Translation Invariance of Finitely Generated Shift-Invariant Spaces

Definitions and mathematical statement

The source asks for a general characterization of finitely generated shift-invariant spaces $V$ invariant under translation by $1/n$. It does not explicitly give the ambient function space, scalar field, dimension, generator conditions or parameter range. The following fixes the customary one-dimensional Hilbert-space reading conditionally, rather than claiming it is the full unstated scope.

Let $T_t f(x)=f(x-t)$ on $L^2({\mathbb{R}})$, fix an integer $n\geq2$, and let $\varphi_1,\ldots,\varphi_r\in L^2({\mathbb{R}})$ with $r<\infty$. Set \[ V=\overline{\operatorname{span}}\{T_k\varphi_j: k\in{\mathbb{Z}},\ 1\leq j\leq r\}. \] Thus finite generation is by integer translates and closure is in $L^2$, with no frame, Riesz-basis, support or regularity assumption added. The target is necessary-and-sufficient, appropriately checkable conditions on the generators or intrinsic data of $V$ for \[ T_{1/n}V=V. \] In this setting inclusion $T_{1/n}V\subseteq V$ is equivalent to equality, because its $n$th power is the already invertible integer translation. The case $n=1$ is automatic. Fixing one $n$ is different from requiring every $n$, or all real translations, and those quantifiers are not interchanged.

Under the Fourier convention $\widehat f(\xi)=\int f(x)e^{-2\pi i x\xi}\,dx$, this is invariance of the Fourier image of $V$ under multiplication by $e^{-2\pi i\xi/n}$. This merely specifies the property; it is not a supplied complete generator classification. The phrase "almost translation-invariant" in the assessment is not given an approximation norm and is not substituted for the exact fractional-translation invariance in the question.

The literature triage checked 21 August 2026 reports special cases and general invariance criteria, while saying it located no source establishing the conjectured comparable characterization at the stated generality. The intended extra scope or form of that characterization is not specified. A failure to locate a source is not an independent proof of mathematical open status, and no new classification or literature judgment is asserted here.

Short English statement

The question seeks a checkable characterization of exact fractional-translation invariance. In the standard $L^2({\mathbb{R}})$ interpretation, the number of generators is finite but no frame or regularity hypotheses are assumed. The source does not specify a broader ambient category or what extra form of characterization is intended.

Sources

• Local UnsolvedMath registry, permanent ID 30000035, source code catalogue entry, supplied Oberwolfach Reports record; updated 16 September 2026.
• 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.
• Supplied report reference: https://ems.press/content/serial-article-files/45879.
• Supplied reference label "Shift-invariant spaces on LCA groups": https://doi.org/10.1016/j.jfa.2009.11.013.
• Literature triage checked 21 August 2026, derived from the supplied OWR curation; independently verify time-sensitive status and cited claims. Full author/theorem entries are not supplied.