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.
Convolution of two functions on
Definition
Let be measurable functions. When, for a given , the function
is measurable and integrable on , the convolution of and at is
For classes, later items make the representative convention explicit.
Depends on
Used by
- The Duhamel heat potential Definition
- The heat evolution Hₜ of initial data Definition
- The inhomogeneous dyadic frequency partition and its Littlewood-Paley operators Definition
- The Lusin area function for a fixed admissible kernel and aperture Definition
- Cotlar's inequality for maximal truncations Lemma
- Fourier pairing for a finite measure and Schwartz data Lemma
- The heat kernel semigroup identity Γₜ*Γₛ=Γₜ₊ₛ Lemma
- Convolution on L¹(ℝⁿ) is bilinear, commutative, and associative Proposition
- Complex-time heat operators form a bounded holomorphic semigroup Theorem
- The support of a convolution lies in the closure of the support sumset Theorem
Dependency tree · two levels
2 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
- Walter Rudin, Real and Complex Analysis, 3rd ed. (standard reference, not scraped)