Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 327 / 1 / c

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- Solution c

Solution

Each continuous

f_{α}

of polynomial growth defines a regular tempered distribution by

⟨ f_{α}, φ ⟩ = \int_{R^{n}} f_{α} (x) φ (x) d x .

Choosing an integer

L > M_{α} + n

gives

∣ ⟨ f_{α}, φ ⟩ ∣ \leq C_{α} \int (1 + ∣ x ∣)^{M_{α} - L} d x sup_{x} (1 + ∣ x ∣)^{L} ∣ φ (x) ∣,

which is bounded by finitely many Schwartz space seminorms. Its distributional derivative satisfies

⟨ D^{α} f_{α}, φ ⟩ = (- 1)^{∣ α ∣} ⟨ f_{α}, D^{α} φ ⟩

and is therefore tempered. A finite sum of continuous linear forms is continuous, so

∣ α ∣ \leq N \sum D^{α} f_{α} \in S^{'} (R^{n}) .

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