Open Problems in Wavelet Theory — Problem 13

Definitions and mathematical statement

The supplied excerpt does not specify the ambient dimension or the dilation and translation conventions. The following statement uses the conventional one-dimensional dyadic interpretation of an orthonormal wavelet, with complex-valued functions; these conventions require confirmation against the full source.

Let ${\mathbb{N}}_0=\{0,1,2,\ldots\}$. Define the Schwartz class by \[ \mathcal S({\mathbb{R}})=\left\{f\in C^\infty({\mathbb{R}};{\mathbb{C}}):\sup_{x\in{\mathbb{R}}}(1+|x|)^m|f^{(n)}(x)|<\infty\text{ for every }m,n\in{\mathbb{N}}_0\right\}. \] For $\psi\in\mathcal S({\mathbb{R}})$, use the Fourier transform convention \[ \widehat\psi(\xi)=\int_{{\mathbb{R}}}\psi(x)e^{-2\pi i x\xi}\,{\,\mathrm{d}} x. \] Call $\psi$ an orthonormal dyadic wavelet if \[ \left\{\psi_{j,k}:j,k\in{\mathbb{Z}}\right\},\qquad \psi_{j,k}(x)=2^{j/2}\psi(2^j x-k), \] is an orthonormal basis of $L^2({\mathbb{R}};{\mathbb{C}})$.

Question (Bownik and Rzeszotnik, Problem 13, 2019). Is it true that every orthonormal dyadic wavelet $\psi\in\mathcal S({\mathbb{R}})$ has compactly supported Fourier transform? Equivalently, does every such $\psi$ satisfy \[ \exists B>0\quad\forall\xi\in{\mathbb{R}},\qquad |\xi|>B\Longrightarrow\widehat\psi(\xi)=0? \] The source motivates the question by stating that the only known construction of Schwartz-class wavelets is a Meyer wavelet construction, which is band-limited. This is the supplied source's historical qualification, not an independently verified assertion about all constructions or the current literature. The registry labels the problem open; that label is supplied metadata, not an independently verified mathematical assessment.

Short English statement

Under the explicitly stated one-dimensional dyadic convention, must every orthonormal wavelet that is smooth and rapidly decreasing have a Fourier transform that vanishes outside a bounded interval? The excerpt leaves the wavelet conventions implicit, so this interpretation requires source confirmation.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $801174$, source problem code catalogue entry, title Open Problems in Wavelet Theory --- Problem 13. The record attributes the problem to Bownik and Rzeszotnik, with proposed year 2019, and labels it open.
• Bownik and Rzeszotnik, Open Problems in Wavelet Theory (2019), Problem 13. Supplied source URL: https://arxiv.org/abs/1905.08230. The record reports access on 2026-10-16 and extraction by arxiv_latex. These are supplied provenance details; the source was not independently visited or verified for this draft.