Open Problems in Wavelet Theory — Problem 8

Definitions and mathematical statement

For $0<b<1$, let \[ E_b=(-1,-b)\cup(b,1),\qquad \widehat{\psi_b}=\mathbf{1}_{E_b}, \] where $\mathbf{1}_{E_b}$ denotes the indicator function of $E_b$, and $\psi_b\in L^2({\mathbb{R}})$.

The supplied excerpt does not specify the Fourier transform normalization or the dilation and translation conventions for a frame wavelet. Consequently, the following explicit formulation is conditional on the standard dyadic, integer-translation interpretation with Fourier transform \[ \widehat f(\xi)=\int_{{\mathbb{R}}}f(x)e^{-2\pi i x\xi}\,{\,\mathrm{d}} x \] for $f\in L^1({\mathbb{R}})\cap L^2({\mathbb{R}})$, extended to $L^2({\mathbb{R}})$. Under this interpretation, define \[ \psi_{b,j,k}(x)=2^{j/2}\psi_b(2^j x-k),\qquad j,k\in{\mathbb{Z}}. \] The function $\psi_b$ is a frame wavelet if there exist real constants $0<A_b\le B_b<\infty$ such that, for every $f\in L^2({\mathbb{R}})$, \[ A_b\|f\|_2^2 \le \sum_{j\in{\mathbb{Z}}}\sum_{k\in{\mathbb{Z}}} \left|\langle f,\psi_{b,j,k}\rangle\right|^2 \le B_b\|f\|_2^2, \qquad \langle f,g\rangle=\int_{{\mathbb{R}}}f(x)\overline{g(x)}\,{\,\mathrm{d}} x. \] Determine exactly the set \[ \left\{b\in\left(\frac18,\frac16\right]: \psi_b\text{ is a frame wavelet}\right\}. \] The lower endpoint $1/8$ is excluded and the upper endpoint $1/6$ is included. The Fourier and affine-system conventions above require confirmation against the original source before this conditional formulation can be accepted as the intended statement.

Short English statement

Determine precisely which parameters $1/8<b\le 1/6$ make the function with Fourier transform $\mathbf{1}_{(-1,-b)\cup(b,1)}$ a frame wavelet for $L^2({\mathbb{R}})$. The explicit frame inequalities use a stated conventional interpretation whose agreement with the original source remains to be checked.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $801169$, source problem code catalogue entry, titled Open Problems in Wavelet Theory --- Problem 8.
• Bownik and Rzeszotnik, Open Problems in Wavelet Theory (2019), Problem 8. Supplied source URL: https://arxiv.org/abs/1905.08230. The record reports access on 2026-10-16 and extraction from arXiv LaTeX; these are supplied provenance claims, not independently verified access.
• The supplied problem text attributes the problem to Weber and the first author, Bownik, with reference label [BW]. No bibliographic details for [BW] are supplied.
• The registry labels the problem open. This is a supplied status label, not an independently verified assessment of the literature.