Frame Measure Function: Concepts & Applications

• Frame measure function is a scalar invariant that quantifies redundancy, tightness, and measurement consistency in diverse settings like reproducing-kernel Hilbert spaces and quantum measurement.
• In harmonic analysis and RKHS, it connects density properties and Fourier frame constructions, providing concrete measures such as the Balan–Landau ultrafilter average and Beurling densities.
• In control theory, the frame measure function emerges as normalized frame potential metrics that relate the tightness of system reachability frames to controllability criteria.

In the literature represented here, the term frame measure function is used in several related ways rather than as a single universal definition. In reproducing-kernel Hilbert spaces it denotes the Balan–Landau ultrafilter average

$$m_\omega(\mathcal F)=\omega\!-\!\lim_{r\to\infty}\frac1{\#(\Lambda\cap B_r)} \sum_{\lambda\in\Lambda\cap B_r}\langle k_\lambda,\widetilde k_\lambda\rangle$$

for a frame $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$, together with the extremal limits $M^-(\mathcal F)$ and $M^+(\mathcal F)$ (Bownik et al., 15 Sep 2025). In the Parseval-frame generalization of Gleason theory, it is “nothing but the mapping” $$\Phi=\{\phi_i\}\mapsto \sum_i f(\phi_i)=\operatorname{tr}A$$ when $f(x)=\langle Ax,x\rangle$ (Benedetto et al., 2020). In control-theoretic frame analysis, it denotes a scale-invariant measure of tightness of the frame generated by the reachability or controllability data, such as $\operatorname{tr}(W^2)/(\operatorname{tr}W)^2$ or the equivalent form $$\operatorname{tr}(W_c(T))/\sqrt{\operatorname{tr}(W_c(T)^2)}$$ (Sheriff et al., 2017, PK et al., 2019). This suggests a common role: assigning a scalar quantity to a frame-like structure in order to quantify redundancy, tightness, or measurement consistency.

1. Frame measures as the ambient concept

A principal background notion is the frame measure for function spaces defined by a measure $\mu$. For a finite Borel measure $\mu$ on $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$0, conjugate exponents $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$1, and $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$2, a Borel measure $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$3 is a $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$4-frame measure if there are constants $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$5 such that

$\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$6

If $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$7, $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$8 is tight; if $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$9, it is a $M^-(\mathcal F)$0-Plancherel measure. In the Hilbert case $M^-(\mathcal F)$1, this is exactly the usual definition of a frame measure in $M^-(\mathcal F)$2 (Farhadi et al., 2019).

The same scheme extends to locally compact abelian groups. If $M^-(\mathcal F)$3 is an LCA group, $M^-(\mathcal F)$4 is a finite positive Borel measure on $M^-(\mathcal F)$5, and $M^-(\mathcal F)$6 is a Borel measure on the dual group $M^-(\mathcal F)$7, then $M^-(\mathcal F)$8 is a $M^-(\mathcal F)$9-frame measure when

$M^+(\mathcal F)$0

equivalently when the analysis map $M^+(\mathcal F)$1 is bounded with closed range and bounded inverse on its range (Birgani et al., 2021).

Several structural facts delimit this setting. Every finite Borel measure $M^+(\mathcal F)$2 is a $M^+(\mathcal F)$3-Bessel measure for any finite $M^+(\mathcal F)$4, so the family of upper-bound-only measures is very large (Farhadi et al., 2019). By contrast, genuine frame measures are constrained: any $M^+(\mathcal F)$5-frame measure $M^+(\mathcal F)$6 must be $M^+(\mathcal F)$7-finite, and on LCA groups the existence of a $M^+(\mathcal F)$8-frame measure forces a strong uniformity condition on $M^+(\mathcal F)$9, namely that its density be essentially bounded above and below on its support (Farhadi et al., 2019, Birgani et al., 2021). The LCA-group theory also identifies an obstruction: if $$\Phi=\{\phi_i\}\mapsto \sum_i f(\phi_i)=\operatorname{tr}A$$0 and $$\Phi=\{\phi_i\}\mapsto \sum_i f(\phi_i)=\operatorname{tr}A$$1 are non-atomic probability measures whose supports form a packing pair, then $$\Phi=\{\phi_i\}\mapsto \sum_i f(\phi_i)=\operatorname{tr}A$$2 admits no $$\Phi=\{\phi_i\}\mapsto \sum_i f(\phi_i)=\operatorname{tr}A$$3-frame measure for any $$\Phi=\{\phi_i\}\mapsto \sum_i f(\phi_i)=\operatorname{tr}A$$4 (Birgani et al., 2021).

