Dual Wavelet Frames and Tight Gabor Frames

Definitions and mathematical statement

Three requests. Characterize when the canonical dual of a wavelet frame has wavelet structure; extend descriptions of dual wavelet frames to subspaces; and construct an explicit tight Gabor frame with exponentially decaying generator and Fourier transform. All three requests are retained. The source does not fix dimension, wavelet dilation, multiplicity of generators, subspace class, or Gabor lattice.

A frame $(f_i)_{i\in I}$ for a Hilbert space $H$ satisfies $A\|f\|^2\le\sum_i|\langle f,f_i\rangle|^2\le B\|f\|^2$ for some $0<A\le B<\infty$ and all $f\in H$. Its frame operator is $Sf=\sum_i\langle f,f_i\rangle f_i$, using the inner product linear in its first argument, and its canonical dual is $(S^{-1}f_i)_i$. For illustration, in the usual one-dimensional dyadic model let $D f(x)=2^{1/2}f(2x)$, $T^k f(x)=f(x-k)$, and $\psi_{j,k}=D^jT^k\psi$, $j,k\in{\mathbb{Z}}$. The first request asks for criteria ensuring a generator $\widetilde\psi$ with $S^{-1}\psi_{j,k}=D^jT^k\widetilde\psi$ for all $j,k$, necessarily $\widetilde\psi=S^{-1}\psi$. This dyadic single-generator notation illustrates wavelet structure without asserting that it is the only intended setting.

For a closed subspace $M\subset H$, a frame for $M$ obeys the frame inequalities on $M$, with its frame operator inverted on $M$. Dual frames in $M$ yield $f=\sum_i\langle f,g_i\rangle f_i$ for every $f\in M$. The second request is for descriptions of dual wavelet frames in the relevant subspaces; the necessary invariance, wavelet structure, and whether the intended duals are ordinary subspace duals or oblique duals are unspecified. No inverse on $M^\perp$ is presumed.

In a standard one-dimensional Gabor formulation, the third request seeks explicit $g\in L^2({\mathbb{R}})$, $a,b>0$, and $A>0$ such that\[\sum_{m,n\in{\mathbb{Z}}}\left|\left\langle f,e^{2\pi i mbx}g(x-na)\right\rangle\right|^2=A\|f\|_2^2\quad\text{for all }f\in L^2({\mathbb{R}}),\] together with exponential decay, for example $|g(x)|\le C_1e^{-c_1|x|}$ and $|\widehat g(\xi)|\le C_2e^{-c_2|\xi|}$, with positive constants and $\widehat g(\xi)=\int_{\mathbb{R}} g(x)e^{-2\pi ix\xi}\,dx$. The source does not specify pointwise versus weighted-norm decay, a prescribed lattice, or what qualifies as an explicit or elementary construction. These displayed conventions are an explanatory formulation, not a claim for every $a,b$.

The supplied literature triage, dated 2026-08-21, labels the multipart problem partially solved: structural characterizations of dual frames and subspace frames and decay criteria are known in important settings, but the requested general subspace extension and explicit elementary tight Gabor construction are described as not jointly settled. Precise theorem scopes are absent, and this dated assessment is not independently verified here.

Short English statement

Determine when canonical duals retain wavelet structure, describe wavelet duals on subspaces, and give an explicit tight Gabor construction with exponential decay in both time and frequency. The excerpt leaves the wavelet and Gabor parameters and the intended subspace and decay conventions unspecified.

Sources

• Local UnsolvedMath registry, permanent ID 30000036, code catalogue entry; Dual Wavelet Frames and Tight Gabor 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 Oberwolfach report: https://ems.press/content/serial-article-files/45879.
• Supplied literature label Christensen--Eldar: https://doi.org/10.1007/s00041-005-4030-0.
• Supplied literature label Gabor decay paper: https://doi.org/10.1007/BF02511155.
• Literature triage checked 2026-08-21, status partially_solved; record updated 2026-09-16. The references' detailed theorem statements are not supplied; verify the dated assessment independently.