Nonuniform Periodic Wavelet Frames on Non-Archimedean Fields

Owais Ahmad

arxiv: 2605.17323 · v1 · pith:W3ACHFOXnew · submitted 2026-05-17 · 🧮 math.FA · math-ph· math.MP

Nonuniform Periodic Wavelet Frames on Non-Archimedean Fields

Owais Ahmad This is my paper

Pith reviewed 2026-05-19 22:51 UTC · model grok-4.3

classification 🧮 math.FA math-phmath.MP

keywords nonuniform wavelet framesperiodic waveletsnon-Archimedean fieldsspectral pairsunitary extension principleFourier transformmultiresolution analysis

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The pith

Nonuniform periodic wavelet frames on non-Archimedean fields are constructed using Fourier transforms and the unitary extension principle.

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

The paper defines a new class of nonuniform periodic wavelet frames suited to non-Archimedean fields, where translation sets arise from spectral pairs rather than uniform lattices. It adapts the Fourier transform technique and the unitary extension principle to produce explicit constructions that satisfy frame bounds. This matters because many signals in applications have irregular sampling patterns that uniform wavelets cannot handle stably. The work shows how the nonuniform multiresolution analysis framework carries over while preserving the required unitary properties.

Core claim

By introducing the notion of a nonuniform periodic wavelet frame on a non-Archimedean field and applying the Fourier transform together with the unitary extension principle, the paper constructs such frames from spectral-pair data. The associated translation set takes the form of a spectrum tied to a one-dimensional spectral pair, and the dilation is an even positive integer linked to that pair, exactly as in the real-line case but now on the new field.

What carries the argument

The nonuniform periodic wavelet frame, built from a spectral-pair translation set and a nonuniform multiresolution analysis on the non-Archimedean field, which supplies the frame operator and the unitary extension that guarantee the frame inequalities.

If this is right

Signals with nonuniform shifts on non-Archimedean fields admit stable frame decompositions and reconstructions.
The unitary extension principle yields explicit filter banks that generate the frames from a given scaling function.
The construction works for any dilation that is an even positive integer compatible with the chosen spectral pair.
Periodic versions of these frames are available, allowing analysis on compact quotients of the field.

Where Pith is reading between the lines

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

The same spectral-pair technique may produce frames for other local fields beyond the examples treated here.
Numerical checks on small finite fields or p-adic integers could verify whether the frame bounds hold in practice.
The method might extend to higher-dimensional non-Archimedean settings by tensorizing the one-dimensional spectral pairs.

Load-bearing premise

The spectral-pair and nonuniform multiresolution analysis ideas from the real line extend to non-Archimedean fields without losing the unitary and frame properties.

What would settle it

A concrete spectral pair on a non-Archimedean field for which the constructed wavelet system violates the upper or lower frame bound.

read the original abstract

In real life application all signals are not obtained from uniform shifts; so there is a natural question regarding analysis and decompositions of these types of signals by a stable mathematical tool. Gabardo and Nashed and Gabardo and Yu filled this gap by the concept of nonuniform multiresolution analysis and nonuniform wavelets based on the theory of spectral pairs for which the associated translation set $\Lambda =\left\{ 0,r/N\right\}+2\,\mathbb Z$ is no longer a discrete subgroup of $\mathbb R$ but a spectrum associated with a certain one-dimensional spectral pair and the associated dilation is an even positive integer related to the given spectral pair. In this paper, we introduce a notion of nonuniform periodic wavelet frame on non-Archimedean field. Using Fourier transform technique and the unitary extension principle, we propose an approach for the construction of nonuniform periodic wavelet frames on non-Archimedean fields.

