Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2014 / Paper 1 / 4 / iv

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- Solution iv

Solution

For sufficiently rapidly decaying functions, substitute the inverse Fourier transform of

g

and interchange the integrals:

\int_{- \infty}^{\infty} f (x)^{*} g (x) d x = \frac{1}{2 π} \int_{- \infty}^{\infty} g (k) [\int_{- \infty}^{\infty} f (x)^{*} e^{ik x} d x] d k = \frac{1}{2 π} \int_{- \infty}^{\infty} f (k)^{*} g (k) d k .

The bracketed integral is the complex conjugate of

f (k)

. This is Parseval's identity for the chosen normalization. One may first prove it for Schwartz functions, where the interchanges are justified, and then extend it to

L^{2}

by the Plancherel theorem.

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