Dynamical Frames in Theory and Applications

• Dynamical Frames are frame constructions defined by dynamic processes rather than predetermined structures, appearing in Hilbert-space theory, astrometry, and general relativity.
• They often involve iterative methods—such as operator or semigroup actions and tensor-product formulations—to generate frame elements that support robust state reconstruction.
• Their applications span dynamical sampling, celestial navigation, and gauge-invariant observables, bridging abstract mathematical theory with practical measurements.

to=arxiv_search.search ាជ্ঞા оЌjson {"^^^^^^^^3query3 frames\"^^^^^^^^3 OR ti:\3^^^^^^^^"dynamical frames\"","max_results":^^^^3all:\3query3^^^^,"sort_by":"submittedDate","sort_order":"descending"}】【：】【“】【assistant to=arxiv_search.searchสดงความคิดเห็น 买天天中彩票json {"^^^^^^^^3query3 OR id:(&&&^^^^3all:\3^^^^&&&) OR id:(&&&^^^^3 OR ti:\3^^^^&&&) OR id:(Kofinas et al., 2021) OR id:(Bailey et al., 25 May 2025) OR id:(Sahu et al., 2023) OR id:(Aldroubi et al., 2024) OR id:(Goeller et al., 2022) OR id:(Aguilera et al., 2022) OR id:(Ballesteros et al., 2020) OR id:(&&&^^^^3all:\3query3^^^^&&&)","max_results":^^^^3 OR ti:\3query3^^^^,"sort_by":"submittedDate","sort_order":"descending"} to=arxiv_search.search 大发彩票官网json {"^^^^^^^^3query3 OR id:(&&&^^^^3all:\3 OR ti:\3^^^^&&&) OR id:(&&&^^^^3all:\33^^^^&&&) OR id:(&&&^^^^3all:\3^^^^4&&&) OR id:(&&&^^^^3all:\3^^^^5&&&) OR id:(&&&^^^^3all:\3^^^^6&&&) OR id:(&&&^^^^3all:\3^^^^7&&&) OR id:(&&&^^^^3all:\3^^^^8&&&) OR id:(&&&^^^^3all:\3^^^^9&&&)","max_results":^^^^3 OR ti:\3query3^^^^,"sort_by":"submittedDate","sort_order":"descending"} Dynamical frames are frame constructions whose defining reference structure is generated, transported, or realized by dynamics rather than fixed a priori. In contemporary research the expression denotes several distinct objects: operator-orbit frames in Hilbert spaces, ephemeris-realized celestial reference frames, physical reference systems in generally covariant theories, and local time-evolving coordinate frames attached to moving objects in dynamical systems (&&&^^^^3all:\3^^^^&&&, &&&^^^^3query3^^^^&&&, &&&^^^^3 OR ti:\3^^^^&&&, Kofinas et al., 2021). This suggests a common organizing idea: the frame is itself part of the dynamics, even though the underlying mathematics ranges from bounded operators and semigroup representations to planetary ephemerides, scalar reference fields, and object-centric rotations.

^^^^3all:\3^^^^. Scope and terminological regimes

The term does not have a single universal meaning. Its interpretation depends on the ambient theory and on what is meant by a “frame”: a spanning system, a celestial reference realization, a physical localization device, or a moving coordinate system.

Domain	Frame object	Representative formulation
Hilbert-space frame theory	Orbit-generated spanning family	PRESERVED_PLACEHOLDER_^^^^3query3^^^^
Dynamical sampling	Time-space sampling system	PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^ or PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^
Astrometry	Dynamical celestial frame	Planetary ephemeris realizing an inertial frame
General relativity	Physical reference system	Congruence, tetrad, dust, or scalar reference fields
Interacting dynamical systems	Local object-centric coordinates	$F_i(t)=(R_i(t),p_i(t))$
Quantum reference frames	Frame-changing unitary symmetry	Canonical transformations between quantum systems

In astrometry, “dynamical frame” is contrasted directly with a kinematic celestial frame: the former is realized by a planetary ephemeris and defined by non-accelerating Solar System motions, whereas the latter is defined as globally nonrotating with respect to the distant Universe and realized by extragalactic sources (&&&^^^^3query3^^^^&&&). In general relativity, by contrast, the central distinction is between coordinates and physical reference frames: coordinates are mathematical labels, while a dynamical reference frame is a material or field-theoretic system that satisfies its own equations of motion and localizes observables relationally (&&&^^^^3 OR ti:\3^^^^&&&).

