ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

How statement and proof provenance work

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An exponential best-error rate transferred to Chebyshev interpolation

Example

Let $f (x) = \exp (x)$ on $[- 1, 1]$ . For every $n \geq 1$ , $E_{n} (f) \leq \frac{e}{(n + 1)!}, ∥ f - I_{n}^{σ} f ∥_{\infty} \leq (1 + C \log (n + 1)) \frac{e}{(n + 1)!}, σ \in {R, L},$ where $C = 4 + (π + 4) / \log 2$ . In particular, both Chebyshev interpolants converge uniformly for this function with the displayed rate.

Facts & Assumptions

Given: $n \geq 1$ , the function $f (x) = \exp (x)$ , and either Chebyshev node family.

[F1]

Best uniform polynomial error and Chebyshev interpolation operators: $E_{n} (f) = \inf_{p \in P_{n}} ∥ f - p ∥_{\infty}$ , and $I_{n}^{σ}$ is its specified interpolation operator.

[F2]

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

[F3]

Chebyshev-root and Lobatto interpolation norms have logarithmic upper bounds: $Λ_{n}^{σ} \leq C \log (n + 1)$ for $n \geq 1$ .

[F4]

The real exponential function and the number $e$ by a power series: $\exp (x) = \sum_{k = 0}^{\infty} x^{k} / k!$ and $e = \exp (1) = \sum_{k = 0}^{\infty} 1 / k!$ .

[F5]

The exponential series converges absolutely for every real argument: the exponential series converges absolutely for every real argument.

[F6]

The factorial $n!$ and the falling factorial $n^{\underline{k}}$ , defined by recursion in $N$ : $0! = 1$ and $(m + 1)! = (m + 1) m!$ .

[F7]

The natural logarithm as the inverse of the exponential function and The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t: $\log$ is the natural logarithm and $\log x = \int_{1}^{x} d t / t$ for $x > 0$ .

[F8]

For $∣ r ∣ < 1$ the sequence $r^{k}$ is null, and for $∣ r ∣ > 1$ the sequence $∣ r ∣^{k}$ diverges to $+ \infty$ : if $0 < q < 1$ , then $q^{n} \to 0$ .

[F9]

The canonical natural $ι (n) = n \cdot 1_{F}$ of a field and Canonical naturals are positive and strictly increasing: natural indices and factorials in real inequalities are read through their canonical real images, and every positive natural has positive real image.

[F10]

The exponential function is strictly increasing: $\exp$ is continuous on $R$ , so $f \in C [- 1, 1]$ and the quantities in [F1]–[F3] apply.

Verification

1.1givenF1F4F5F6F10algebra

By [F10], $f = \exp$ is continuous on $[- 1, 1]$ . Put $p_{n} (x) = \sum_{k = 0}^{n} x^{k} / k!$ , which belongs to $P_{n}$ . For $∣ x ∣ \leq 1$ , [F4]–[F6] give $∣ \exp (x) - p_{n} (x) ∣ \leq \sum_{m = 0}^{\infty} 1 / (n + 1 + m)! \leq \frac{1}{(n + 1)!} \sum_{m = 0}^{\infty} 1 / m! = e / (n + 1)!$ . The middle inequality follows from $(n + 2) \dots (n + 1 + m) \geq 1 \dots m = m!$ , including the empty product for $m = 0$ . Taking the infimum over $P_{n}$ proves $E_{n} (\exp) \leq e / (n + 1)!$ .

2.1givenF2F3step 1.1algebra

Apply the near-best estimate [F2] and then the norm bound [F3] to step 1.1. This gives $∥ \exp - I_{n}^{σ} \exp ∥_{\infty} \leq (1 + Λ_{n}^{σ}) E_{n} (\exp) \leq (1 + C \log (n + 1)) e / (n + 1)!$ for each $σ \in {R, L}$ .

3.1givenF7F8F9step 2.1algebra∎

By [F7], $\log (n + 1) = \int_{1}^{n + 1} d t / t \leq n$ and $\log 2 = \int_{1}^{2} d t / t > 0$ , so $C > 0$ . For $n \geq 1$ , $(n + 1)! = 2 \cdot 3 \dots (n + 1) \geq 2^{n}$ because this product has $n$ factors, each at least $2$ , so the bound in step 2.1 is at most $e (1 + C n) / 2^{n}$ . For $n \geq 2$ , the ratio of successive terms of $n 2^{- n}$ is $(n + 1) / (2 n) \leq 3 / 4$ , hence induction gives $n 2^{- n} \leq 2 (3 / 4)^{n - 2}$ . The geometric-sequence limit [F8] now shows both $2^{- n}$ and $n 2^{- n}$ tend to zero, so the interpolation errors tend to zero.

Depends on

Used by

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Dependency tree · two levels

83 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Lloyd N. Trefethen, Approximation Theory and Approximation Practice, Ch. 2 (standard reference, not scraped)
Tobin A. Driscoll and Richard J. Braun, Fundamentals of Numerical Computation, Problems and conditioning (standard reference, not scraped)