Wavelet Structure of Canonical Tight Frames

Definitions and mathematical statement

Question. Does the canonical normalized tight frame associated with every wavelet frame always have wavelet structure? For a frame $(f_i)_{i\in I}$ in a Hilbert space $H$, let $Sf=\sum_i\langle f,f_i\rangle f_i$ be its positive invertible frame operator, with inner product linear in the first argument. Its canonical normalized tight frame is $(S^{-1/2}f_i)_i$, where the inverse square root is the positive one; normalized tight means Parseval:\[\sum_i|\langle f,S^{-1/2}f_i\rangle|^2=\|f\|^2\quad(f\in H).\] This is different from the canonical dual $(S^{-1}f_i)_i$. The question concerns retention of wavelet structure, not existence of the canonical Parseval frame.

In the usual one-dimensional dyadic single-generator interpretation, put $D f(x)=2^{1/2}f(2x)$, $T^k f(x)=f(x-k)$, and suppose $(D^jT^k\psi)_{j,k\in{\mathbb{Z}}}$ is a frame for $L^2({\mathbb{R}})$. The question then asks whether there is a generator $\psi^{\mathrm{tight}}\in L^2({\mathbb{R}})$ such that\[S^{-1/2}D^jT^k\psi=D^jT^k\psi^{\mathrm{tight}}\quad\text{for every }j,k\in{\mathbb{Z}}.\] With this fixed indexing, $\psi^{\mathrm{tight}}=S^{-1/2}\psi$, so the condition is equivalently $S^{-1/2}D^jT^k\psi=D^jT^kS^{-1/2}\psi$ for all indices. The source does not specify the dimension, dilation, number of generators, or whether wavelet structure permits reindexing; this standard interpretation is therefore explicit and conditional, rather than a silent restriction of the general wording.

The supplied literature triage checked 2026-08-21 labels the item open. It reports that canonical dual wavelet frames need not preserve wavelet structure in general, but that no theorem or counterexample settling the distinct canonical-tight question was found in the checked literature. A negative result about $S^{-1}$ is not asserted here to settle the stated $S^{-1/2}$ question. The assessment is dated and requires independent verification.

Short English statement

Applying the inverse square root of a wavelet frame operator always gives a Parseval frame. The question asks whether this particular frame is again generated by a wavelet with the same dilation and translation structure. It must be distinguished from the canonical dual-frame question.

Sources

• Local UnsolvedMath registry, permanent ID 30000043, code catalogue entry; Wavelet Structure of Canonical Tight Frames, Oberwolfach Reports Open Problems.
• Mini-Workshop: Wavelets and Frames (2004), Oberwolfach Reports, DOI 10.4171/owr/2004/10, pp. 479--544. https://doi.org/10.4171/owr/2004/10. Proposed year 2004; no proposer named.
• Supplied literature label OWR 10/2004: https://doi.org/10.4171/OWR/2004/10.
• Supplied literature label Canonical dual wavelet frames: https://doi.org/10.1006/acha.2002.0381.
• Supplied literature label Wavelet frame survey: https://arxiv.org/abs/1901.08024.
• Literature triage checked 2026-08-21, status open; record updated 2026-09-16. Verify the dated assessment and cited claims independently.