^^^^3 OR ti:\3^^^^. Operator-generated frames in Hilbert spaces

In frame theory, a sequence $\{f_k\}\subset H$ in a separable Hilbert space is a frame if there exist $0<A\le B<\infty$ such that

$$A\|x\|^2 \le \sum_k |\langle x,f_k\rangle|^2 \le B\|x\|^2,\qquad x\in H.$$

A dynamical frame is obtained when the frame vectors are iterates of an operator. For a bounded operator $A:H\to H$, a countable generator set $G\subset H$, and iteration counts $L(g)$, the dynamical set is

PRESERVED_PLACEHOLDER_^^^^3all:\3query3^^^^

and it is called a dynamical frame when it is a frame for PRESERVED_PLACEHOLDER_^^^^3all:\3all:\3^^^^ (&&&^^^^3all:\3^^^^&&&).

A basic structural result is that the canonical dual preserves the iterative form. If PRESERVED_PLACEHOLDER_^^^^3all:\3 OR ti:\3^^^^ is the frame operator of PRESERVED_PLACEHOLDER_^^^^3all:\33^^^^, then with

PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^4

the canonical dual is again dynamical: PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^5 This places dynamical frames alongside Gabor and wavelet systems, where duality preserves an underlying generating mechanism. The same paper also shows invariance under invertible changes of variables and gives a normal-operator characterization of scalability in finite dimensions by reducing the existence of positive scalings to explicit equations in the eigenbasis (&&&^^^^3all:\3^^^^&&&).

For single-generator orbits PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^6, bounded representability is highly rigid. A frame PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^7 can be written as PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^8 for a bounded operator PRESERVED_PLACEHOLDER_^^^^3all:\3^^^^9 if and only if the kernel of its synthesis operator is invariant under the right shift on PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3query3^^^^; equivalently, for a dual frame PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3all:\3^^^^,

PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3 OR ti:\3^^^^

In the overcomplete case the image chain PRESERVED_PLACEHOLDER_^^^^3 OR ti:\33^^^^ has finite length, and after a finite index the tail spans a fixed finite-codimensional subspace PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^4 on which PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^5 is surjective; moreover,

PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^6

is still a frame for all PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^7 (&&&^^^^3all:\33^^^^&&&). This makes dynamical frames unusually sensitive to ordering and to perturbations of PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^8 or PRESERVED_PLACEHOLDER_^^^^3 OR ti:\3^^^^9, while also revealing a specific redundancy pattern.

3. Semigroup, tensor-product, and cyclic extensions

The orbit picture extends from a single operator to semigroup representations. For a countable unital semigroup $F_i(t)=(R_i(t),p_i(t))$^^^^3query3^^^^ and a representation $F_i(t)=(R_i(t),p_i(t))$^^^^3all:\3^^^^, the orbit

$F_i(t)=(R_i(t),p_i(t))$^^^^3 OR ti:\3^^^^

is a frame when it satisfies the frame inequalities. A representation is called central if all frame generators are equivalent via invertible operators in the commutant $F_i(t)=(R_i(t),p_i(t))$3. The general characterization is that centrality is equivalent to the analysis range $F_i(t)=(R_i(t),p_i(t))$4 being co-hyperinvariant for the weak-operator-topology closed algebra generated by the left regular representation. In particular, every frame representation of $F_i(t)=(R_i(t),p_i(t))$5 is central, so for any commuting $F_i(t)=(R_i(t),p_i(t))$6-tuple $F_i(t)=(R_i(t),p_i(t))$7 all frame generators are equivalent up to an invertible operator commuting with every $F_i(t)=(R_i(t),p_i(t))$8 (Bailey et al., 25 May 2025).

