 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2007/ib/paper-2/10g/iii/solution

Codex  0 Created 2026-09-24 Updated 2026-10-14

For

d

variables, integrate over the product of

d

⟨ f, g ⟩ = \frac{1}{( 2 π ) ^{d}} \int_{[0, 2 π]^{d}} f (e^{i θ_{1}}, \dots, e^{i θ_{d}}) \overline{g (e^{i θ_{1}}, \dots, e^{i θ_{d}})} d θ_{1} \dots d θ_{d} .

α, β \in Z_{\geq 0}^{d}

, applying one-variable Fourier orthogonality in each variable gives

⟨ z^{α}, z^{β} ⟩ = \prod_{j = 1}^{d} δ_{α_{j} β_{j}} .

f = \sum_{α} a_{α} z^{α}

, its squared norm is

\sum_{α} ∣ a_{α} ∣^{2}

, a finite sum. The form is therefore a positive definite Hermitian form and

{z_{1}^{α_{1}} \dots z_{d}^{α_{d}} : α_{j} \geq 0} is an orthonormal basis .

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