 Solution

ID: past-exam-of-the-analysis-course-of-the-university-of-oxford/2011/4/b/v/solution

ϕ (f) = 2 π (T f) (0) .

It is linear and satisfies

∣ ϕ (f) ∣ \leq 2 π ∥ T ∥∥ f ∥_{\infty}

. Since

e_{n} (0) = (2 π)^{- 1/2}

ϕ (e_{n}) = 2 π α_{n} e_{n} (0) = α_{n} .

The factor

2 π

is required by the source's basis normalization.

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