Woven weighted exponentials

Abstract: Let $f$ and $g$ be nonzero functions in $L<sup>2([0,1])$. The \emph{woven weighted exponential system} (associated with $f$ and $g$) is defined by $$\Wc(f,g)=\bigset{\set{fe^{2πi nt}}<em>{n\in J} \cup \set{ge^{2πi nt}}</em>{n\in J<sup>c}\,|\,J\subset\Z}.$$ We say that $\Wc(f,g)$ is \emph{wovenly complete}, (resp. \emph{wovenly minimal}, a \emph{woven frame}) if the weaving $$\set{fe^{2πi nt}}<em>{n\in J} \cup \set{ge^{2πi nt}}</em>{n\in J<sup>c}$$ is complete, (resp. minimal, a frame) for all $J\subseteq \Z.$ In this paper, we study conditions that imply certain approximation properties of $\Wc(f,g)$, such as completeness, minimality and the frame property. We first provide a complete characterization of the woven weighted exponential systems that are wovenly complete. We also show that $\Wc(f,g)$ is a woven frame if $f/g$ is strictly positive or strictly negative over $[0,1].$ Additionally, several counterexamples are provided to show that certain seemingly correct conditions do not imply the desired approximation properties of $\Wc(f,g).$ All results presented in this paper apply equivalently to systems of regular translates and Gabor systems at critical density in $L<sup>2(\R)$.

• The paper characterizes woven completeness spectrally: both weights must be nonzero almost everywhere, with no nonzero function separating their Fourier supports into complementary index sets.
• It proves that frame generators with a strictly positive or negative ratio produce woven Riesz bases, and establishes quantitative Paley–Wiener perturbation bounds for stability under controlled changes.
• Through fiberization and the Zak transform, the results extend to regular translates and critical-density Gabor systems, while open problems remain about minimality and completeness beyond the frame setting.

Overview and motivation

This paper studies how two systems of weighted exponentials in $L^2[0,1]$ interact under "weaving," the operation of partitioning their index set $\mathbb{Z}$ into a subset $J$ and its complement and taking the union $$\mathcal{E}(f,g,J) = \{fe^{2\pi i n t}\}_{n\in J} \cup \{ge^{2\pi i n t}\}_{n\in J^c}$$. The woven system $(f,g)$ is the collection of all such weavings over all $J \subseteq \mathbb{Z}$, and it is called wovenly complete, wovenly minimal, or a woven frame when every weaving retains the corresponding property. The notion of weaving was introduced by Bemrose et al. for frames generally; this paper specializes to weighted exponentials, where the one-weight theory is completely understood via Heil's characterization: $(f)$ is complete iff $f \neq 0$ a.e., minimal iff $1/f \in L^2$, Bessel iff $f \in L^\infty$, and a frame (indeed a Riesz basis) iff $\mathbb{Z}$0 a.e.

The choice of setting is not merely technical convenience. Via the fiberization map, systems of regular translates $\mathbb{Z}$1 in $\mathbb{Z}$2 are unitarily equivalent to weighted exponential systems with generator $\mathbb{Z}$3, where $\mathbb{Z}$4; via the Zak transform, Gabor systems at critical density are equivalent to two-dimensional weighted exponential systems. Consequently every result proved here transfers verbatim to those settings, which is where the paper's applications lie.

Woven Bessel sequences and woven orthonormal bases

The Bessel case admits a clean characterization: $\mathbb{Z}$5 is a woven Bessel sequence if and only if both $\mathbb{Z}$6 and $\mathbb{Z}$7 belong to $\mathbb{Z}$8. The paper also records an instructive failure of stability: if $\mathbb{Z}$9 and $J$0 in $J$1 with uniformly bounded $J$2 norms, then $J$3 is woven Bessel, but without that uniform bound the conclusion fails — the example $J$4 shows that woven Bessel sequences (even woven frames) are not stable under arbitrary $J$5 limits.

For orthonormal bases, the rigidity of the one-weight condition $J$6 a.e. yields a complete characterization: $J$7 is a woven orthonormal basis if and only if $J$8 a.e. and $J$9 for some constant phase $$\mathcal{E}(f,g,J) = \{fe^{2\pi i n t}\}_{n\in J} \cup \{ge^{2\pi i n t}\}_{n\in J^c}$$0. The proof exploits Fourier uniqueness: cross-orthogonality between elements indexed by $$\mathcal{E}(f,g,J) = \{fe^{2\pi i n t}\}_{n\in J} \cup \{ge^{2\pi i n t}\}_{n\in J^c}$$1 and $$\mathcal{E}(f,g,J) = \{fe^{2\pi i n t}\}_{n\in J} \cup \{ge^{2\pi i n t}\}_{n\in J^c}$$2 forces $$\mathcal{E}(f,g,J) = \{fe^{2\pi i n t}\}_{n\in J} \cup \{ge^{2\pi i n t}\}_{n\in J^c}$$3 to have only the zeroth Fourier coefficient nonzero. In contrast to the Bessel case, woven orthonormal bases are stable under $$\mathcal{E}(f,g,J) = \{fe^{2\pi i n t}\}_{n\in J} \cup \{ge^{2\pi i n t}\}_{n\in J^c}$$4 limits of generators.

