DefinitionDefinition: Literature-sourcedProof: 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.

Smooth functions on the unit interval with bounded derivative data

Definition

Let $q \geq 1$ be an integer and let $U \subseteq R$ be an open interval containing $[0, 1]$ . A real function $f : U \to R$ belongs to the smooth derivative-bound class $S^{q}$ when it is $C^{q}$ on $U$ in the sense of Continuity and derivative on a real interval. For $0 \leq r \leq q$ define $M_{r} (f) = \sup_{0 \leq t \leq 1} ∣ f^{(r)} (t) ∣$ , with $f^{(0)} = f$ . These derivative bounds are finite because the derivatives are continuous on the compact interval. They are bounds on the restriction to $[0, 1]$ , not a norm on arbitrary extensions to $U$ ; no claim that this class exhausts $W^{q, \infty} (0, 1)$ is part of the definition.

Depends on

Continuity and derivative on a real interval

Used by

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

Gilbert Strang and Kevin Amaratunga, Wavelets and Filter Banks, MIT OpenCourseWare 18.327, Handout 13 (standard reference, not scraped)
Stéphane Mallat, A Wavelet Tour of Signal Processing, 3rd ed., Chapter 9 (standard reference, not scraped)