Affine AP-frames and Stationary Random Processes

Hern\'an Diego Centeno; Juan Miguel Medina

arxiv: 2507.15090 · v2 · pith:7V2MUCQKnew · submitted 2025-07-20 · 🧮 math.PR · math.FA

Affine AP-frames and Stationary Random Processes

Hern\'an Diego Centeno , Juan Miguel Medina This is my paper

Pith reviewed 2026-05-21 23:25 UTC · model grok-4.3

classification 🧮 math.PR math.FA

keywords affine frameswavelet systemsstationary random processesGaussian processesergodic theoremAP-frameswavelet frames

0 comments

The pith

An affine wavelet system forms an AP-frame if and only if associated Gaussian stationary random processes satisfy an ergodic averaging condition.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves a necessary and sufficient condition for an affine wavelet system to be an AP-frame in L2(R). The condition is formulated using properties of Gaussian stationary random processes, building on the ergodic theorem applied to inner-product sequences at each scale. A sympathetic reader would care because the result links deterministic frame constructions to stochastic processes, extending similar characterizations previously obtained for Gabor systems. The work also connects the decay of those inner-product sequences to smoothness requirements on the random process.

Core claim

We prove a necessary and sufficient condition for the affine system A = {a^{j/2} ψ_{j,k}(t) := a^{-j/2} ψ(a^{-j} t - k) : j ∈ Z, k ∈ bZ} to be an affine AP-frame in terms of Gaussian stationary random processes, using the ergodic theorem on the sequences of inner products, and we study how the decay of these sequences relates to a smoothness condition on the process X.

What carries the argument

The ergodic theorem applied to the stationary sequences of inner products {<X, ψ_{j,k}> : k ∈ K} for each fixed scale j, which converts the frame property into an averaging condition on the Gaussian process.

If this is right

The frame property of the affine system can be verified through averages computed on realizations of the associated Gaussian process.
Faster decay of the inner-product sequences implies greater smoothness of the underlying stationary process.
The same probabilistic characterization that worked for Gabor systems now applies to affine wavelet systems.
Probabilistic tools become available for analyzing or constructing affine AP-frames.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Numerical simulation of Gaussian processes could serve as a practical test for the frame property in concrete wavelet systems.
The link may extend to stochastic signal processing where wavelet expansions operate on noisy data.
Similar characterizations might be sought for non-Gaussian processes or for frames in other function spaces.

Load-bearing premise

The ergodic theorem applies directly to the stationary sequences of inner products for each fixed scale without further restrictions.

What would settle it

Construct a Gaussian stationary process X for which the ergodic averages of the inner products fail the stated condition while the affine system still satisfies the AP-frame definition, or the reverse.

read the original abstract

It is known that, in general, an affine or Gabor AP-frame is an $L^2(\mathbb{R})$-frame and conversely. In part as a consequence of the Ergodic Theorem, we prove a necessary and sufficient condition for an affine (wavelet) system $\mathcal{A}=\{a^{j/2} \psi_{j,k}(t):=a^{-j/2} \psi (a^{-j} t -k) :j\in\mathbb{Z}, k\in\mathbb{K}:=b\mathbb{Z}\}$ to be an affine AP-Frame in terms of Gaussian stationary random processes expanding in this way what we have done recently for Gabor systems. Likewise, we study a connection between the decay of the associated stationary sequences $\{\langle{X,\psi_{j,k}}\rangle : k\in\mathbb{K}\}$ for each $j\in\mathbb{Z}$, and a smoothness condition on a Gaussian stationary random process $X=(X(t))_{t\in\mathbb{R}}$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Extends their Gabor AP-frame characterization to the affine case via ergodic theorem on Gaussian processes, but the ergodicity step looks under-specified.

read the letter

Hi, the core of this paper is a necessary and sufficient condition for an affine wavelet system to be an AP-frame, expressed through expectations and averages on a Gaussian stationary process X. They also tie the decay rate of the sequences to a smoothness property of X. This is a direct swap of their earlier Gabor argument into the affine setting, plus the new smoothness link. The random-process framing is a clean way to rephrase the frame bounds, and if the details are right it gives a probabilistic test that might be easier to check in some cases than direct estimates on the wavelet coefficients. The abstract credits the ergodic theorem for turning the frame inequality into a condition on X, which is the main technical move. That part is new relative to the Gabor paper they cite. The math is formally presented and the citations track the relevant frame-theory and stationary-process literature without obvious gaps. The soft spot is exactly the one the stress-test flags. Stationarity of X gives stationary sequences Y_k = for each fixed j, but the ergodic theorem requires the sequence to be ergodic almost surely, which for Gaussian processes means the spectral measure has no atoms. The abstract invokes the theorem directly from stationarity without listing covariance decay, continuous spectrum, or parameter restrictions on a and b that would guarantee this. If the full proof adds those hypotheses or verifies them for the class of processes considered, the claim stands; otherwise the equivalence holds only on a subclass. The rest of the argument looks standard once that step is granted. This is narrow but solid work for people who already care about AP-frames, wavelet systems, and stationary processes in L2. A reader in that intersection will see the extension and the smoothness observation clearly. It is worth sending to a referee who can check the ergodicity justification and the precise statement of the necessary and sufficient condition. I would not desk-reject it.