Woven completeness: a full characterization

The central structural result is a complete characterization of woven completeness. The paper first establishes the sufficient sign condition: if $$\mathcal{E}(f,g,J) = \{fe^{2\pi i n t}\}_{n\in J} \cup \{ge^{2\pi i n t}\}_{n\in J^c}$$5 a.e. or $$\mathcal{E}(f,g,J) = \{fe^{2\pi i n t}\}_{n\in J} \cup \{ge^{2\pi i n t}\}_{n\in J^c}$$6 a.e., then $$\mathcal{E}(f,g,J) = \{fe^{2\pi i n t}\}_{n\in J} \cup \{ge^{2\pi i n t}\}_{n\in J^c}$$7 is wovenly complete. This condition is sufficient but not necessary — the pair $$\mathcal{E}(f,g,J) = \{fe^{2\pi i n t}\}_{n\in J} \cup \{ge^{2\pi i n t}\}_{n\in J^c}$$8, $$\mathcal{E}(f,g,J) = \{fe^{2\pi i n t}\}_{n\in J} \cup \{ge^{2\pi i n t}\}_{n\in J^c}$$9 yields Riesz bases for every weaving by the Kadets 1/4-Theorem, yet $(f,g)$0 is not real-valued.

The characterization itself is spectral. Writing $(f,g)$1 for the $(f,g)$2-th Fourier coefficient of $(f,g)$3, the theorem states that $(f,g)$4 is wovenly complete if and only if both $(f,g)$5 and $(f,g)$6 are nonzero a.e. and there is no nonzero $(f,g)$7 and no subset $(f,g)$8 with $(f,g)$9 and $J \subseteq \mathbb{Z}$0 in $J \subseteq \mathbb{Z}$1. The intuition is that weaving permutes all possible spectral shifts of the generators, so completeness of every weaving requires the spectra of $J \subseteq \mathbb{Z}$2 and $J \subseteq \mathbb{Z}$3 to overlap "highly" — no weight can separate them onto complementary frequency sets. A useful corollary of the negative direction: if $J \subseteq \mathbb{Z}$4 for all $J \subseteq \mathbb{Z}$5, then $J \subseteq \mathbb{Z}$6 fails to be wovenly complete; in particular, trigonometric polynomials with disjoint spectra never form a wovenly complete system.

As a concrete application, the paper proves that for $J \subseteq \mathbb{Z}$7 and $J \subseteq \mathbb{Z}$8 with $J \subseteq \mathbb{Z}$9, the system $(f)$0 is wovenly complete. The argument is a Riemann–Lebesgue iteration: any separating weight $(f)$1 would satisfy $(f)$2 along arithmetic progressions of step $(f)$3, forcing all Fourier coefficients of $(f)$4 to vanish.

Woven minimality and its relation to completeness

For frames of weighted exponentials, the paper proves a duality lemma: $(f)$5 is wovenly complete (respectively wovenly minimal) if and only if $(f)$6 is. This enables the key equivalence: for frames, $(f)$7 is wovenly complete if and only if it is wovenly $(f)$8-minimal, meaning that no nontrivial $(f)$9 scalar sequence produces a vanishing woven linear combination. Since ordinary minimality implies $f \neq 0$0-minimality, this yields the corollary that woven minimality implies woven completeness for frames of weighted exponentials — so a wovenly minimal pair of frames is automatically wovenly exact.

Two caveats qualify this picture. First, the converse fails without the frame hypothesis: $f \neq 0$1 and $f \neq 0$2 generate a wovenly complete but not wovenly minimal system, since neither $f \neq 0$3 nor $f \neq 0$4 is a frame. Second, individual exactness does not force woven minimality even in principle: the paper constructs explicit piecewise-defined weights $f \neq 0$5 and $f \neq 0$6 for which both $f \neq 0$7 and $f \neq 0$8 are exact, yet a specific weaving $f \neq 0$9 fails to be minimal, using a half-periodic function $1/f \in L^2$0 with $1/f \in L^2$1 to exhibit a redundant element. The authors explicitly leave open whether woven minimality always implies woven completeness, and whether woven completeness implies woven minimality, for general systems of weighted exponentials.

