 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2003/iii/paper-69/2/1/solution

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

Use degree-indexed Fourier partial sums and Fejér sums, as in this paper. For a

2 π

f_{k} = \frac{1}{2 π} \int_{- π}^{π} f (t) e^{- ik t} d t, s_{n} (f, x) = \sum_{k = - n}^{n} f_{k} e^{ik x}, σ_{n} (f) = \frac{1}{n + 1} \sum_{j = 0}^{n} s_{j} (f) .

Equivalently,

σ_{n} (f, x) = \sum_{∣ k ∣ \leq n} (1 - \frac{∣ k ∣}{n + 1}) f_{k} e^{ik x} .

These are trigonometric polynomials of degree at most

n

. Some treatments index the Fejér sum by the number of terms rather than its degree; the degree convention here is why the formulas below contain

σ_{2 n - 1}

and

σ_{n - 1}

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