Past exam of the mathematics course of the University of Cambridge / 2008 / ii / Paper 4 / 30C / i

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- Solution i

Solution

Choose the Fourier transform convention

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 .

It maps the Schwartz space to itself. For a tempered distribution

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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