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

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Floater–Hormann error bounds for a bounded-variation highest derivative

Statement

Let $n \geq 1$ , $0 \leq d < n$ , and $x_{0} < \dots < x_{n}$ be real nodes in $[a, b] = [x_{0}, x_{n}]$ . Put $h = \max_{0 \leq j < n} (x_{j + 1} - x_{j})$ , $m = n - d$ , and $ϵ = 1$ if $m$ is even and $ϵ = 0$ if $m$ is odd. Let $f \in C^{d + 1} ([a, b])$ , set $g = f^{(d + 1)}$ , and suppose $g$ has finite total variation $V = {Var}_{[a, b]} (g)$ as defined in Bounded variation and total variation on an interval. For a continuous $q$ on $[a, b]$ , write $∥ q ∥_{\infty} = \sup_{x \in [a, b]} ∣ q (x) ∣$ . If $d \geq 1$ , then $∥ f - r_{d} ∥_{\infty} \leq \frac{h^{d + 1}}{d + 1} (V + ϵ ∥ g ∥_{\infty}),$ If $d = 0$ , let $β$ be the local mesh ratio in Local mesh ratio for degree-zero Floater–Hormann interpolation. Then $∥ f - r_{0} ∥_{\infty} \leq h (1 + β) (V + ϵ ∥ f^{'} ∥_{\infty}) .$ No mesh-ratio restriction is needed when $d \geq 1$ . For $d = 0$ , a fixed global mesh-ratio bound $ρ = h / h_{\min}$ implies $β \leq ρ$ by A global mesh ratio bounds the local degree-zero ratio.

Facts & Assumptions

Given: Ordered distinct nodes $x_{0}, \dots, x_{n}$ , $n \geq 1$ , $0 \leq d < n$ , $f \in C^{d + 1} ([x_{0}, x_{n}])$ , and $g = f^{(d + 1)}$ of finite total variation $V$ .

[F1]

Floater–Hormann weights equal the local-polynomial blend weights and Newton remainder for a Floater–Hormann window polynomial: off the nodes, $f (x) - r_{d} (x) = (\sum_{i = 0}^{n - d} (- 1)^{i} G_{i}) / D (x)$ with $G_{i} = f [x_{i}, \dots, x_{i + d}, x]$ and $D (x) = \sum_{i} λ_{i} (x)$ .

[F2]

Bounded variation controls adjacent Hermite–Genocchi windows: the adjacent divided differences satisfy $\sum_{i = 0}^{n - d - 1} ∣ G_{i + 1} - G_{i} ∣ \leq V / (d + 1)!$ .

[F3]

Hermite–Genocchi simplex formula for divided differences: an order- $(d + 1)$ divided difference is the simplex integral of $g$ ; the simplex volume is $1 / (d + 1)!$ .

[F4]

Floater–Hormann barycentric denominator lower bound: on an open gap, $∣ D (x) ∣ \geq 1 / (d! h^{d + 1})$ if $d \geq 1$ , and $∣ D (x) ∣ \geq 1 / [h (1 + β)]$ if $d = 0$ .

[F5]

Floater–Hormann interpolants have no real poles: the quotient is continuous on $[a, b]$ and interpolates the data at every node.

[F6]

Bounded variation and total variation on an interval: $V$ is the supremum of the sums of absolute successive increments over all finite partitions of $[a, b]$ .

[F7]

Higher derivatives and the classes $C^{k}$ and $C^{\infty}$ : $C^{d + 1}$ means derivatives through order $d + 1$ exist and are continuous on the closed interval; under the additional condition $f \in C^{d + 2}$ , the recursive definition gives $g^{'} = f^{(d + 2)}$ .

[F8]

Mean value theorem on a closed real interval: on every partition subinterval, a function continuous on the closed subinterval and differentiable on its interior has an increment equal to its derivative at an interior point times the interval length.

[F9]

The natural numbers $N$ (von Neumann): $N$ contains $0$ and is closed under successor, so $d + 1 \in N$ .

[F10]

Extreme value theorem: a continuous real function on a nonempty compact subset of $R$ attains a greatest and a least value: a continuous real function on a nonempty compact interval has a finite attained maximum and minimum.

Proof

1.1givenF1F2F9F10algebra

By [F9], $d + 1 \in N$ . Fix $x$ in an open node gap. By [F7], the derivatives in [F1] exist and are continuous. By [F1], $f (x) - r_{d} (x) = N (x) / D (x)$ with $N (x) = \sum_{i = 0}^{m} (- 1)^{i} G_{i}$ . [F10] makes the derivative sup norm in the statement finite. Pair successive terms as $(G_{0} - G_{1}) + (G_{2} - G_{3}) + \dots$ . The sum of the absolute values of these pair differences is bounded by the full adjacent-increment sum in [F2], hence by $V / (d + 1)!$ . If $m$ is even there is one unpaired term $G_{m}$ ; otherwise there is none.

2.1givenF3F10step 1.1algebra

The integral formula [F3] and its simplex volume give $∣ G_{m} ∣ \leq ∥ g ∥_{\infty} / (d + 1)!$ . Thus in both parity cases $∣ N (x) ∣ \leq \frac{V + ϵ ∥ g ∥_{\infty}}{(d + 1)!} .$ This bounds the numerator in either parity case.

3.1givenF1F4step 2.1algebra

If $d \geq 1$ , divide step 2.1 by the lower bound $∣ D (x) ∣ \geq 1 / (d! h^{d + 1})$ from [F4] to obtain $∣ f (x) - r_{d} (x) ∣ \leq \frac{h^{d + 1}}{d + 1} (V + ϵ ∥ g ∥_{\infty}) .$ This is the required open-gap bound for $d \geq 1$ .

3.2givenF1F4step 2.1algebra

If $d = 0$ , $g = f^{'}$ and [F4] instead gives $∣ f (x) - r_{0} (x) ∣ \leq h (1 + β) (V + ϵ ∥ f^{'} ∥_{\infty}) .$ This is the required open-gap bound for $d = 0$ .

4.1givenF5F6F7F8F10step 3.1step 3.2algebra∎

At every node the error is zero by [F5]. Since $r_{d}$ has no real poles, $f - r_{d}$ is continuous on the compact interval and has a finite supremum by [F10]. The estimates on the open gaps therefore imply the stated supremum bounds. If $f \in C^{d + 2}$ , then for any partition [F8] gives $∣ g (t_{i + 1}) - g (t_{i}) ∣ \leq ∥ f^{(d + 2)} ∥_{\infty} (t_{i + 1} - t_{i})$ ; summing and taking the supremum yields $V \leq (b - a) ∥ f^{(d + 2)} ∥_{\infty}$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

55 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

Michael S. Floater and Kai Hormann, Barycentric rational interpolation with no poles and high rates of approximation, Sections 2–3 (standard reference, not scraped)