 Solution

ID: past-exam-of-the-options-ii-course-of-the-university-of-oxford/2005/d/4/iii/solution

For fixed real

t

e^{i t x}

does not change the absolute value. The product integrability proved in (ii) therefore justifies Fubini theorem in the following calculation. Set

u = x - y

in the inner integral:

g * h (t) = \int_{R} \int_{R} g (x - y) h (y) e^{i t x} d y d x = \int_{R} h (y) [\int_{R} g (x - y) e^{i t x} d x] d y = \int_{R} h (y) e^{i t y} [\int_{R} g (u) e^{i t u} d u] d y = g (t) h (t) .

Thus

g * h = g h .

The positive exponential convention in the source is used throughout; no normalization factor is introduced under that convention.

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