Woven frames

The main positive result on frames states: if $1/f \in L^2$2 and $1/f \in L^2$3 are frames and $1/f \in L^2$4 is strictly positive or strictly negative a.e., then $1/f \in L^2$5 is a woven frame — indeed every weaving is a Riesz basis. The proof is a Neumann-series argument: after rescaling so that $1/f \in L^2$6 (legitimate because frames impose uniform bounds on $1/f \in L^2$7), the synthesis operator of any weaving factors as $1/f \in L^2$8, and the perturbation term has operator norm strictly less than one, uniformly in $1/f \in L^2$9. An immediate corollary is that any two real-valued continuous frame generators form a woven frame. Two further consequences deserve note:

• Reciprocal pairs: if $f \in L^\infty$0 is a frame, then $f \in L^\infty$1 is a woven Riesz basis. Combined with the completeness characterization, this forces the spectra $f \in L^\infty$2 and $f \in L^\infty$3 to be highly overlapping whenever $f \in L^\infty$4 has absolutely summable Fourier coefficients and never vanishes — in particular they can never be disjoint.
• Perturbation stability: if $f \in L^\infty$5 with $f \in L^\infty$6 is a woven frame, and more generally a Paley–Wiener type criterion holds: if $f \in L^\infty$7 has bounds $f \in L^\infty$8 and $f \in L^\infty$9 satisfies $\mathbb{Z}$00 with $\mathbb{Z}$01, then $\mathbb{Z}$02 is a woven frame with explicit bounds $\mathbb{Z}$03 and $\mathbb{Z}$04.

It should be emphasized that the strict-sign hypothesis is again sufficient rather than necessary, as the Kadets-type example above already produces woven Riesz bases from complex phases.

Applications to translates and Gabor systems

Via fiberization, the results translate directly to shift-invariant systems. For $\mathbb{Z}$05 with $\mathbb{Z}$06 a.e., the paper answers affirmatively a question left open by prior work: if $\mathbb{Z}$07 lie in the closed span of $\mathbb{Z}$08 with $\mathbb{Z}$09, then every partial-translate union $\mathbb{Z}$10 is complete in $\mathbb{Z}$11. Under the stronger quantitative hypothesis $\mathbb{Z}$12 and $\mathbb{Z}$13, every such weaving is a frame for the closed span and is $\mathbb{Z}$14-minimal, since $\mathbb{Z}$15 is strictly positive. Both statements extend to dilated translates $\mathbb{Z}$16 through the $\mathbb{Z}$17-fiberization.

For Gabor systems at critical density, the Zak transform reduces matters to two-dimensional weighted exponentials. For $\mathbb{Z}$18 in the Feichtinger algebra $\mathbb{Z}$19 whose Fourier transforms are compactly supported in $\mathbb{Z}$20 with $\mathbb{Z}$21 of constant strict sign a.e., every weaving $\mathbb{Z}$22 is complete in $\mathbb{Z}$23.

Limitations and open questions

Several restrictions bound the scope of the results. The sign conditions ($\mathbb{Z}$24 or $\mathbb{Z}$25 of constant strict sign) are sufficient only, and the Kadets example shows genuinely complex-phase woven frames exist outside their reach. The equivalence of woven completeness and woven $\mathbb{Z}$26-minimality, and hence the implication from woven minimality to woven exactness, require both constituent systems to be frames; the counterexample with $\mathbb{Z}$27, $\mathbb{Z}$28 shows the frame hypothesis cannot simply be dropped. Individual exactness of the two systems does not imply woven minimality, as the explicit piecewise construction demonstrates. The authors state two unresolved questions explicitly: whether woven minimality implies woven completeness for general systems of weighted exponentials, and whether woven completeness implies woven minimality for frames. Finally, the perturbation theorem's smallness condition $\mathbb{Z}$29 is quantitative and may be far from sharp.

Conclusion

The paper delivers a complete spectral characterization of woven completeness for weighted exponential systems, a Neumann-series proof that consistent-sign frame generators weave into frames (indeed Riesz bases) with uniform bounds, an equivalence between woven completeness and woven $\mathbb{Z}$30-minimality in the frame setting, and Paley–Wiener stability of woven frames. Through fiberization and the Zak transform, these results apply equally to regular translates and critical-density Gabor systems, yielding new completeness and frame guarantees for partial translates within shift-invariant spaces. The remaining gaps — necessity of sign conditions, the minimality–completeness implications outside the frame setting — delineate precisely what remains to be understood about woven approximation properties.