 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2009/iii/paper-69/3/2/a/solution

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

For unit-spaced knots, the quadratic cardinal B-spline is a translate of

N_{j, 3} (t) = ⎩ ⎨ ⎧ (t - j)^{2} /2, 3/4 - (t - j - 3/2)^{2}, (j + 3 - t)^{2} /2, 0, j \leq t \leq j + 1, j + 1 \leq t \leq j + 2, j + 2 \leq t \leq j + 3, otherwise .

For example, the Cox-de Boor recurrence gives

N_{j, 3} (t) = (t - j) N_{j, 2} (t) /2 + (j + 3 - t) N_{j + 1, 2} (t) /2

from the two linear hats. Substituting their linear pieces gives the three expressions above. At

x_{i} = i + 3/2

, only the B-splines with indices

j = i - 1, i, i + 1

have nonzero values, and their values are respectively

N_{j} (x_{i}) = ⎩ ⎨ ⎧ 3/4, 1/8, 0, j = i, ∣ j - i ∣ = 1, ∣ j - i ∣ \geq 2.

Only indices

1 \leq j \leq n

belong to the given basis, so entries outside that range are omitted at the boundary.

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