 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2018/paper-2/5/a/solution

A sufficient assumption is that

f

is absolutely continuous,

f, f^{'} \in L^{1} (R)

, and

f (t) \to 0

f^{'} (ω) = [f (t) e^{- iω t}]_{- \infty}^{\infty} + iω \int_{- \infty}^{\infty} f (t) e^{- iω t} d t = iω f (ω) .

The decay assumption removes the boundary term, and integrability makes both transforms ordinary convergent integrals. The same derivative identity extends to tempered distributions.

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