 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2024/paper-1/4/b/solution

Insert the inverse Fourier transforms of

f

and its complex conjugate, initially under enough decay to justify interchanging integrals:

\int_{R} ∣ f (x) ∣^{2} d x = \frac{1}{( 2 π ) ^{2}} \int_{R} \int_{R} f (k) \overline{f (ℓ)} [\int_{R} e^{i (k - ℓ) x} d x] d k d ℓ = \frac{1}{2 π} \int_{R} \int_{R} f (k) \overline{f (ℓ)} δ (k - ℓ) d k d ℓ = \frac{1}{2 π} \int_{R} ∣ f (k) ∣^{2} d k .

The middle step uses the stated Dirac delta distribution identity, and the last uses its sampling property. This is the Parseval identity for the chosen normalization. Approximation extends the result to square-integrable functions, including the function used below.

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