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.
Gibbs phenomenon
Dirichlet partial sums are controlled by the oscillatory kernel , whose sign changes create the persistent overshoot quantified in Gibbs overshoot at a piecewise C^1 jump. By contrast, The Fejer kernel is a positive approximate identity and The Poisson kernel on the circle is a positive approximate identity show that Fejer and Poisson kernels are positive and have total mass one. Their summation methods therefore average across a jump instead of amplifying it, and the preceding convergence theorems return the midpoint value rather than a fixed overshoot.
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Dependency tree · two levels
9 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
- Loukas Grafakos, Classical Fourier Analysis, 3rd ed. (standard reference, not scraped)
- Richard S. Laugesen, Harmonic Analysis Lecture Notes (standard reference, not scraped)