A different extension appears for two commuting bounded operators. The system

$F_i(t)=(R_i(t),p_i(t))$9

is characterized by similarity to canonical model tuples built from the bilateral shift on $\{f_k\}\subset H$^^^^3query3^^^^ and the compression of the unilateral shift on vector-valued Hardy spaces. In the unilateral case the model space is a closed $\{f_k\}\subset H$^^^^3all:\3^^^^-reducing, $\{f_k\}\subset H$^^^^3 OR ti:\3^^^^-invariant subspace $\{f_k\}\subset H$3, and the resulting basic tuple is Parseval. In the bilateral case the model is a reducing subspace of $\{f_k\}\subset H$4 (Aguilera et al., 2022).

Tensor-product dynamical frames arise when one combines iterative representations in two Hardy spaces. For diagonal operators $\{f_k\}\subset H$5 and $\{f_k\}\subset H$6, the tensorized orbit

$\{f_k\}\subset H$7

is a frame for $\{f_k\}\subset H$8 if and only if the product sequence $\{f_k\}\subset H$9 satisfies a Carleson condition in the unit disc. The paper also proves a tensor-product interpolation theorem and a ratio criterion implying the Carleson condition (Sahu et al., 2023).

In finite dimensions, cyclic frames form a distinguished subclass of dynamical frames. A single-generator dynamical frame $0<A\le B<\infty$^^^^3query3^^^^ is cyclic when $0<A\le B<\infty$^^^^3all:\3^^^^. The finite-dimensional characterization states that cyclicity forces $0<A\le B<\infty$^^^^3 OR ti:\3^^^^ to be diagonalizable with distinct $0<A\le B<\infty$3th roots of unity as eigenvalues, and in the eigenbasis the generator must have no zero coordinates. An equivalent criterion is invariance of the synthesis-kernel under the right shift on $0<A\le B<\infty$4. Tight cyclic frames are automatically unitary and equal-norm, and the canonical dual remains dynamical, with $0<A\le B<\infty$5 (&&&^^^^3all:\3^^^^9&&&).

4. Dynamical sampling and inverse problems

Dynamical frames arise naturally in dynamical sampling, where one seeks to reconstruct a state or source term from spatiotemporal samples. The basic problem is to determine when measurements of the form $0<A\le B<\infty$6 or, more concretely, time samples from an orbit, define a frame. This motivation underlies much of the frame-theoretic literature (&&&^^^^3all:\3^^^^&&&).

A precise source-recovery formulation is given for the discrete system

$0<A\le B<\infty$7

where $0<A\le B<\infty$8, $0<A\le B<\infty$9 is an unknown constant source, and measurements are

$$A\|x\|^2 \le \sum_k |\langle x,f_k\rangle|^2 \le B\|x\|^2,\qquad x\in H.$$^^^^3query3^^^^

for a Bessel family $$A\|x\|^2 \le \sum_k |\langle x,f_k\rangle|^2 \le B\|x\|^2,\qquad x\in H.$$^^^^3all:\3^^^^. In the full-space finite-time case, $$A\|x\|^2 \le \sum_k |\langle x,f_k\rangle|^2 \le B\|x\|^2,\qquad x\in H.$$^^^^3 OR ti:\3^^^^ can be recovered stably from finitely many time samples if and only if $$A\|x\|^2 \le \sum_k |\langle x,f_k\rangle|^2 \le B\|x\|^2,\qquad x\in H.$$3 is a frame for $$A\|x\|^2 \le \sum_k |\langle x,f_k\rangle|^2 \le B\|x\|^2,\qquad x\in H.$$4. In the infinite-time subspace case, under $$A\|x\|^2 \le \sum_k |\langle x,f_k\rangle|^2 \le B\|x\|^2,\qquad x\in H.$$5, stable recovery for all $$A\|x\|^2 \le \sum_k |\langle x,f_k\rangle|^2 \le B\|x\|^2,\qquad x\in H.$$6 is equivalent to the frame property of

$$A\|x\|^2 \le \sum_k |\langle x,f_k\rangle|^2 \le B\|x\|^2,\qquad x\in H.$$7

on $$A\|x\|^2 \le \sum_k |\langle x,f_k\rangle|^2 \le B\|x\|^2,\qquad x\in H.$$8, and reconstruction is given by

$$A\|x\|^2 \le \sum_k |\langle x,f_k\rangle|^2 \le B\|x\|^2,\qquad x\in H.$$9

for any dual frame $A:H\to H$^^^^3query3^^^^ of $A:H\to H$^^^^3all:\3^^^^ (Aldroubi et al., 2024).

