Optimal K-Dual Frames

• The paper shows that optimal K-dual frames rely on factorizing the operator K through analysis and synthesis operators to achieve reconstruction on the range of K.
• It demonstrates that canonical inverses and minimal-norm Douglas factorizations are key in reducing erasure errors and minimizing reconstruction energy.
• The study highlights that optimality can be canonical, non-unique, or explicitly non-canonical, depending on the structure of the frame operator and the chosen error minimization criteria.

Optimal $K$-dual frames are dual reconstruction systems associated with $K$-frames, $K$-$g$-frames, $K$-fusion frames, and their continuous analogues, in which the reconstruction target is $Kf$ rather than $f$. Their common operator-theoretic core is a factorization of $K$ through analysis and synthesis operators, such as $K=T_FT_G^*$, $K=TR^*$, or $K$0, together with the fact that the relevant frame operator is often invertible only on $K$1 rather than on the whole ambient space (Neyshaburi et al., 2017, Cheshmavar et al., 2020, Mondal et al., 4 Aug 2025). In this literature, “optimal” can mean several distinct things: a canonical inverse-type choice built from the frame operator, a minimal-norm Douglas factorization, a minimizer of convex spectral functionals, or a dual minimizing the effect of coefficient erasures. A recurrent misconception is that optimality is always synonymous with the canonical dual; the cited work shows a more differentiated picture, with canonical optimality in some regimes, non-uniqueness in others, and explicit non-canonical optima in still others.

1. Operator-theoretic foundation

A $K$2-frame for a Hilbert space $K$3 is a sequence $K$4 satisfying

$K$5

with $K$6. The lower estimate is tied to $K$7, so the frame controls reconstruction on the geometry induced by $K$8, not necessarily on all of $K$9 in the ordinary frame sense (Shamsabadi et al., 2018). Associated synthesis, analysis, and frame operators are defined exactly as in ordinary frame theory, but the frame operator need not be invertible on all of $K$0; if $K$1 has closed range, it is invertible on $K$2 (Shamsabadi et al., 2018).

The notion of $K$3-dual is not entirely uniform across the literature. In one Hilbert-space formulation, a Bessel sequence $K$4 is a $K$5-dual of $K$6 if

$K$7

whereas another formulation uses

$K$8

At the operator level these are variants of the same guiding idea: $K$9 is reconstructed through an overview–analysis factorization, and the difference lies in whether the synthesis side is explicitly projected to $g$0 (Shamsabadi et al., 2018, Neyshaburi et al., 2017).

The same pattern persists in generalized settings. For $g$1-$g$2-frames and continuous $g$3-$g$4-frames, the defining factorization is

$g$5

with $g$6 and $g$7 the synthesis operators of the primal and dual systems. In the continuous $g$8-$g$9-frame setting, the existence criterion is the range inclusion

$K$0

and the totality of duals is the set of adjointable solutions of $K$1 (Cheshmavar et al., 2020). This factorization viewpoint is the basic language in which optimality is later formulated.

2. Canonical and structurally preferred $K$2-duals

The canonical $K$3-dual is the most systematic generalization of the canonical dual frame. For a $K$4-frame $K$5 with frame operator $K$6, one canonical $K$7-dual is

$K$8

and $K$9 is a $Kf$0-dual in the sense

$Kf$1

This formula makes explicit that the inverse is taken on $Kf$2, not on the whole space (Shamsabadi et al., 2018).

For $Kf$3-fusion frames $Kf$4, the canonical $Kf$5-dual fusion frame is

$Kf$6

and the defining duality identity is

$Kf$7

In the operator form used by the paper, this is

$Kf$8

The same work shows that a Bessel fusion sequence is a $Kf$9-fusion frame iff $f$0, so dual existence is again a range-factorization statement (Shamsabadi et al., 2018).

A stronger optimality statement appears in the $f$1-fusion literature through the Douglas minimal solution. If $f$2 is a $f$3-fusion frame and $f$4, then the distinguished solution $f$5 minimizes the norm among all such factorizations. Under the condition

$f$6

the canonical $f$7-dual coincides with the dual generated by this minimal-norm Douglas solution. The same paper also proves a least-squares statement for $f$8-resolutions of $f$9: among all families $K$0 resolving $K$1, the one induced by $K$2 minimizes the $K$3-energy of the representing vector in $K$4 (Neyshaburi et al., 2017). This is an explicit operator-theoretic sense in which the canonical dual is optimal.

