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.

Periodic multilevel transform using the four-tap Daubechies filter

Definition

Let $J \geq 3$ and $N = 2^{J}$ . Starting with $a^{(0)} = x \in R^{N}$ , for $ℓ = 1, \dots, J - 2$ apply one-level periodic analysis with the D4 pair of Explicit four-tap Daubechies filter and its high-pass partner to the current approximation vector $a^{(ℓ - 1)}$ , whose length is $N / 2^{ℓ - 1} \geq 8$ . Store the detail vector $d^{(ℓ)}$ and set the next approximation to $a^{(ℓ)}$ . Stop with the length-four vector $a^{(J - 2)}$ . The output consists of that coarse vector and all detail vectors, ordered from coarsest details to finest. At every stage indices are reduced modulo that stage's current length; there is no padding or symmetric extension.

Depends on

Used by

Dependency tree · two levels

4 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, Handouts 8 and 15 (standard reference, not scraped)
Stéphane Mallat, A Wavelet Tour of Signal Processing, 3rd ed., Chapter 9 (standard reference, not scraped)