This ambient theory places frame measure functions in a broader analytical context. A plausible implication is that many scalar “frame measure functions” are best viewed not as standalone objects, but as summaries of how a frame interacts with an analysis operator, a dual frame, or a Fourier transform inequality.

2. Parseval frames, Gleason functions, and POVMs

For a finite sequence $$\Phi=\{\phi_i\}\mapsto \sum_i f(\phi_i)=\operatorname{tr}A$$5, the Parseval condition

$$\Phi=\{\phi_i\}\mapsto \sum_i f(\phi_i)=\operatorname{tr}A$$6

for every $$\Phi=\{\phi_i\}\mapsto \sum_i f(\phi_i)=\operatorname{tr}A$$7 is equivalent to

$$\Phi=\{\phi_i\}\mapsto \sum_i f(\phi_i)=\operatorname{tr}A$$8

Since each $$\Phi=\{\phi_i\}\mapsto \sum_i f(\phi_i)=\operatorname{tr}A$$9, the family $f(x)=\langle Ax,x\rangle$0 is exactly a finite positive-operator-valued measure on $f(x)=\langle Ax,x\rangle$1 (Benedetto et al., 2020).

Within this framework, a Gleason function of weight $f(x)=\langle Ax,x\rangle$2 for $f(x)=\langle Ax,x\rangle$3-element Parseval frames is a function $f(x)=\langle Ax,x\rangle$4 or $f(x)=\langle Ax,x\rangle$5 such that

$f(x)=\langle Ax,x\rangle$6

for every Parseval frame $f(x)=\langle Ax,x\rangle$7. The central theorem states that if $f(x)=\langle Ax,x\rangle$8 and $f(x)=\langle Ax,x\rangle$9 is bounded, or non-negative, or continuous, and is a Gleason function of weight $\operatorname{tr}(W^2)/(\operatorname{tr}W)^2$0 for all finite Parseval frames in $\operatorname{tr}(W^2)/(\operatorname{tr}W)^2$1 or $\operatorname{tr}(W^2)/(\operatorname{tr}W)^2$2, then there exists a unique self-adjoint operator $\operatorname{tr}(W^2)/(\operatorname{tr}W)^2$3 with $\operatorname{tr}(W^2)/(\operatorname{tr}W)^2$4 such that

$\operatorname{tr}(W^2)/(\operatorname{tr}W)^2$5

Equivalently,

$\operatorname{tr}(W^2)/(\operatorname{tr}W)^2$6

for every Parseval frame (Benedetto et al., 2020).

In this setting, the frame measure function is explicitly identified as

$\operatorname{tr}(W^2)/(\operatorname{tr}W)^2$7

The same paper gives a quantum-measurement interpretation: for the rank-one POVM $\operatorname{tr}(W^2)/(\operatorname{tr}W)^2$8, the associated probabilities are

$\operatorname{tr}(W^2)/(\operatorname{tr}W)^2$9

and $$\operatorname{tr}(W_c(T))/\sqrt{\operatorname{tr}(W_c(T)^2)}$$0, which equals $$\operatorname{tr}(W_c(T))/\sqrt{\operatorname{tr}(W_c(T)^2)}$$1 when $$\operatorname{tr}(W_c(T))/\sqrt{\operatorname{tr}(W_c(T)^2)}$$2 is normalized (Benedetto et al., 2020).

A further structural issue concerns frame length. If $$\operatorname{tr}(W_c(T))/\sqrt{\operatorname{tr}(W_c(T)^2)}$$3 denotes the bounded Gleason functions for $$\operatorname{tr}(W_c(T))/\sqrt{\operatorname{tr}(W_c(T)^2)}$$4-element Parseval frames, then a trivial zero-padding argument gives $$\operatorname{tr}(W_c(T))/\sqrt{\operatorname{tr}(W_c(T)^2)}$$5, but bounded functions in $$\operatorname{tr}(W_c(T))/\sqrt{\operatorname{tr}(W_c(T)^2)}$$6 need not extend to longer frames. The decisive result is that for $$\operatorname{tr}(W_c(T))/\sqrt{\operatorname{tr}(W_c(T)^2)}$$7, every bounded Gleason function for $$\operatorname{tr}(W_c(T))/\sqrt{\operatorname{tr}(W_c(T)^2)}$$8-element Parseval frames extends, with weight shift $$\operatorname{tr}(W_c(T))/\sqrt{\operatorname{tr}(W_c(T)^2)}$$9, to all $\mu$0-element Parseval frames, so $\mu$1 for all $\mu$2 (Benedetto et al., 2020). The same mechanism weakens Busch’s finite-dimensional analog of Gleason’s theorem by replacing countable additivity with additivity on every $\mu$3-element POVM for some $\mu$4 (Benedetto et al., 2020).