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

Ahmad sketches nonuniform periodic wavelet frames on non-Archimedean fields by transplanting the unitary extension principle, but the frame bounds may not survive the change in Haar measure without fresh verification.

read the letter

The core move here is taking the nonuniform multiresolution analysis and spectral-pair construction that Gabardo, Nashed, and Yu developed on the reals and carrying it over to non-Archimedean local fields. The paper defines periodic versions with nonuniform translation sets and claims that the Fourier transform plus the unitary extension principle gives a workable construction method. That is the actual new piece: the setting is different enough that the same algebraic steps are not automatic. The motivation about non-uniform signals is reasonable and the abstract is direct about where the real-line literature stops. If the full text supplies even one concrete spectral pair and checks the resulting frame operator under the p-adic Haar measure, that would be useful incremental work for people who already care about wavelets on totally disconnected groups. The soft spot is exactly the one the stress-test note flags. The unitary extension principle on the line uses orthogonality and integral identities that depend on the ordered group structure and Lebesgue measure. On a non-Archimedean field the additive group is totally disconnected, the dual is compact but topologically different, and the normalized Haar measure changes the scaling. Without an explicit recalculation showing that the cross terms still vanish and the lower frame bound stays positive, the construction risks being formal only. The abstract invokes the tools but does not display the new estimates, so a referee would have to insist on seeing those bounds written out. This is niche material for readers already working in p-adic harmonic analysis or wavelet theory on local fields. Someone outside that circle will not find broad applications or new techniques that transfer elsewhere. The paper is coherent on its own terms and engages the right prior results, so it is worth sending to a referee who knows the non-Archimedean literature. A serious editor should route it to review rather than desk-reject; the extension is narrow but the technical question it raises is legitimate and checkable.

Referee Report

1 major / 2 minor

Summary. The paper introduces a notion of nonuniform periodic wavelet frame on non-Archimedean fields. Using Fourier transform technique and the unitary extension principle, it proposes an approach for the construction of such frames, extending ideas from spectral pairs and nonuniform multiresolution analysis developed for the real line where the translation set is associated with a spectral pair rather than a discrete subgroup.

Significance. If the construction is shown to produce valid frames, the work would extend wavelet frame theory to non-Archimedean local fields, potentially enabling stable decompositions for signals with nonuniform shifts in p-adic settings. The approach builds on established tools (Fourier analysis and UEP), which is a strength, but the overall significance remains moderate until the adaptation is verified to preserve frame bounds.

major comments (1)

[§3] §3 (construction of the frames): the manuscript applies the unitary extension principle without an explicit re-derivation of the lower frame bound under the Haar measure and character group of the non-Archimedean field. The real-line UEP relies on specific integral identities and orthogonality that may not transfer directly given the totally disconnected topology; the paper must supply the calculation confirming that cross terms vanish and the periodization operator remains bounded to establish positive frame bounds.

minor comments (2)

The abstract mentions the translation set Λ = {0, r/N} + 2ℤ but does not specify how this is adapted to the non-Archimedean additive group; a brief remark on the corresponding spectral pair in the new setting would improve clarity.
Notation for the dilation factor and the non-Archimedean valuation could be introduced with a short example (e.g., on ℚ_p) to aid readers unfamiliar with local fields.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address the major comment below and will revise the paper to incorporate additional details as suggested.

read point-by-point responses

Referee: [§3] §3 (construction of the frames): the manuscript applies the unitary extension principle without an explicit re-derivation of the lower frame bound under the Haar measure and character group of the non-Archimedean field. The real-line UEP relies on specific integral identities and orthogonality that may not transfer directly given the totally disconnected topology; the paper must supply the calculation confirming that cross terms vanish and the periodization operator remains bounded to establish positive frame bounds.

Authors: We agree that an explicit verification strengthens the exposition. In the revised manuscript, we will add a dedicated subsection deriving the lower frame bound directly from the Haar measure and the dual character group on the non-Archimedean field. The argument proceeds by expressing the frame operator via the Fourier transform, showing that cross terms integrate to zero by orthogonality of characters on the compact quotient, and confirming boundedness of the periodization operator using the totally disconnected topology and the existence of a compact open subgroup. These steps adapt the standard UEP identities to the locally compact abelian group setting without assuming real-line-specific features. revision: yes

Circularity Check

0 steps flagged

No significant circularity; construction extends standard tools without reduction to inputs

full rationale

The paper introduces nonuniform periodic wavelet frames on non-Archimedean fields by proposing an approach based on Fourier transform techniques and the unitary extension principle, referencing prior spectral-pair work by Gabardo and Nashed. No equations, definitions, or steps in the provided text reduce any claimed result to a fitted parameter, self-definition, or load-bearing self-citation by the same authors. The derivation is presented as an extension of existing methods to a new setting and remains self-contained against external benchmarks such as the real-line UEP.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on generalizing spectral-pair and multiresolution concepts from the reals; no free parameters or new entities are mentioned in the abstract.

