 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/ii/paper-4/23g/a/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

(1 + ∣ ξ ∣^{4}) u (ξ) = f (ξ) .

Thus

u (ξ) = \frac{f ( ξ )}{1 + ∣ ξ ∣ ^{4}} .

The multiplier and all its polynomially weighted derivatives have at most polynomial growth, so this defines a tempered distribution. In fact it maps

H^{s}

continuously into

H^{s + 4}

. Since the multiplier never vanishes, any homogeneous tempered solution has zero Fourier transform; uniqueness follows.

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