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.

Haar cell-average projection on the unit interval

Definition

For an integer $J \geq 0$ , put $h = 2^{- J}$ and $I_{J, k} = [k h, (k + 1) h]$ for $0 \leq k < 2^{J}$ . For an integrable real function $f$ on $[0, 1]$ , its Haar cell-average projection is the piecewise constant function $P_{J} f (x) = \frac{1}{h} \int_{I_{J, k}} f (t) d t when x \in [k h, (k + 1) h) .$ The value at $x = 1$ is defined by the average on the last cell. The half-open intervals give an unambiguous partition for point values; changing endpoint values does not change the integrals or the $L^{2}$ error.

Used by

Haar cell averages approximate smooth bounded-derivative data at first order Theorem

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

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