TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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Chebyshev–Lobatto interpolation inherits a weighted near-best bound

Statement

For $f \in C [- 1, 1]$ and $N \geq 0$ , let $E_{N} (f)$ and $Λ_{N}^{L}$ be the best uniform error and the Lebesgue constant from Best uniform polynomial error and Chebyshev interpolation operators. Then

∥ f - I_{N}^{L} f ∥_{C} \leq \sqrt{π} (1 + Λ_{N}^{L}) E_{N} (f),

where $∥ \cdot ∥_{C}$ is the arcsine-weighted norm of Arcsine-weighted inner product on the Chebyshev interval.

Facts & Assumptions

Given: A continuous real function $f$ on $[- 1, 1]$ and $N \geq 0$ .

[F1]

Chebyshev interpolation error is controlled by best polynomial error: $∥ f - I_{N}^{L} f ∥_{\infty} \leq (1 + Λ_{N}^{L}) E_{N} (f)$ .

[F2]

Arcsine-weighted inner product on the Chebyshev interval: $∥ h ∥_{C}^{2} = \int_{0}^{π} ∣ h (\cos θ) ∣^{2} d θ$ , so $∥ h ∥_{C} \leq \sqrt{π} ∥ h ∥_{\infty}$ .

Proof

1.1givenF2algebra

For every $θ \in [0, π]$ , $∣ h (\cos θ) ∣ \leq ∥ h ∥_{\infty}$ . Integrating its square in [F2] gives $∥ h ∥_{C}^{2} \leq π ∥ h ∥_{\infty}^{2}$ and hence $∥ h ∥_{C} \leq \sqrt{π} ∥ h ∥_{\infty}$ .

2.1givenF1step 1.1algebra∎

Apply step 1.1 to $h = f - I_{N}^{L} f$ , then use the uniform near-best bound [F1]. The result is $∥ f - I_{N}^{L} f ∥_{C} \leq \sqrt{π} (1 + Λ_{N}^{L}) E_{N} (f)$ .

Depends on

Used by

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Dependency tree · two levels

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Sources

Jie Shen and Tao Tang, Spectral and High-Order Methods with Applications, §1.8 (standard reference, not scraped)
Lloyd N. Trefethen, Approximation Theory and Approximation Practice, Ch. 5 (standard reference, not scraped)