Woven Frames in Hilbert Spaces

• Woven frames are families of frames in a Hilbert space whose mixed interlacings always form frames with universal bounds.
• The theory extends classical frame operators with operator-theoretic, geometric, and perturbative methods to guarantee stability under partitioning.
• Applications range from time-frequency analysis to fusion and K-frame settings, ensuring robust reconstruction in various signal processing contexts.

Woven frames are families of frames indexed over a common set whose coordinatewise interlacings remain frames with uniform bounds. In the standard two-frame formulation, frames $\Phi=\{\varphi_i\}_{i\in I}$ and $\Psi=\{\psi_i\}_{i\in I}$ for a Hilbert space $H$ are woven if there exist constants $0<A\le B$ such that for every subset $\sigma\subset I$, the mixed family $$\{\varphi_i\}_{i\in \sigma}\cup \{\psi_i\}_{i\in \sigma^c}$$ is a frame for $H$ with the same bounds $A,B$. More generally, for finitely many frames $F_j=\{f_{ij}\}_{i\in I}$, $j\in[m]$, one requires that every partition $\Psi=\{\psi_i\}_{i\in I}$0 of $\Psi=\{\psi_i\}_{i\in I}$1 produce a mixed frame $\Psi=\{\psi_i\}_{i\in I}$2 with universal bounds. The subject was introduced in “Weaving Frames” and has since developed into a substantial branch of frame theory with operator-theoretic, geometric, perturbative, time-frequency, fusion, and $\Psi=\{\psi_i\}_{i\in I}$3-relative variants (Bemrose et al., 2015, Rahimi et al., 2018).

1. Foundational definition and basic equivalences

A countable family $\Psi=\{\psi_i\}_{i\in I}$4 is a frame if there exist constants $\Psi=\{\psi_i\}_{i\in I}$5 such that

$\Psi=\{\psi_i\}_{i\in I}$6

For finitely many frames $\Psi=\{\psi_i\}_{i\in I}$7, $\Psi=\{\psi_i\}_{i\in I}$8, the woven condition requires universal constants $\Psi=\{\psi_i\}_{i\in I}$9 such that for every partition $H$0 of $H$1,

$H$2

A mixed family associated with a fixed partition is a weaving frame, and if only the uniform upper inequality holds, the family is Bessel woven (Rahimi et al., 2018).

For pairs, the definition specializes to

$H$3

for every $H$4. The universal character of the constants is the substantive requirement: woven-ness is stronger than the assertion that each individual interlacing is merely a frame (Bemrose et al., 2015, Calderón et al., 2021).

An apparently weaker notion, weakly woven, asks only that every weaving be a frame, with bounds allowed to depend on the partition. For two frames this distinction collapses: weakly woven if and only if woven. This equivalence is one of the foundational structural facts of the theory and is repeatedly used in later perturbative and geometric arguments (Bemrose et al., 2015, Calderón et al., 2021).

In finite dimensions, woven-ness reduces to spanning. If $H$5 is finite-dimensional and $H$6 are frames, then they are woven if and only if every mixed family $H$7 spans $H$8. In particular, if both frames are full spark and $H$9, then they are necessarily woven (Bemrose et al., 2015). This finite-dimensional criterion should not be transferred naively to infinite dimensions: order and labeling matter, and even two orthonormal bases need not be woven if their labels are mismatched (Dörfler et al., 2017).

2. Operator-theoretic framework

A major development in the theory is the extension of the standard operator machinery of frame theory to woven families. For a fixed partition $0<A\le B$0, the weaving analysis, synthesis, and frame operators are

$0<A\le B$1

$0<A\le B$2

For the full family $0<A\le B$3, one uses the coefficient Hilbert space $0<A\le B$4 and defines

$0<A\le B$5

$0<A\le B$6

The Gram operator is understood in the induced form $0<A\le B$7 (Rahimi et al., 2018).

This framework yields direct analogues of classical frame characterizations. A family of Bessel sequences $0<A\le B$8 is a woven frame with universal bounds $0<A\le B$9 if and only if its woven frame operator satisfies

$\sigma\subset I$0

Equivalently,

$\sigma\subset I$1

Thus positivity and bounded invertibility of $\sigma\subset I$2 characterize woven frames exactly as in ordinary frame theory (Rahimi et al., 2018).

