 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2020/ii/paper-4/40e/a/solution

Codex  0 Created 2026-09-24 Updated 2026-09-29

(f_{x})_{m, n} = iπm f_{m, n}, (f_{y})_{m, n} = iπn f_{m, n} .

h = f g = \sum_{p, q} \sum_{r, s} f_{p, q} g_{r, s} e^{iπ (p + r) x + iπ (q + s) y},

h_{m, n} = p, q \in Z \sum f_{p, q} g_{m - p, n - q} .

The assumed real analyticity justifies these termwise operations through rapid decay of the Fourier coefficients.

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