3. Control-theoretic frame measure functions

For the discrete-time LTI system

$\mu$5

the reachability matrix at horizon $\mu$6 is

$\mu$7

Its columns $\mu$8, with $\mu$9, span the reachable subspace, so when $\mu$0 is controllable and $\mu$1 they form a finite frame for $\mu$2. The associated frame operator is the controllability Gramian

$\mu$3

A frame is tight exactly when $\mu$4, equivalently

$\mu$5

(Sheriff et al., 2017).

Three standard measures of quality are built from $\mu$6: $\mu$7, $\mu$8, and $\mu$9. For vectors of fixed lengths $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$00, each optimization attains its unique optimum exactly when

$\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$01

that is, exactly when the $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$02 form a tight frame (Sheriff et al., 2017).

The resulting frame-theoretic measure is the normalized frame potential

$\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$03

Its universal lower bound is

$\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$04

with equality if and only if $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$05 is a tight frame. The frame measure function for the LTI system is then defined by

$\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$06

Moreover, if $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$07, then the system is classically controllable (Sheriff et al., 2017).

For the continuous-time system

$\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$08

the endpoint map

$\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$09

sends an orthonormal basis $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$10 of the control space $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$11 to a sequence $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$12. The frame operator of $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$13 is exactly the controllability Gramian

$\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$14

and $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$15 is a frame if and only if the system is controllable on $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$16 (PK et al., 2019).

The continuous-time paper prefers the equivalent tightness-based quantity

$\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$17

It satisfies $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$18, with equality if and only if $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$19, hence if and only if the frame is tight. It is homogeneous under $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$20, invariant under orthonormal similarity, and related to the normalized frame potential by

$\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$21

If the reachable subspace has dimension $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$22, then $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$23; in particular, $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$24 guarantees controllability (PK et al., 2019).

4. Redundancy and the Balan–Landau frame-measure function

In reproducing-kernel Hilbert spaces on metric-measure spaces, the most literal use of the term frame-measure function is the Balan–Landau construction. Let $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$25 be a reproducing-kernel Hilbert space with kernels $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$26, let $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$27 be discrete, and assume $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$28 is a frame with canonical dual $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$29. Fixing a free ultrafilter $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$30 on $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$31, the frame-measure function is

$\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$32

The associated extremal quantities are

$\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$33

$\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$34

and these satisfy

$\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$35

The diagonal terms $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$36 are averaged precisely to quantify infinite-frame redundancy (Bownik et al., 15 Sep 2025).

The relevant density parameters are the Beurling densities

$\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$37

and, when the kernel diagonal is not constant, the dimension-free densities

$\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$38

Under the standing assumptions of diagonal bounds, weak localization, and the homogeneous approximation property, the main theorem is

$\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$39

In particular,

$\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$40

recovering the necessary lower density condition in this RKHS setting (Bownik et al., 15 Sep 2025).

This identity turns the frame-measure function into a quantitative redundancy invariant. Defining

$\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$41

one has $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$42, $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$43 if and only if $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$44 is a Riesz basis, and if $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$45 has density $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$46, then $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$47 (Bownik et al., 15 Sep 2025). The same paper proves a subframe theorem: if $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$48, then for every $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$49 there exists a subset $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$50 such that $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$51 is still a frame and $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$52 (Bownik et al., 15 Sep 2025).

The stated applications are to exponential frames on possibly unbounded spectra and to arbitrary nonlocalized Gabor frames. In both cases the conclusion is the existence of frames with lower density at most $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$53, which the paper interprets as confirming that the Balan–Landau frame-measure function is a meaningful quantitative definition of redundancy for a large class of infinite frames (Bownik et al., 15 Sep 2025).

