 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2008/ii/paper-4/30c/i/solution

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

f (ξ) = \int_{R^{n}} e^{- i x \cdot ξ} f (x) d x, f (x) = (2 π)^{- n} \int e^{i x \cdot ξ} f (ξ) d ξ .

u

, define the transform by bilinear distributional duality:

⟨ u, ϕ ⟩ = ⟨ u, ϕ ⟩ (ϕ \in S) .

For an ordinary integrable function this agrees with the first formula by interchanging its integrals. The absence of complex conjugation is the distribution-pairing convention.

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