 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2013/paper-1/4/iii/solution

The direct squared L2 norm is the length of the interval, namely

2

f (k) = \int_{- 1}^{1} e^{- ik x} d x = \frac{2 s i n k}{k}, f (0) = 2.

For the transformed norm, let

I = \int_{0}^{\infty} sin^{2} k / k^{2} d k

. Integration by parts, with vanishing endpoint term

- sin^{2} k / k

, gives

I = 2 \int_{0}^{\infty} \frac{s i n k c o s k}{k} d k = \frac{π}{2} .

The supplied oscillatory integral is understood as an improper integral. Therefore

\frac{1}{2 π} \int_{- \infty}^{\infty} \frac{2 sin k}{k}^{2} d k = \frac{4}{π} I = 2,

which agrees with the direct norm and verifies the Parseval identity with the specified normalization.

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