Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 2 / 23H / b

Table of contents
- Solution b

Solution

Take the extension from part (a), and choose

M > N + n + 1

. The positive function

w (x) = (1 + ∣ x ∣^{2})^{- M /2}

belongs to

X_{N, k}

. Positivity extends from nonnegative Schwartz functions to nonnegative elements obtained as

X_{N, k}

limits. Since

- ∥ ϕ ∥_{M, 0} w \leq ϕ \leq ∥ ϕ ∥_{M, 0} w,

positivity gives

∣ v [ϕ] ∣ \leq v [w] ∥ ϕ ∥_{M, 0} .

Taking

K = v [w]

proves that every positive distribution has order zero.

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