5. Infinite-dimensional measure spaces and frame analysis

A different but closely related strand studies frame measures in infinite-dimensional Hilbert spaces. Let $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$54 be an infinite-dimensional separable Hilbert space. A positive measure $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$55 on a measure space $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$56 is a frame measure with bounds $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$57 if every $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$58 extends to a measurable function $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$59 on $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$60 and

$\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$61

Equivalently, the analysis operator $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$62, $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$63, satisfies

$\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$64

If $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$65, the measure is tight (Jorgensen et al., 2016).

The crucial negative result is that, in infinite dimension, no finite Borel measure on $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$66 itself can satisfy these inequalities. The remedy is to pass to a larger space via a Gelfand triple

$\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$67

with $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$68 carrying the relevant $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$69-algebra. In the model construction $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$70, Minlos’ theorem produces a unique centered Gaussian measure $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$71 on $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$72 with characteristic functional

$\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$73

and differentiation gives

$\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$74

so $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$75 (Jorgensen et al., 2016).

The same paper develops three principal classes: Gaussian frame measures, Markov path-space measures, and determinantal measures. Gaussian measures are parametrized by positive self-adjoint covariance operators $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$76 with $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$77, yielding

$\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$78

Markov path-space measures are built from a countable frame $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$79 by transition probabilities

$\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$80

on $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$81. Determinantal measures arise from the Gram operator $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$82 of a frame with an upper bound, with finite-set weights $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$83 extending to a determinantal point process (Jorgensen et al., 2016).

Although these constructions do not themselves define a frame measure function in the Balan–Landau sense, they show that measure-theoretic formulations of frame analysis persist beyond finite-dimensional or discrete settings. This suggests that scalar frame measure functions are one layer within a larger probabilistic and operator-theoretic framework.

6. Harmonic-analysis examples and fractal context

The harmonic-analysis theory of frame measures supplies many examples in which frame measure functions, redundancy functionals, or tightness measures become meaningful. For finite measures on $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$84, the $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$85-theory produces explicit frame-measure examples: the Bernoulli measure

$\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$86

has $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$87 as a $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$88-Plancherel measure, and by interpolation this $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$89 is a tight $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$90-frame measure for every $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$91; for $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$92, the counting measure $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$93 is the classical Fourier-frame measure with $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$94, and again is a $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$95-frame measure for every $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$96 (Farhadi et al., 2019). The same paper lists the $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$97-Cantor measure and other spectral affine-IFS measures as further cases admitting infinite discrete $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$98-frame measures (Farhadi et al., 2019).

On LCA groups, the construction theory is stable under shifts, convolution, and $\mathcal F=\{k_\lambda:\lambda\in\Lambda\}$99-reweighting. If $M^-(\mathcal F)$00 is a $M^-(\mathcal F)$01-frame measure for $M^-(\mathcal F)$02, then so is $M^-(\mathcal F)$03 for any finite measure $M^-(\mathcal F)$04 on $M^-(\mathcal F)$05; if $M^-(\mathcal F)$06 with $M^-(\mathcal F)$07, then $M^-(\mathcal F)$08 is a $M^-(\mathcal F)$09-frame measure for $M^-(\mathcal F)$10 if and only if it is one for Lebesgue measure, with frame bounds $M^-(\mathcal F)$11 and $M^-(\mathcal F)$12 (Birgani et al., 2021). These permanence properties explain why scalar functionals attached to a frame or frame measure often emphasize invariance or normalization.

A particularly notable fractal result is the existence of a Fourier frame for the uniform middle-third Cantor measure. If $M^-(\mathcal F)$13 is any locally and uniformly $M^-(\mathcal F)$14-dimensional measure supported on an $M^-(\mathcal F)$15-quasi-regular set $M^-(\mathcal F)$16, then $M^-(\mathcal F)$17 admits a frame of exponentials. In particular, for the uniform middle-third Cantor measure $M^-(\mathcal F)$18, there exists a countable set $M^-(\mathcal F)$19 such that $M^-(\mathcal F)$20 is a frame for $M^-(\mathcal F)$21, so $M^-(\mathcal F)$22 admits a generalized spectrum (Cabrelli et al., 2018). Within the present topic, this is significant because the existence of a frame is the prerequisite for any subsequent scalar quantification of redundancy, tightness, or measure assignment.

Taken together, these examples show that frame measure functions occur at the intersection of Fourier analysis, operator theory, quantum measurement, and control. The exact formula depends on the framework, but the recurring pattern is the reduction of a structured frame problem to a single scalar invariant that encodes how evenly, redundantly, or canonically the frame represents the underlying space.