The same operator language gives reconstruction and normalization. When $\sigma\subset I$3 is woven,

$\sigma\subset I$4

and $\sigma\subset I$5 is the standard dual woven family. Moreover,

$\sigma\subset I$6

is a tight woven frame with universal bound $\sigma\subset I$7, so every woven frame can be Parsevalized (Rahimi et al., 2018).

Several operator characterizations coexist in the literature. One formulation states that woven-ness is equivalent to the existence, for every partition, of an overview operator $\sigma\subset I$8 satisfying $\sigma\subset I$9; another states that a weaving is tight exactly when its mixed frame operator is $$\{\varphi_i\}_{i\in \sigma}\cup \{\psi_i\}_{i\in \sigma^c}$$0 (Chern et al., 2019, Bhandari et al., 2018). In the pair case, surjectivity of the mixed synthesis operator is also decisive: two frames are woven if and only if the synthesis operator of each weaving has range $$\{\varphi_i\}_{i\in \sigma}\cup \{\psi_i\}_{i\in \sigma^c}$$1 (Bhandari et al., 2018).

Operator transport laws are equally important. If $$\{\varphi_i\}_{i\in \sigma}\cup \{\psi_i\}_{i\in \sigma^c}$$2 is bounded and invertible, then

$$\{\varphi_i\}_{i\in \sigma}\cup \{\psi_i\}_{i\in \sigma^c}$$3

with transformed bounds $$\{\varphi_i\}_{i\in \sigma}\cup \{\psi_i\}_{i\in \sigma^c}$$4 and $$\{\varphi_i\}_{i\in \sigma}\cup \{\psi_i\}_{i\in \sigma^c}$$5. In particular, unitary operators preserve the woven bounds (Rahimi et al., 2018). At the same time, applying an invertible operator to a single frame need not produce a woven pair with the original frame; this distinction is explicitly illustrated by counterexamples (Bhandari et al., 2018).

3. Geometry, Riesz bases, and duality

The theory becomes especially rigid for Riesz bases. If two Riesz bases are woven as frames, then every weaving is in fact a Riesz basis. Consequently, a Riesz basis cannot be woven with a redundant frame: if $$\{\varphi_i\}_{i\in \sigma}\cup \{\psi_i\}_{i\in \sigma^c}$$6 is a Riesz basis and $$\{\varphi_i\}_{i\in \sigma}\cup \{\psi_i\}_{i\in \sigma^c}$$7 is a woven partner, then $$\{\varphi_i\}_{i\in \sigma}\cup \{\psi_i\}_{i\in \sigma^c}$$8 must also be a Riesz basis (Bemrose et al., 2015). This is one of the central qualitative differences between woven Riesz bases and general woven frames.

A geometric characterization of woven Riesz bases is given in terms of separation of subspaces. For Riesz bases $$\{\varphi_i\}_{i\in \sigma}\cup \{\psi_i\}_{i\in \sigma^c}$$9 and $H$0, woven-ness is equivalent to the existence of a uniform positive lower bound on the distance, or equivalently angle, between

$H$1

for all $H$2 (Bemrose et al., 2015). This subspace-transversality viewpoint is sharpened in later work.

A recent characterization reformulates woven Riesz bases entirely in terms of the change-of-basis operator. In finite dimensions, if $H$3 and $H$4 are bases and $H$5 is the change-of-basis matrix from $H$6 to $H$7, then the bases are woven if and only if every central submatrix $H$8 indexed by $H$9 is invertible. In the infinite-dimensional Riesz-basis setting, the analogue is that all central compressions $A,B$0 must be uniformly boundedly invertible. This provides a sharp operator-theoretic criterion for woven Riesz bases (Cabrelli et al., 2024).

The same result admits a reconstruction interpretation. In finite dimensions, if $A,B$1, meaning all central submatrices are invertible, then for every $A,B$2 one can reconstruct $A,B$3 uniquely from the mixed data

$A,B$4

For the Fourier matrix $A,B$5, the paper shows $A,B$6 when $A,B$7 is prime, so any vector in $A,B$8 can be reconstructed from any weaving of its coordinates and Fourier coefficients (Cabrelli et al., 2024).

Duality supplies another axis of structure. Every Riesz basis is woven with its canonical dual (Neyshaburi et al., 2019). More generally, if a frame $A,B$9 satisfies

$F_j=\{f_{ij}\}_{i\in I}$0

then $F_j=\{f_{ij}\}_{i\in I}$1 has infinitely many dual frames woven with it; analogous statements hold for approximate duals under conditions involving $F_j=\{f_{ij}\}_{i\in I}$2 (Neyshaburi et al., 2019). A separate operator criterion shows that if $F_j=\{f_{ij}\}_{i\in I}$3 and $F_j=\{f_{ij}\}_{i\in I}$4 commutes with all partial frame operators $F_j=\{f_{ij}\}_{i\in I}$5, then $F_j=\{f_{ij}\}_{i\in I}$6 is woven with its canonical dual (Neyshaburi et al., 2019).