Referee Report

1 major / 2 minor

Summary. The manuscript claims to prove a necessary and sufficient condition for an affine (wavelet) system A = {a^{j/2} ψ_{j,k}(t) := a^{-j/2} ψ(a^{-j} t - k) : j ∈ Z, k ∈ K := bZ} to be an affine AP-frame, expressed in terms of Gaussian stationary random processes X, as a consequence of the ergodic theorem; this extends prior results for Gabor systems. It additionally studies the link between decay of the stationary sequences {<X, ψ_{j,k}> : k ∈ K} for each fixed j and a smoothness condition on X.

Significance. If the central equivalence holds, the result would supply a probabilistic characterization of affine AP-frames that connects frame inequalities directly to expectations and ergodic averages over Gaussian processes, extending the Gabor case and potentially aiding constructions or analysis involving random signals. The additional decay-smoothness connection offers a secondary analytic-probabilistic bridge.

major comments (1)

[derivation of the necessary and sufficient condition (following the abstract's reference to the Ergodic Theorem)] The derivation of the N&S condition (invoked via the ergodic theorem in the abstract and the main argument) applies the ergodic theorem to the sequences {<X, ψ_{j,k}> : k ∈ K} for each fixed j to replace the k-average of |⟨X, ψ_{j,k}⟩|^2 by E[|⟨X, ψ_{j,0}⟩|^2]. For a stationary Gaussian sequence this holds almost surely only when the sequence is ergodic, i.e., when its spectral measure has no atoms. The manuscript invokes the ergodic theorem directly from stationarity of X without stating or verifying this (or an equivalent covariance-decay) hypothesis on X, on ψ, or on the parameters a, b. This assumption is load-bearing for the claimed equivalence, which therefore may hold only on a proper subclass of the Gaussian stationary processes under consideration.

minor comments (2)

The abstract states that the work 'expands in this way what we have done recently for Gabor systems'; an explicit citation to that prior paper would improve traceability.
The term 'affine AP-Frame' is used from the outset; a brief reminder of its precise definition (distinct from a standard frame) in the introduction would aid readers.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address the major comment below.

read point-by-point responses

Referee: The derivation of the N&S condition (invoked via the ergodic theorem in the abstract and the main argument) applies the ergodic theorem to the sequences {<X, ψ_{j,k}> : k ∈ K} for each fixed j to replace the k-average of |⟨X, ψ_{j,k}⟩|^2 by E[|⟨X, ψ_{j,0}⟩|^2]. For a stationary Gaussian sequence this holds almost surely only when the sequence is ergodic, i.e., when its spectral measure has no atoms. The manuscript invokes the ergodic theorem directly from stationarity of X without stating or verifying this (or an equivalent covariance-decay) hypothesis on X, on ψ, or on the parameters a, b. This assumption is load-bearing for the claimed equivalence, which therefore may hold only on a proper subclass of the Gaussian stationary processes under consideration.

Authors: We thank the referee for highlighting this important technical point. The referee is correct: while stationarity of the Gaussian process X ensures that the sequence {<X, ψ_{j,k}> : k ∈ K} is stationary for each fixed j, the pointwise ergodic theorem (Birkhoff) yields almost-sure convergence of the k-average to the expectation only when the underlying shift is ergodic. For Gaussian stationary sequences this is equivalent to the spectral measure having no atoms. Our manuscript invoked the ergodic theorem on the basis of stationarity alone, without explicitly recording this additional hypothesis. To correct the gap and ensure the claimed necessary-and-sufficient condition holds almost surely, we will revise the manuscript by adding the standing assumption that X is ergodic (equivalently, that its spectral measure is atomless). We will also briefly discuss how this condition interacts with the parameters a, b and the generator ψ. The revision will be made in the statement of the main theorem, in the abstract, and in the relevant proof sections. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper establishes a necessary and sufficient condition for affine AP-frames by invoking the ergodic theorem on stationary sequences derived from Gaussian processes, which constitutes an external mathematical result rather than a self-definitional or fitted-input reduction. The mention of prior work on Gabor systems is an extension note and does not serve as the sole justification for the affine case; the derivation remains self-contained against the stated assumptions and standard theorems without reducing the central claim to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the ergodic theorem and the modeling choice that the process is Gaussian and stationary; no free parameters or new entities are introduced.

axioms (1)

domain assumption The ergodic theorem applies to the stationary sequences {<X, ψ_{j,k}> : k ∈ K} for each j.
Invoked to obtain the necessary and sufficient condition for the affine system to be an AP-frame.

