Open Problems in Wavelet Theory — Problem 7

Definitions and mathematical statement

Let $H$ denote the ambient Hilbert space intended by the source, and write $W_{j,k}\psi=\psi_{j,k}$, for $(j,k)\in{\mathbb{Z}}^2$, for its wavelet generation rule. The supplied record does not specify $H$, the dilation and translation parameters, or the normalization of $W_{j,k}$. The formulation below is relative to that intended setup; it does not quantify over arbitrary choices of these data.

A generator $\psi\in H$ is Bessel with bound strictly less than one if there exists $B\in[0,1)$ such that \[ \sum_{(j,k)\in{\mathbb{Z}}^2}\left|\langle f,W_{j,k}\psi\rangle_H\right|^2\le B\|f\|_H^2 \qquad\text{for every }f\in H. \] A countable indexed family $(u_\lambda)_{\lambda\in\Lambda}\subset H$ is a Parseval frame for $H$ if \[ \sum_{\lambda\in\Lambda}|\langle f,u_\lambda\rangle_H|^2=\|f\|_H^2 \qquad\text{for every }f\in H. \] Problem. For every $\psi\in H$ satisfying the Bessel inequality above for some $B<1$, does there exist a single generator $\psi'\in H$, using the same wavelet generation rule, such that \[ \sum_{(j,k)\in{\mathbb{Z}}^2}|\langle f,W_{j,k}\psi\rangle_H|^2+ \sum_{(j,k)\in{\mathbb{Z}}^2}|\langle f,W_{j,k}\psi'\rangle_H|^2 =\|f\|_H^2 \qquad\text{for every }f\in H? \] Here the combined system is interpreted as the indexed family consisting of both wavelet systems, with multiplicities retained. The source displays a set union and does not explicitly discuss multiplicities. Specification of the ambient wavelet setup remains necessary for a fully determined mathematical statement.

Short English statement

Can every Bessel wavelet with an upper bound strictly below one be supplemented by one further wavelet generator so that the two wavelet systems together form a Parseval frame? The ambient space and wavelet conventions are absent from the supplied excerpt. The registry labels the problem open; this is a supplied status, not an independently verified assessment.

Sources

• Local UnsolvedMath registry, permanent ID $801168$, source code catalogue entry, entitled Open Problems in Wavelet Theory --- Problem 7. The supplied record labels it open and lists Bownik and Rzeszotnik as proposers, with proposed year 2019.
• Bownik and Rzeszotnik, Open Problems in Wavelet Theory (2019), Problem 7. Supplied source URL: https://arxiv.org/abs/1905.08230. The provenance records access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance claims, not independently verified access or source checks.
• The source passage attributes the problem to Christensen and his collaborators and cites the labels [CKK1] and [CKK2]. Their bibliographic entries and links are not supplied, so these references cannot be identified reliably from this record alone.