 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2019/paper-1/4/c/solution

Differentiate the defining Fourier integral

n

k

\frac{d ^{n} f}{d k ^{n}} = \int_{R} (- i x)^{n} f (x) e^{- ik x} d x .

For a nonnegative integer

n

, this gives

F [x^{n} f (x)] (k) = i^{n} \frac{d ^{n} f ( k )}{d k ^{n}} .

x^{n} f

together with suitable lower moments justifies differentiation under the integral. The same multiplication-derivative identity holds more generally for tempered distributions.

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