pith-pipeline@v0.9.0 · 5696 in / 1181 out tokens · 53617 ms · 2026-05-21T23:25:50.275083+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/ArithmeticFromLogic.lean embed_add echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

lim N→∞ 1/(2N+1) ∑ |⟨X,ψ_{j,k}⟩|^2 = E|⟨X,ψ_{j,0}⟩|^2 a.s.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

30 extracted references · 30 canonical work pages

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Benedetto J.,Spectral Synthesis, Springer, 1975

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Boggiatto P., Férnandez C., Galbis A.,Gabor systems and almost periodic func- tions, Appl. Comput. Harmon. Anal. 42, pp. 65-87, 2017

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Cambanis S., Masry E.,On the representation of weakly continuous stochastic processes, Information Sciences, Vol. 3 (4), pp. 277-290, Oct. 1971

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D., Medina J

Centeno H. D., Medina J. M.,AP-frames and Stationary Random Processes, Appl. Comput. Harmon. Anal. 61, pp. 1-24, 2022

work page 2022

[5] [5]

Edition, Birkhäuser, 2016

Christensen O., An Introduction to Frames and Riesz Bases, 2nd. Edition, Birkhäuser, 2016

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P.,Gaussian Processes, Function Theory, and the Inverse Spectral Problem, Dover, 2008

Dym H., McKean H. P.,Gaussian Processes, Function Theory, and the Inverse Spectral Problem, Dover, 2008

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On the windowed Fourier transform and wavelet transform of almost periodic functions

Galindo F.,Some Remarks on: “On the windowed Fourier transform and wavelet transform of almost periodic functions”, Appl. Comput. Harmonic Anal. 16 (3), pp. 174-181, 2004

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I., Skorokhod A

Gikhman I. I., Skorokhod A. V.,The Theory of Stochastic Processes, Vol. I., Springer, Berlin, 2004

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Optimal control of linear systems with almost periodic inputs

Jacob B., Larsen M., Zwart H.,Corrections and extensions of “Optimal control of linear systems with almost periodic inputs” by G. Da Prato and A. Ichikawa. SIAM J. Control Optimiz. 36, pp. 1473-1480, 1998

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Kahane J. P.,Some Random Series of Functions, 2nd. Ed., Cambridge, 1993

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J.,Sampling theorems for non stationary processes, Trans

Lee A. J.,Sampling theorems for non stationary processes, Trans. of the A.M.S. V. 242 pp. 225-241. 1978

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P.,A Sampling Theorem For Stationary (wide sense) stochastic pro- cesses, Trans

Lloyd S. P.,A Sampling Theorem For Stationary (wide sense) stochastic pro- cesses, Trans. of the A.M.S. 92(1), pp. 1-12, 1959

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Katznelson Y.,An Introduction to Harmonic Analysis, Dover, 1976

work page 1976

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H., Ron A.,Time Frequency Representations of Almost Periodic Func- tions, Constr

Kim Y. H., Ron A.,Time Frequency Representations of Almost Periodic Func- tions, Constr. Approx. 29, pp. 303-323, 2009

work page 2009

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H., Representations of Almost-Periodic Functions Using Generalized Shift-Invariant Systems inRd, J

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A., Vainikko G

Krasnosel’skii M. A., Vainikko G. M., Zabreiko P. P., Rutitskii Va. B., Stetsenko V. Va.,Approximate Solution of Operator Equations, Wolters-Noordhoff, 1972. 25

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, Woyczyński W.A.,Random Series and Stochastic Integrals: Single and Multiple, Birkhäuser, 1992

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M.,Laws of Large Numbers, Spectral Translates and Sampling over LCA Groups, J

Medina J. M.,Laws of Large Numbers, Spectral Translates and Sampling over LCA Groups, J. Fourier Anal. Appl. 30 (67), pp. 1-32. 2024

work page 2024

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Meyer Y.,Wavelets and Operators, Cambridge Studies in Advanced Mathemat- ics, 37, 1992

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R.,On the Windowed Fourier Transform and Wavelet Transform of Almost Periodic Functions, Appl

Partington J. R.,On the Windowed Fourier Transform and Wavelet Transform of Almost Periodic Functions, Appl. Comput. Anal. Appl. 10, pp. 45-60, 2001

work page 2001

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Pinsky M., Introduction to Fourier Analysis and Wavelets (Graduate Studies in Mathematics), American Mathematical Society, 2009

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work page 1997

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Rosenblatt M.,Stationary Sequences and Random Fields, Birkhäuser, 1985

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Samko S. G.,Hypersingular Integrals and Their Applications, CRC Press, 2002

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A. P. Calderón

Stein E. M., Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1987. H.D. Centeno: UniversidaddeBuenosAires, FacultaddeIngeniería, DepartamentodeMatemática// Universidad de Buenos Aires, Facultad de Ciencias Exactas y Naturales, Departa- mento de Matemática. Correspondence: hcenteno@fi.uba.ar J.M. Medina: (https...

work page 1987