Alphabeta MathHow 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.
Completeness and spectral rates require law-specific hypotheses
Remarks
The finite moment and Gram hypotheses in Finite product-law polynomial-chaos input data construct only the declared finite polynomial family. Completeness in requires a separate density proof for that law; no such conclusion is inferred from moments alone. Likewise, an exponential or algebraic spectral rate requires quantified mixed-derivative or analyticity bounds. The weighted least-squares procedure in this page is a separately defined collocation method. No general equality with a spectral Galerkin method is asserted; exact agreement is limited to the polynomial degree conditions stated in the Gaussian-quadrature corollary.
Used by
Nothing in the library uses this result yet.
Dependency tree · 0 levels
Nothing. This result depends on no other item in the library.
Sources
- Dongbin Xiu and George Em Karniadakis, The Wiener–Askey Polynomial Chaos for Stochastic Differential Equations, §§4–6 (standard reference, not scraped)
- Ivo Babuška, Fabio Nobile, and Raúl Tempone, A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data, §4 (standard reference, not scraped)