-of-oxford/2009/a/2/c/solution

The monic degree-zero polynomial is

ϕ_{0} = 1

x - a

to one gives

a = \int_{0}^{1} x d x = 1/2

, so

ϕ_{1} = x - 1/2

The relevant integrals are

∥ ϕ_{0} ∥_{2}^{2} = 1, ∥ ϕ_{1} ∥_{2}^{2} = \frac{1}{12}, (ϕ_{1}, x ϕ_{1}) = \frac{1}{24} .

ϕ_{2} = (x - \frac{1}{2}) (x - \frac{1}{2}) - \frac{1}{12} = x^{2} - x + \frac{1}{6} .

Thus

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

Their mutual orthogonality can also be checked directly by integrating on

[0, 1]

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