These positive duality results are not exhaustive. The literature presents sufficient conditions, not a complete characterization, for when a frame is woven with its canonical, alternate, or approximate dual (Neyshaburi et al., 2019).

4. Perturbation, stability, and construction methods

Perturbation theory is central because woven-ness is intended to model robust reconstruction under changes of sensing or representation systems. A clean operator criterion is the following: if $F_j=\{f_{ij}\}_{i\in I}$7 is a frame with optimal lower frame bound $F_j=\{f_{ij}\}_{i\in I}$8, and $F_j=\{f_{ij}\}_{i\in I}$9 is a Bessel sequence whose synthesis operator satisfies

$j\in[m]$0

then $j\in[m]$1 is a woven pair with explicit universal bounds

$j\in[m]$2

This formulation allows $j\in[m]$3 to be merely Bessel, and every weaving is handled uniformly because its synthesis operator is $j\in[m]$4 (Calderón et al., 2021).

The same paper derives a corollary for perturbations induced by an operator: if $j\in[m]$5 and

$j\in[m]$6

then $j\in[m]$7 and $j\in[m]$8 are woven (Calderón et al., 2021). It also proves that if $j\in[m]$9 is already woven and $\Psi=\{\psi_i\}_{i\in I}$00 are invertible operators sufficiently close in the sense that

$\Psi=\{\psi_i\}_{i\in I}$01

where $\Psi=\{\psi_i\}_{i\in I}$02 is the optimal lower woven bound, then $\Psi=\{\psi_i\}_{i\in I}$03 and $\Psi=\{\psi_i\}_{i\in I}$04 remain woven (Calderón et al., 2021).

A geometric necessary condition for a woven pair is expressed through angles between subspaces. If $\Psi=\{\psi_i\}_{i\in I}$05 is woven, then

$\Psi=\{\psi_i\}_{i\in I}$06

where $\Psi=\{\psi_i\}_{i\in I}$07 and $\Psi=\{\psi_i\}_{i\in I}$08 is an oblique projection. The paper explicitly emphasizes that this condition is necessary, not sufficient; it guarantees only a uniform frame-sequence type lower bound, not spanning of the whole space (Calderón et al., 2021).

Earlier perturbation criteria remain influential. One sufficient condition says that if

$\Psi=\{\psi_i\}_{i\in I}$09

with

$\Psi=\{\psi_i\}_{i\in I}$10

then $\Psi=\{\psi_i\}_{i\in I}$11 and $\Psi=\{\psi_i\}_{i\in I}$12 are woven (Neyshaburi et al., 2019). Another operator-closeness result states that if a frame $\Psi=\{\psi_i\}_{i\in I}$13 and a bounded operator $\Psi=\{\psi_i\}_{i\in I}$14 satisfy

$\Psi=\{\psi_i\}_{i\in I}$15

then $\Psi=\{\psi_i\}_{i\in I}$16 and $\Psi=\{\psi_i\}_{i\in I}$17 are woven (Bemrose et al., 2015).

Construction theory complements perturbation theory. One finite-dimensional method shows that if $\Psi=\{\psi_i\}_{i\in I}$18 is a frame with $\Psi=\{\psi_i\}_{i\in I}$19, then $\Psi=\{\psi_i\}_{i\in I}$20 is also a frame and the two are woven (Bhandari et al., 2018). Another construction uses idempotent operators: if $\Psi=\{\psi_i\}_{i\in I}$21 satisfies $\Psi=\{\psi_i\}_{i\in I}$22, has closed range, and $\Psi=\{\psi_i\}_{i\in I}$23, then for a frame $\Psi=\{\psi_i\}_{i\in I}$24 of $\Psi=\{\psi_i\}_{i\in I}$25, the family $\Psi=\{\psi_i\}_{i\in I}$26 is also a frame for $\Psi=\{\psi_i\}_{i\in I}$27 and is woven with $\Psi=\{\psi_i\}_{i\in I}$28 (Bhandari et al., 2018). The hypotheses are essential; the same paper provides counterexamples when $\Psi=\{\psi_i\}_{i\in I}$29 fails or $\Psi=\{\psi_i\}_{i\in I}$30 (Bhandari et al., 2018).

