Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 318 / 5 / 2 / b

Table of contents
- Solution b

Solution

Substitute the B-spline expansion

s = \sum_{j = 1}^{n} a_{j} M_{j}

into

(M_{i}, s) = k! γ_{i}

. The coefficients solve

G a = b,

with the Gram matrix and right-hand side

G_{ij} = (M_{i}, M_{j}) = \int_{a}^{b} M_{i} (t) M_{j} (t) d t, b_{i} = k! γ_{i} .

The linearly independent B-splines have a positive-definite Gram matrix, so this system has a unique coefficient vector.

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