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

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- Solution c

Solution

Differentiate the defining Fourier integral

n

times with respect to

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}} .

Integrability of

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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