5. Extensions beyond ordinary frames

The woven paradigm extends systematically to operator-valued, fusion, and $\Psi=\{\psi_i\}_{i\in I}$31-relative settings. In fusion-frame language, a weighted family of closed subspaces $\Psi=\{\psi_i\}_{i\in I}$32 is a fusion frame if

$\Psi=\{\psi_i\}_{i\in I}$33

Two such families are woven fusion frames if every partition-induced mixture is again a fusion frame with universal bounds (Rahimi et al., 2018). The main bridge theorem states that woven fusion frames are equivalent to woven ordinary frames obtained from weighted local frame sequences or weighted orthonormal bases spanning the constituent subspaces (Rahimi et al., 2018). The same paper develops perturbation criteria, restriction to closed subspaces by intersection, and a woven Riesz decomposition notion for families of subspaces (Rahimi et al., 2018).

For $\Psi=\{\psi_i\}_{i\in I}$34-frames, the lower bound is measured by $\Psi=\{\psi_i\}_{i\in I}$35: $\Psi=\{\psi_i\}_{i\in I}$36 A family of $\Psi=\{\psi_i\}_{i\in I}$37-frames is $\Psi=\{\psi_i\}_{i\in I}$38-woven if every partitioned mixture is again a $\Psi=\{\psi_i\}_{i\in I}$39-frame with uniform $\Psi=\{\psi_i\}_{i\in I}$40-frame bounds. The central operator characterization is

$\Psi=\{\psi_i\}_{i\in I}$41

where $\Psi=\{\psi_i\}_{i\in I}$42 is the synthesis operator of the weaving determined by $\Psi=\{\psi_i\}_{i\in I}$43. In the two-frame case, weakly $\Psi=\{\psi_i\}_{i\in I}$44-woven and $\Psi=\{\psi_i\}_{i\in I}$45-woven are equivalent, and the paper proves a Paley–Wiener type perturbation theorem in which quantitative closeness of two $\Psi=\{\psi_i\}_{i\in I}$46-frames yields $\Psi=\{\psi_i\}_{i\in I}$47-woven-ness (Deepshikha et al., 2017).

The corresponding subspace-based theory is that of $\Psi=\{\psi_i\}_{i\in I}$48-fusion frames. A weighted collection $\Psi=\{\psi_i\}_{i\in I}$49 is a $\Psi=\{\psi_i\}_{i\in I}$50-fusion frame if

$\Psi=\{\psi_i\}_{i\in I}$51

Two such families are woven if

$\Psi=\{\psi_i\}_{i\in I}$52

for every $\Psi=\{\psi_i\}_{i\in I}$53. The theory includes transport under operators $\Psi=\{\psi_i\}_{i\in I}$54, equivalence with ordinary woven fusion frames on $\Psi=\{\psi_i\}_{i\in I}$55 when $\Psi=\{\psi_i\}_{i\in I}$56 is closed, Paley–Wiener type operator perturbation, and erasure criteria (Bhandari et al., 2019).

The generalized-frame, or $\Psi=\{\psi_i\}_{i\in I}$57-frame, setting displays both continuity with and divergence from ordinary weaving. A $\Psi=\{\psi_i\}_{i\in I}$58-frame is a sequence of operators $\Psi=\{\psi_i\}_{i\in I}$59 satisfying

$\Psi=\{\psi_i\}_{i\in I}$60

Two $\Psi=\{\psi_i\}_{i\in I}$61-frames are woven when every subset $\Psi=\{\psi_i\}_{i\in I}$62 satisfies

$\Psi=\{\psi_i\}_{i\in I}$63

A key result is that weaving $\Psi=\{\psi_i\}_{i\in I}$64-frames are characterized by weaving of the induced ordinary frames $\Psi=\{\psi_i\}_{i\in I}$65 and $\Psi=\{\psi_i\}_{i\in I}$66. Another is that a $\Psi=\{\psi_i\}_{i\in I}$67-frame and any dual $\Psi=\{\psi_i\}_{i\in I}$68-frame are woven, with universal bounds

$\Psi=\{\psi_i\}_{i\in I}$69

At the same time, some rigidities of ordinary Riesz-basis weaving fail in the generalized setting: a $\Psi=\{\psi_i\}_{i\in I}$70-frame can weave with a $\Psi=\{\psi_i\}_{i\in I}$71-Riesz basis without itself being a $\Psi=\{\psi_i\}_{i\in I}$72-Riesz basis, and a weaving of two $\Psi=\{\psi_i\}_{i\in I}$73-exact frames need not be $\Psi=\{\psi_i\}_{i\in I}$74-exact (Deepshikha et al., 2020).

