Exponential Frames for Measurable Sets

Definitions and mathematical statement

For a measurable set $A\subset{\mathbb{R}}$ with $0<|A|<\infty$, equip $L^2(A)$ with Lebesgue measure and set $e_\gamma(x)=e^{2\pi\mathrm i\gamma x}$ on $A$. An unweighted exponential frame indexed by a countable discrete set $\Gamma\subset{\mathbb{R}}$ means that there are $0<a\leq b<\infty$ such that \[ a\|f\|_{L^2(A)}^2\leq \sum_{\gamma\in\Gamma}\left|\int_A f(x)e^{-2\pi\mathrm i\gamma x}\,dx\right|^2 \leq b\|f\|_{L^2(A)}^2 \qquad\forall f\in L^2(A). \] The frame is tight if $a=b$ can be chosen. The two source questions, under the finite-measure interpretation supplied by the assessment, are whether every such $A$ admits an exponential frame, and whether every such $A$ admits a tight exponential frame. Frequencies and bounds may depend on $A$; boundedness of $A$ is not assumed. Weighted exponentials, an orthogonal basis, and a merely continuous frame are different demands.

The source also phrases the question as existence of $\Gamma$ and $\eta\in H$ whose translates form a frame or a tight frame. An abstract Hilbert space alone does not define translations. One precise realization of its stated equivalence is the translation-invariant space \[ H_A=\{g\in L^2({\mathbb{R}}):\widehat g=0\text{ a.e. outside }A\}, \qquad \widehat\eta=1_A, \] with $T_\gamma\eta(t)=\eta(t-\gamma)$. The unitary Fourier transform takes these translates to $e^{-2\pi\mathrm i\gamma x}1_A(x)$, so their frame and tight-frame properties are exactly the corresponding exponential properties with frequency set $-\Gamma$. This explains the equivalence in a specified model, without claiming that arbitrary $H$ or arbitrary cyclic vectors give the same unweighted question. The extracted sentence leaves the original $H$, action, vector restrictions and assumptions on $A$ unstated.

The supplied literature triage checked 21 August 2026 reports existence of discrete exponential frames for every finite-measure set, including unbounded ones, as proved, but says that the stronger general tight-frame demand is not resolved. These strength levels and their dated supplied status must remain separate.

Short English statement

Does every positive finite-measure subset of the real line admit a discrete exponential frame, and can one always choose it tight? The supplied assessment reports the ordinary-frame assertion proved, while retaining the general tight-frame question. The translation formulation needs a specified Hilbert-space representation.

Sources

• Local UnsolvedMath registry, permanent ID 30000044, exact source code catalogue entry; Exponential Frames for Measurable Sets. Proposed year 2004, status partially_solved; corrected_verified extraction preserves both frame strengths and the asserted equivalence.
• 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: "Exponential frames for unbounded sets", https://arxiv.org/abs/1410.5693; "Proceedings of the AMS version", https://doi.org/10.1090/proc/12868. This dated OWR-derived curation requests independent verification of time-sensitive status and cited claims.