This source-recovery viewpoint sharpens the operational meaning of a dynamical frame. It is not merely a spanning set generated by iterations; it is the exact stability criterion for identifying latent variables from dynamical measurements. The same paper also shows that finite-time recovery can fail even for one-dimensional source spaces in infinite-dimensional $A:H\to H$^^^^3 OR ti:\3^^^^, so the distinction between finite and infinite observation horizons is intrinsic rather than technical (Aldroubi et al., 2024).

5. Astrometric and celestial dynamical frames

In astrometry, a dynamical reference frame is the inertial celestial frame realized by a planetary ephemeris. Pulsar timing positions depend on the chosen ephemeris because topocentric arrival times are transformed to barycentric times using that ephemeris. A timing position therefore realizes the orientation of a specific dynamical frame such as DE^^^^3 OR ti:\3query3query3^^^^, DE4^^^^3query3^^^^5, DE4^^^^3 OR ti:\3all:\3^^^^, DE43^^^^3query3^^^^, or DE436. Kinematic frames, by contrast, are realized from distant extragalactic sources: the International Celestial Reference Frame by VLBI and Gaia-CRF3 by optical astrometry (&&&^^^^3query3^^^^&&&).

A direct comparison can be made because some pulsars have sub-milliarcsecond positions from timing, VLBI, and Gaia. The frame tie is estimated by propagating Gaia or VLBI positions to the timing epoch and fitting the residuals to a global small-angle rotation,

$A:H\to H$3

Using 49 Gaia pulsars, 6^^^^3 OR ti:\3^^^^ VLBI pulsars, and ^^^^3 OR ti:\3^^^^83 timing solutions for 93 pulsars, the analysis finds that DE^^^^3 OR ti:\3query3query3^^^^ shows an orientation offset at roughly the $A:H\to H$4–$A:H\to H$5 mas level relative to Gaia-CRF3 and the VLBI pulsar frame, while DE4^^^^3query3^^^^5 and DE4^^^^3 OR ti:\3all:\3^^^^ are broadly consistent with no significant rotation and sub-mas ties to modern ICRS realizations. The DE^^^^3 OR ti:\3query3query3^^^^ vs Gaia-CRF3 MSP-only solution gives $A:H\to H$6, $A:H\to H$7, $A:H\to H$8 mas; the DE^^^^3 OR ti:\3query3query3^^^^ vs VLBI MSP-only solution gives $A:H\to H$9, $G\subset H$^^^^3query3^^^^, $G\subset H$^^^^3all:\3^^^^ mas (&&&^^^^3query3^^^^&&&).

The same study emphasizes limitations: archival timing uncertainties can be underestimated, timing error ellipses near the ecliptic have unavailable RA–Dec covariances, Gaia non-MSP associations may be contaminated, and the newest ephemerides are sample-limited. Even so, pulsars provide a three-way link—timing to the dynamical frame, VLBI to ICRF, Gaia to Gaia-CRF3—that directly probes the orientation of ephemeris-based frames. Simulations in the paper indicate that with at least 5 well-timed MSPs a sub-mas dynamical–kinematic tie is achievable, and with about ^^^^3all:\3^^^^5 MSPs of about ^^^^3query3^^^^.^^^^3 OR ti:\3^^^^ mas timing precision plus Gaia or VLBI astrometry, about ^^^^3query3^^^^.^^^^3 OR ti:\3^^^^ mas alignment precision is realistic (&&&^^^^3query3^^^^&&&).