In continuous $K$5-$K$6-frame theory, the canonical choice is expressed through the frame operator $K$7. If $K$8 is a $K$9-$K=T_FT_G^*$0-$K=T_FT_G^*$1-frame, then for every $K=T_FT_G^*$2,

$K=T_FT_G^*$3

is a $K=T_FT_G^*$4-$K=T_FT_G^*$5-$K=T_FT_G^*$6-dual pair, and the family $K=T_FT_G^*$7 is singled out as the $K=T_FT_G^*$8-$K=T_FT_G^*$9-$K=TR^*$0-canonical dual (Cheshmavar et al., 2020). In the locally $K=TR^*$1-algebra setting, a canonical-looking $K=TR^*$2-dual $K=TR^*$3-frame is obtained from the restriction of the frame operator to $K=TR^*$4: $K=TR^*$5 again emphasizing that inversion is performed on $K=TR^*$6 rather than globally (Eljazzar et al., 2024).

3. Generalizations and non-uniqueness mechanisms

The theory extends in several directions without changing its algebraic core. In Hilbert $K=TR^*$7-modules, continuous $K=TR^*$8-$K=TR^*$9-frames are defined by

$K$00

and duality is encoded by

$K$01

The key structural theorem states that a $K$02-$K$03-Bessel system is a $K$04-$K$05-$K$06-frame iff $K$07, provided $K$08 is orthogonally complemented (Cheshmavar et al., 2020).

In Hilbert modules over locally $K$09-algebras, fixing a $K$10-orthonormal basis $K$11 produces a $K$12-operator $K$13 such that $K$14. The $K$15-$K$16-frame condition is then exactly

$K$17

and a $K$18-dual $K$19-frame sequence $K$20 with $K$21 is characterized by

$K$22

This formulation makes the family of all duals transparent: every admissible $K$23 solving $K$24 yields a dual (Eljazzar et al., 2024).

Non-uniqueness is not exceptional but structural. In the same locally $K$25-module setting, if $K$26 is a $K$27-dual $K$28-frame of $K$29 and $K$30 is another $K$31-frame sequence with $K$32, then $K$33 is again a $K$34-dual $K$35-frame. Affine combinations of duals are also duals under $K$36 (Eljazzar et al., 2024). In Hilbert spaces, approximate $K$37-$K$38-duals are defined by

$K$39

and the Neumann series then produces an exact dual

$K$40

from any approximate one (Cheshmavar et al., 2018). For continuous $K$41-frames, a $K$42-dual pair satisfies

$K$43

and when $K$44 has closed range the paper constructs the dual

$K$45

which is the continuous analogue of canonical inversion on the range (Rahimlou et al., 2019).

4. Other optimality paradigms

One influential optimality paradigm does not start from $K$46-duality, but from the spectrum of dual frame operators. For an ordinary frame $K$47, the set of dual frame operators is

$K$48

so the canonical dual operator $K$49 is minimal in the Löwner order. Imposing a trace lower bound $K$50, the paper identifies a vector $K$51 that is submajorization-minimal in the spectral feasible set, and proves that every convex increasing tracial functional

$K$52

is minimized exactly at duals with spectrum $K$53 (Massey et al., 2011). The paper explicitly notes that $K$54-duals are not treated there; the natural extension is an affine slice

$K$55

and this suggests a majorization-based theory of optimal $K$56-duals.

A different line of work emphasizes Parseval and equal-norm $K$57-frames. For a Parseval $K$58-frame, the frame operator is

$K$59

and the paper establishes a Naimark-type model

$K$60

for an orthonormal basis $K$61 of a larger Hilbert space and the projection $K$62 onto $K$63 (Sadri et al., 2021). It also proves that any finite set of $K$64-norm vectors can be extended to a $K$65-norm frame, and under suitable orthogonality hypotheses a Parseval $K$66-frame has infinitely many equal-norm $K$67-dual frames (Sadri et al., 2021). For a Parseval $K$68-frame $K$69 with synthesis operator $K$70 and a $K$71-dual $K$72 that is a Parseval $K$73-frame with synthesis operator $K$74, the identity

$K$75

holds for all $K$76, isolating a duality error term determined entirely by $K$77 (Sadri et al., 2021).

