DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge 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.

Absolute summability of a multi-index detail family

Definition

Let $A \subseteq N_{0}^{d}$ and let $a_{ℓ} \geq 0$ for $ℓ \in A$ . The nonnegative sum $\sum_{ℓ \in A} a_{ℓ}$ is defined when the set ${\sum_{ℓ \in F} a_{ℓ} : F \subseteq A finite}$ is bounded above; its value is the supremum of that set in the complete ordered field $R$ (Complete ordered field (least-upper-bound property)). The empty finite sum is $0$ . This definition does not require an enumeration of $A$ .

For the Smolyak details $δ_{ℓ} (f)$ of Smolyak cubature on a finite downward-closed index set, absolute detail summability means that $\sum_{ℓ \in A} ∣ δ_{ℓ} (f) ∣$ is defined for $A = N_{0}^{d}$ . For an omitted set $A = N_{0}^{d} ∖ Λ$ , the same definition gives the omitted absolute-detail tail.

Depends on

Used by

Absolute detail reconstruction and omitted-tail error bound Theorem

Dependency tree · two levels

16 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

T. Gerstner and M. Griebel, Sparse Grids, Encyclopedia of Quantitative Finance, §Classical Sparse Grids (standard reference, not scraped)
T. Gerstner and M. Griebel, Dimension-Adaptive Tensor-Product Quadrature, §2 (standard reference, not scraped)