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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2012/iii/paper-82/1/1/solution

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Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 82 / 1 / 1 / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-14

The Fejér kernel for the printed normalization is

F_{N} (t) = \frac{1}{N} \sum_{j = 0}^{N - 1} e^{2 πij t}^{2} = \sum_{∣ j ∣ \leq N} (1 - \frac{∣ j ∣}{N}) e^{2 πij t} .

(1)

Its endpoint coefficients at

\pm N

are zero. It is nonnegative, and integrating its Fourier series gives

\int_{T} F_{N} = 1

. Hence

S_{N} (f) = f * F_{N}

and

∣ S_{N} (f, t) ∣ \leq ∥ f ∥_{\infty} \int F_{N} \leq 1, S^{*} (f, t) \leq 1.

(2)

Thus the absolute constant can be $C = 1$ . Positivity of the Fejér kernel, rather than mere boundedness of its individual Fourier coefficients, is the reason this holds.

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