5. Erasure-optimal $K$78-duals

The most explicit finite-dimensional theory of optimal $K$79-dual frames appears in the study of erasures. For an $K$80 $K$81-dual pair $K$82 in a finite-dimensional Hilbert space and a single-erasure mask $K$83, the error operator is

$K$84

and its operator norm equals $K$85 when the erased index is $K$86. If $K$87 is positive, the global one-erasure operator-norm optimum over all $K$88 $K$89-dual pairs is

$K$90

and a pair is optimal iff

$K$91

The same optimal value occurs for the one-erasure spectral-radius problem, and in that case optimality is equivalent to 1-uniformity,

$K$92

Thus 1-uniform $K$93-dual pairs are simultaneously the spectrally optimal and operator-norm optimal objects for one erasure (Mondal et al., 4 Aug 2025).

For two erasures, the same paper introduces 2-uniform $K$94-dual pairs, defined by the constancy of

$K$95

When $K$96 and a 2-uniform $K$97-dual pair exists, the lower bound for the optimal two-erasure spectral radius is attained, with the explicit formula depending on $K$98 and $K$99 (Mondal et al., 4 Aug 2025).

The fixed-frame problem is subtler. For a Parseval $K$00-frame $K$01, the canonical $K$02-dual is $K$03. The paper shows that $K$04 is a one-erasure operator-norm optimal $K$05-dual under the geometric condition

$K$06

where $K$07 and $K$08 are spans of the frame elements indexed by the maxima and non-maxima of $K$09. With an additional linear independence hypothesis, $K$10 remains optimal, but if $K$11 there are uncountably many optimal $K$12-duals; uniqueness occurs only under a sharper independence condition (Mondal et al., 4 Aug 2025). The spectral-radius problem has an analogous criterion, now formulated with the maxima of $K$13 and the corresponding subspaces $K$14 (Mondal et al., 4 Aug 2025).

Related work on ordinary duals generated by graphs gives a particularly rigid model of erasure optimality. For connected graph-generated frames, the canonical dual is the unique spectrally optimal dual for all $K$15-erasures, whereas for disconnected graphs the canonical dual remains spectrally optimal but is not unique (Deepshikha et al., 27 Jul 2025). This is not a $K$16-frame result, but it isolates a structural dichotomy—irreducible versus decomposable geometry—that plausibly extends to $K$17-dual problems.

6. Probabilistic erasures and the geometry of optimal sets

A further optimization layer appears in probabilistic erasure models. For ordinary frames, one line of work measures the error operator by the weighted average

$K$18

and shows that for fixed frames the optimal dual set is closed and convex, and for $K$19 is nonempty and compact (Arati et al., 2024). Another uses the Hilbert–Schmidt norm of probabilistic error operators

$K$20

and proves that, for each $K$21, the set of $K$22-erasure probabilistic optimal duals is a nonempty compact convex subset of the affine dual space (Mondal et al., 12 Jun 2026).

The Hilbert–Schmidt theory also identifies the one-erasure functional

$K$23

proves the lower bound $K$24, and gives geometric conditions under which the canonical dual is optimal or uniquely optimal, again through a decomposition of the index set into maximizers and non-maximizers and the subspace condition $K$25 (Mondal et al., 12 Jun 2026). These constructions are not stated for $K$26-frames, but the paper’s operator-level setup depends only on affine dual constraints and on norms of error operators. That strongly suggests a direct $K$27-frame analogue with $K$28 in place of $K$29.

Across these strands, the topic resolves into a stable set of themes. Optimal $K$30-dual frames are governed by factorization of $K$31, inversion on $K$32, canonical constructions built from restricted frame operators, and error minimization under deterministic or probabilistic erasures. Canonical $K$33-duals are structurally preferred and sometimes provably optimal, but finite-dimensional erasure theory shows that optimality can also be non-unique or even non-canonical. The current literature therefore supports a plural notion of optimality: canonical when range geometry and symmetry dominate, Douglas-minimal when least-energy factorization is primary, and erasure-optimal when the decisive object is the error operator rather than the inverse alone.