Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 318 / 3 / b

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- Solution b

Solution

Write the interpolating spline as

P_{x} f = \sum_{j = 1}^{n} c_{j} N_{j} .

The interpolation equations are

A_{x} c = f_{x}

, where

(f_{x})_{i} = f (x_{i})

, so

∥ c ∥_{ℓ_{\infty}} \leq ∥ A_{x}^{- 1} ∥_{ℓ_{\infty}} ∥ f_{x} ∥_{ℓ_{\infty}} \leq ∥ A_{x}^{- 1} ∥_{ℓ_{\infty}} ∥ f ∥_{\infty} .

The B-splines are nonnegative and form a partition of unity on the spline interval. Consequently

∣ P_{x} f (t) ∣ \leq \sum_{j} ∣ c_{j} ∣ N_{j} (t) \leq ∥ c ∥_{ℓ_{\infty}} .

Taking the supremum over

t

and then over

∥ f ∥_{\infty} \leq 1

proves the operator norm bound

∥ P_{x} ∥_{L_{\infty}} \leq ∥ A_{x}^{- 1} ∥_{ℓ_{\infty}} .

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