Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2019 / Paper 1 / 4 / a

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Solution

For the stated Fourier transform convention, Fourier inversion is

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

For example, the pointwise formula holds at continuity points under the usual integrability hypotheses; the same normalization applies to the

L^{2}

or distributional inverse.

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