 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/i/solution

First take

f, g \in L^{1} (R)

s = x + y

h (k) = \int_{R} \int_{R} e^{- ik x} \overline{f (y)} g (x + y) d y d x = [\int_{R} \overline{f (y)} e^{ik y} d y] [\int_{R} g (s) e^{- ik s} d s] = \overline{f (k)} g (k) .

x + y

are both essential to this convention.

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