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

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Weighted Chebyshev projection has its spectral Sobolev rate

Statement

Let $s \geq 0$ , $N \geq 0$ , and $u \in H_{C}^{s}$ as defined in Chebyshev weighted spectral Sobolev scale and projection. Then

∥ u - Π_{N} u ∥_{C} \leq (1 + (N + 1)^{2})^{- s / 2} ∥ u ∥_{H_{C}^{s}} .

Thus the weighted projection error is $O (N^{- s})$ when the data belong to this explicitly defined weighted spectral Sobolev space.

Facts & Assumptions

Given: $s \geq 0$ , an integer $N \geq 0$ , and a Chebyshev coefficient expansion $u = \sum_{n \geq 0} c_{n} T_{n}$ with finite $H_{C}^{s}$ norm.

[F1]

Chebyshev weighted spectral Sobolev scale and projection: $Π_{N} u = \sum_{n = 0}^{N} c_{n} T_{n}$ and the $H_{C}^{s}$ norm weights coefficient $c_{n}$ by $(1 + n^{2})^{s}$ ; these weights are nondecreasing in $n$ for $s \geq 0$ .

[F2]

Chebyshev polynomials are orthogonal for the arcsine weight: the squared $C$ -norm of $T_{0}$ is $π$ and that of each $T_{n}$ , $n \geq 1$ , is $π / 2$ .

Proof

1.1F1F2algebra

By [F1], $u - Π_{N} u = \sum_{n > N} c_{n} T_{n}$ . Orthogonality from [F2] gives $∥ u - Π_{N} u ∥_{C}^{2} = \frac{π}{2} \sum_{n > N} ∣ c_{n} ∣^{2} .$ Since $n > N$ implies $(1 + n^{2})^{s} \geq (1 + (N + 1)^{2})^{s}$ , this is at most $(1 + (N + 1)^{2})^{- s} \frac{π}{2} \sum_{n > N} (1 + n^{2})^{s} ∣ c_{n} ∣^{2} \leq (1 + (N + 1)^{2})^{- s} ∥ u ∥_{H_{C}^{s}}^{2} .$ Taking square roots proves the result for finite sums.

2.1F1step 1.1algebra∎

Finite sums are dense in the completion defining $H_{C}^{s}$ . The truncation $Π_{N}$ is a contraction in the coefficient norm and in the $C$ -norm, so both sides of the finite-sum inequality pass to the limit along any Cauchy sequence of finite sums. This yields the stated estimate for every $u \in H_{C}^{s}$ .

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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, Finite Difference and Spectral Methods for Ordinary and Partial Differential Equations, Ch. 8 (standard reference, not scraped)