Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 318 / 3 / a

Solution

M_{i} (t)

(t_{j} - t)_{+}^{k - 1}

. It is therefore polynomial of degree at most

k - 1

between knots and globally

C^{k - 2}

. For

t < t_{i}

, the nodal data come from a polynomial of degree

k - 1

, whose order-

k

t \geq t_{i + k}

all truncated powers vanish. Thus

supp M_{i} = [t_{i}, t_{i + k}],

and normalization does not change the degree, smoothness, knots, or support.

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