Alphabeta MathDefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-01
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.
Translation of a function on
Definition
Let and let be a function. The translate of by is the function defined by
This fixes the sign convention used throughout the page: translating by shifts the graph of in the positive -direction.
Used by
- Expanding bumps lose tightness Counterexample
- Rellich compactness fails on ℝⁿ by translations Counterexample
- Strong continuity does not imply operator-norm continuity Counterexample
- The tightness hypothesis of the Fr'echet--Kolmogorov criterion cannot be dropped Counterexample
- The translation semigroup is not analytic Counterexample
- The translation semigroup is not strongly continuous on L-infinity Counterexample
- Difference quotients on a shrunken domain Definition
- Affine flux reduces the entropy semigroup to translation Example
- The right-translation semigroup on Lp has the weak derivative as generator Example
- Translation and differentiation symbols Example
- Centring by translation and modulation preserves the variance product Lemma
- Complex translation, convolution, approximate identities, and mollification Lemma
- Continuous compactly supported functions are translation-continuous in Lᵖ Lemma
- Difference-quotient calculus: integration by parts, product rule, commutation Lemma
- Mollification rates for compactly supported Slobodeckij functions Lemma
- Relative compactness forces uniform translation continuity in Lᵖ Lemma
- The cutoff difference-quotient commutator estimate Lemma
- The translation estimate for W^1,p functions on ℝⁿ Lemma
- ‖τ_h f - f‖ₚ → 0 in Lᵖ(ℝⁿ) as h → 0, for 1 ≤ p < ∞ Theorem
- Poincare-Wirtinger on bounded connected extension domains by Rellich compactness Theorem
- The Fr'echet--Kolmogorov compactness criterion in Lᵖ(ℝⁿ) Theorem
- The negative Laplacian of the fundamental solution is the unit Dirac distribution Theorem
- Translation, modulation, linear dilation and reflection laws Theorem
Dependency tree · 0 levels
Nothing. This result depends on no other item in the library.
Sources
- Richard L. Wheeden and Antoni Zygmund, Measure and Integral: An Introduction to Real Analysis (standard reference, not scraped)