Open Problems in Wavelet Theory — Problem 3

Definitions and mathematical statement

The supplied excerpt does not define its wavelet system or its realization of $H^2({\mathbb{R}})$. Under the standard dyadic, integer-translation convention and the positive-frequency realization of the Hardy space, the question has the following precise formulation. These conventions require confirmation against the cited source.

Use the Fourier transform $\widehat f(\xi)=\int_{{\mathbb{R}}}f(x)e^{-2\pi i x\xi}\,{\,\mathrm{d}} x$, extended to $L^2({\mathbb{R}};{\mathbb{C}})$, and set \[ H^2({\mathbb{R}})=\{f\in L^2({\mathbb{R}};{\mathbb{C}}):\widehat f(\xi)=0\text{ for almost every }\xi<0\}. \] The Schwartz class is \[ \mathcal S({\mathbb{R}})=\left\{f\in C^\infty({\mathbb{R}};{\mathbb{C}}):\sup_{x\in{\mathbb{R}}}|x^m f^{(n)}(x)|<\infty\text{ for all }m,n\in{\mathbb{Z}}_{\geq0}\right\}. \] For $\psi\in H^2({\mathbb{R}})$, define \[ \psi_{j,k}(x)=2^{j/2}\psi(2^j x-k),\qquad j,k\in{\mathbb{Z}}. \] This family is a Riesz basis of $H^2({\mathbb{R}})$ if its closed linear span is $H^2({\mathbb{R}})$ and there exist constants $0<A\leq B<\infty$ such that, for every finitely supported family $(c_{j,k})_{(j,k)\in{\mathbb{Z}}^2}$ of complex numbers, \[ A\sum_{j,k\in{\mathbb{Z}}}|c_{j,k}|^2 \leq \left\|\sum_{j,k\in{\mathbb{Z}}}c_{j,k}\psi_{j,k}\right\|_{L^2({\mathbb{R}})}^2 \leq B\sum_{j,k\in{\mathbb{Z}}}|c_{j,k}|^2. \] Equivalently, a Riesz basis is the image of an orthonormal basis under a bounded invertible linear operator on the Hilbert space.

Question, with the conventions above. Does there exist \[ \psi\in\mathcal S({\mathbb{R}})\cap H^2({\mathbb{R}}) \] such that $(\psi_{j,k})_{j,k\in{\mathbb{Z}}}$ is a Riesz basis of $H^2({\mathbb{R}})$?

The source also defines Riesz wavelets for $L^2({\mathbb{R}})$, but the question here specifically concerns $H^2({\mathbb{R}})$. Every Riesz basis has a dual Riesz basis; the source emphasizes that the dual of a wavelet Riesz basis need not itself be a wavelet system. When it is, the source calls the wavelet biorthogonal. The question above imposes no such additional requirement on the dual.

Short English statement

Does the real-line Hardy space admit a wavelet Riesz basis generated by a Schwartz function? The displayed formulation uses the standard dyadic wavelet system and positive-frequency Hardy space, conventions that the supplied excerpt leaves unspecified. No requirement that the dual basis also be a wavelet system is imposed.

Sources

• Local UnsolvedMath registry, permanent ID $801164$, exact source problem code catalogue entry, titled Open Problems in Wavelet Theory --- Problem 3. The supplied registry labels this problem open; that label is supplied curation, not an independently verified current literature assessment.
• Bownik and Rzeszotnik, Open Problems in Wavelet Theory (2019), Problem 3, https://arxiv.org/abs/1905.08230. The supplied provenance records extraction from arXiv LaTeX and an access date of 2026-10-16. This access information is supplied provenance; no independent source visit or verification is asserted.
• The excerpt attributes the existence problem to Seip through the reference label [Seip]. No bibliographic entry for [Seip] is supplied, so the original reference and its precise scope remain unverified. The registry separately lists Bownik and Rzeszotnik as proposers and 2019 as the proposed year.