$L^p$ Extensions for Shift-Invariant Spaces

Definitions and mathematical statement

For a finite family $\Phi=(\phi_1,\ldots,\phi_N)$ of complex-valued functions in $L^p({\mathbb{R}}^d)$, a conventional norm-closed shift-generated space, when $1\leq p<\infty$, is \[ S_p(\Phi)=\overline{\operatorname{span}\{\phi_j(\,\cdot-k): 1\leq j\leq N,\ k\in{\mathbb{Z}}^d\}}^{\,L^p}. \] Such a space is finitely generated and invariant under integer translations. For $A=(a_{ij})\in{\mathbb{C}}^{m\times N}$ define the linear combinations \[ \psi_i=\sum_{j=1}^N a_{ij}\phi_j\quad(1\leq i\leq m), \qquad \Psi=A\Phi. \] The source asks for an extension to $L^p({\mathbb{R}}^d)$, $p\ne2$, of an $L^2$ theorem about generic linear combinations of generators. It does not state that theorem, the allowed number $m$, or what property the combinations must preserve.

For example, if the intended property is generation, define the length of a nonzero finitely generated space $V$ to be the minimum number $\ell_p(V)$ of generators. One possible precise target would ask, for appropriate $\ell_p(V)\leq m\leq N$, whether \[ S_p(A\Phi)=S_p(\Phi)=V \quad\text{for Lebesgue-almost every }A\in{\mathbb{C}}^{m\times N}. \] This is an explicitly conditional interpretation, not a recovered statement of the missing theorem. Preservation of frame bounds or stable translates is stronger than preservation of the closed span. Likewise "generic" could mean full measure, a residual set, or a specified algebraic condition; the source does not choose one. The displayed norm-closure convention does not decide the additional issues for $p=\infty$ or quasi-Banach $0<p<1$, and the original $p$ range is unspecified.

The supplied literature triage checked 21 August 2026 calls the general $L^p$ extension open while recording substantial results for $p=1$ and specialized shift-invariant settings. Thus the record must not be read as claiming the entire area is untouched, nor as supplying a full theorem for every $p\ne2$.

Short English statement

Extend a generic-linear-combination result for finitely generated $L^2$ shift-invariant spaces to other $L^p$ spaces. The missing $L^2$ theorem, meaning of genericity and preservation property prevent a unique precise target. The supplied assessment records partial $p=1$ and specialized results.

Sources

• Local UnsolvedMath registry, permanent ID 30000038, exact source code catalogue entry; $L^p$ Extensions for Shift-Invariant Spaces. Proposed year 2004, status partially_solved; corrected_verified extraction restores function-space notation and $p\ne2$.
• 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 literature triage checked 2026-08-21: "Oberwolfach report", https://doi.org/10.4171/owr/2004/10; "Shift-invariant spaces in $L^1$", https://doi.org/10.1006/jath.2002.3711; "Related $L^p$ work", https://doi.org/10.1016/j.jmaa.2005.05.055. The dated OWR-derived triage requires independent verification of time-sensitive status and cited claims. Its description of later work is retained as supplied, without independently resolving publication chronology.