6. Relational frames in general relativity and covariant field theory

In general relativity, a reference frame is not a coordinate chart but a physical system represented by fields. One formulation distinguishes three regimes: Idealised Reference Frames, in which both the frame dynamics and its stress-energy are neglected; Dynamical Reference Frames, in which the frame obeys its own equations of motion but its stress-energy is neglected; and Real Reference Frames, in which the frame both evolves and backreacts on the metric. In this terminology a DRF is a test-like physical system—such as a congruence of timelike worldlines, a tetrad field, dust, or four scalar fields $G\subset H$^^^^3 OR ti:\3^^^^ obeying $G\subset H$3—whose dynamics acts as a physical gauge fixing and permits local Dirac observables (&&&^^^^3 OR ti:\3^^^^&&&).

The central construction is relational localization. If $G\subset H$4 are scalar reference fields, then a local observable associated with a field $G\subset H$5 is written schematically as

$G\subset H$6

For the metric, one obtains relational components $G\subset H$7. A nonperturbative generalization uses a universal dressing space $G\subset H$8 of gauge-covariant spacetime points and defines a dynamical frame as an injective map $G\subset H$9; the relational observable of a covariant field $L(g)$^^^^3query3^^^^ is then $L(g)$^^^^3all:\3^^^^. Collections of such frames form relational atlases, and the resulting notion of relational locality satisfies bulk microcausality in the Peierls-bracket sense (Goeller et al., 2022).

In the covariant phase-space treatment of finite subregions, dynamical frames appear as edge modes. A field-dependent map $L(g)$^^^^3 OR ti:\3^^^^ identifies a physical subregion and its timelike boundary in gauge-invariant terms, while the Maurer–Cartan form $L(g)$3 defines the dressed variation $L(g)$4. The extended presymplectic potential is $L(g)$5, and conservation of the subregion presymplectic structure under post-selected dynamics fixes an essentially unique boundary prescription. In GR with Dirichlet boundary conditions, the resulting relational boundary charges reduce to Brown–York quasi-local charges (&&&^^^^3all:\3query3^^^^&&&). A later extension uses DRFs to implement soft cutoffs in covariant phase space: smearing functions are generated by the frame, covariance is recovered only for restricted DRFs and associated Maurer–Cartan forms, and an additional pointwise dependence is required to make the soft-cutoff charges integrable under fluctuating boundary conditions (&&&^^^^3all:\3 OR ti:\3^^^^&&&).

These constructions bear directly on the Hole Argument. One analysis distinguishes coupled reference frames, for which diagonal $L(g)$6 acts and relational observables are gauge-invariant and deterministic, from uncoupled reference frames, for which only reshuffling-invariance holds and a “New Hole Argument” arises unless one restricts physical quantities to gauge-invariant observables. The same work identifies an “Arbitrariness Problem” associated with choosing different physical frames, and resolves it by external diffeomorphisms relating the resulting gauge-invariant observables (&&&^^^^3all:\3all:\3^^^^&&&).

A related but distinct usage of “frame” occurs in conformally connected cosmological formulations. In $L(g)$7 gravity, the Jordan and Einstein frames are dynamically equivalent only on the domain where $L(g)$8 and $L(g)$9; these are also the conditions excluding anisotropic singularities in Bianchi I cosmology (&&&^^^^3all:\3^^^^6&&&). For single-field conformal transformations, attractor behavior is preserved across frames even though the decay rates differ because the e-fold clocks differ; in particular, the duration of inflation in a Jordan frame is always higher than in the Einstein frame (&&&^^^^3all:\3^^^^7&&&).

7. Moving, learned, and quantum frames

In classical mechanics, a dynamical frame can mean a non-inertial frame whose origin translates and whose axes rotate. If the inertial position is written in a carried-translation convention PRESERVED_PLACEHOLDER_^^^^3all:\3query3query3^^^^, then the equation of motion in the accelerating frame contains not only the translational inertial force PRESERVED_PLACEHOLDER_^^^^3all:\3query3all:\3^^^^ and the standard Coriolis, Euler, and centrifugal terms, but also a mixed rotation–translation contribution

PRESERVED_PLACEHOLDER_^^^^3all:\3query3 OR ti:\3^^^^

This term vanishes only in special cases, such as PRESERVED_PLACEHOLDER_^^^^3all:\3query33^^^^ or when the translation is treated in inertial components rather than as