The continuous end of the theory is represented by woven continuous controlled $\Psi=\{\psi_i\}_{i\in I}$75-g-fusion frames. These are measurable families $\Psi=\{\psi_i\}_{i\in I}$76 over a measure space $\Psi=\{\psi_i\}_{i\in I}$77 satisfying, for every partition $\Psi=\{\psi_i\}_{i\in I}$78 of $\Psi=\{\psi_i\}_{i\in I}$79,

$\Psi=\{\psi_i\}_{i\in I}$80

The theory includes operator characterizations, invariance under invertible maps, restriction to subspaces, deletion principles, sufficient conditions via positivity and orthogonality, and a perturbation theorem (Ghosh et al., 2021).

6. Time-frequency constructions and broader packetized generalizations

Time-frequency analysis has supplied some of the most concrete woven constructions. For single-window Gabor systems

$\Psi=\{\psi_i\}_{i\in I}$81

woven-ness asks that every partition of the common index set $\Psi=\{\psi_i\}_{i\in I}$82 produce a frame. This idea is extended to multi-window Gabor systems

$\Psi=\{\psi_i\}_{i\in I}$83

where at each phase-space point one chooses an entire local window family rather than a single atom (Dörfler et al., 2017).

The decisive tool in this setting is the time-frequency localization operator

$\Psi=\{\psi_i\}_{i\in I}$84

The paper establishes two sufficient criteria ensuring that families of variable-window localization operators satisfy

$\Psi=\{\psi_i\}_{i\in I}$85

One is a Hilbert-space norm criterion: if $\Psi=\{\psi_i\}_{i\in I}$86, then corresponding families of localization operators remain uniformly stable. The other is a pointwise phase-space criterion comparing $\Psi=\{\psi_i\}_{i\in I}$87 to the Gaussian STFT $\Psi=\{\psi_i\}_{i\in I}$88 (Dörfler et al., 2017).

After spectral truncation of the localization operators, the retained eigenfunctions generate multi-window Gabor frames, and because the criteria hold for arbitrary assignments $\Psi=\{\psi_i\}_{i\in I}$89, all such frames are woven. This yields explicit woven examples from Hermite functions associated with localization on elliptic domains; the same framework extends to chirped or rotated Gaussians and generalized Hermite-type eigenfunctions (Dörfler et al., 2017).

A newer generalization replaces vectors by subspaces and introduces information packets: families of closed subspaces $\Psi=\{\psi_i\}_{i\in I}$90 such that every $\Psi=\{\psi_i\}_{i\in I}$91 admits an unconditionally convergent expansion

$\Psi=\{\psi_i\}_{i\in I}$92

Frames and fusion frames occur as special cases, but information packets are strictly more general. Woven information packets are defined by requiring that every partition-based recombination of $\Psi=\{\psi_i\}_{i\in I}$93 such families remain an information packet (Christensen et al., 9 Sep 2025).

This broader notion yields explicit wavelet and Gabor constructions. For wavelets, if $\Psi=\{\psi_i\}_{i\in I}$94 is bounded, compactly supported, and satisfies local two-sided power bounds near the origin, then for every $\Psi=\{\psi_i\}_{i\in I}$95, the $\Psi=\{\psi_i\}_{i\in I}$96 collections

$\Psi=\{\psi_i\}_{i\in I}$97

are woven information packets, where

$\Psi=\{\psi_i\}_{i\in I}$98

For Gabor systems, the paper proves that for any fixed $\Psi=\{\psi_i\}_{i\in I}$99, one can find a Gabor frame split into $H$00 woven information packets, but no single Gabor frame can work for all $H$01 because the density condition forces $H$02 (Christensen et al., 9 Sep 2025).

These time-frequency and packetized constructions underscore a recurring theme of the field: woven-ness is easiest to verify when a highly structured geometry permits one to control arbitrary recombinations uniformly. This suggests a broader interpretation of woven frames as a theory of partition-uniform stability, with ordinary frames, Riesz bases, Gabor systems, fusion frames, $H$03-relative systems, and information packets all appearing as instances of the same organizing principle.