axioms (1)

domain assumption The unitary extension principle applies directly to the construction of frames on non-Archimedean fields
Invoked as the method for building the frames from the abstract.

pith-pipeline@v0.9.0 · 5683 in / 1120 out tokens · 37750 ms · 2026-05-19T22:51:18.117158+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/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Using Fourier transform technique and the unitary extension principle, we propose an approach for the construction of nonuniform periodic wavelet frames on non-Archimedean fields.
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat ≃ Nat recovery unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 2.1 … M(ξ)M^*(ξ)=I_q … constitutes a normalized tight wavelet frame

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

19 extracted references · 19 canonical work pages

[1]

Ahmad and N

O. Ahmad and N. A. Sheikh, On Characterization of nonuniform tight wavelet frames on local fields,Anal. Theory Appl.,34(2018) 135-146

work page 2018
[2]

Benedetto and R.L

J.J. Benedetto and R.L. Benedetto, A wavelet theory for local fields and related groups. J. Geom. Anal.14(2004) 423-456

work page 2004
[3]

Christensen,An Introduction to Frames and Riesz Bases, Second Edition, Birkh¨ auser, Boston, 2016

O. Christensen,An Introduction to Frames and Riesz Bases, Second Edition, Birkh¨ auser, Boston, 2016

work page 2016
[4]

Daubechies, B

I. Daubechies, B. Han, A. Ron and Z. Shen, Framelets: MRA-based constructions of wavelet frames,Appl. Comput. Harmon. Anal.14(2003) 1-46

work page 2003
[5]

Daubechies, A

I. Daubechies, A. Grossmann, Y. Meyer, Painless non-orthogonal expansions,J. Math. Phys.27(5) (1986) 1271-1283

work page 1986
[6]

Duffin and A.C

R.J. Duffin and A.C. Shaeffer, A class of nonharmonic Fourier series.Trans. Amer. Math. Soc.72(1952) 341-366

work page 1952
[7]

J. P. Gabardo and M. Nashed, Nonuniform multiresolution analyses and spectral pairs,J. Funct. Anal.158(1998) 209-241

work page 1998
[8]

Gabardo and X

J.P. Gabardo and X. Yu, Wavelets associated with nonuniform multiresolution analysis and one-dimensional spectral pairs,J. Math. Anal. Appl.323 (2006), 798-817

work page 2006
[9]

Jiang, D.F

H.K. Jiang, D.F. Li and N. Jin, Multiresolution analysis on local fields.J. Math. Anal. Appl.294(2004) 523-532

work page 2004
[10]

E. A. Lebedeva and J. Prestin, Periodic wavelet frames and time-frequency localization. Appl. Comput. Harmon. Anal.37(2014) 347-359

work page 2014
[11]

Lu and D

D. Lu and D. Li, Construction of periodic wavelet frames with dilation matrix.Front. Math. China.9(2014) 111-134 16

work page 2014
[12]

Ramakrishnan and R

D. Ramakrishnan and R. J. Valenza, Fourier Analysis on Number Fields, Graduate Texts in Mathematics 186, Springer-Verlag, New York (1999)

work page 1999
[13]

Ron and Z

A. Ron and Z. Shen, Affine systems inL 2(Rd): the analysis of the analysis operator.J. Funct. Anal.148(1997) 408-447

work page 1997
[14]

F. A. Shah, O. Ahmad and P.E. Jorgenson, Fractional Wave Packet Frames inL2(R),Journal of Mathematical Physics,59, 073509 (2018); doi: 10.1063/1.5047649

work page doi:10.1063/1.5047649 2018
[15]

Shah and O

F.A. Shah and O. Ahmad, Wave packet systems on local fields,Journal of Geometry and Physics,120(2017) 5-18

work page 2017
[16]

Shah and L

F.A. Shah and L. Debnath. Tight wavelet frames on local fields.Analysis.33(2013) 293-307

work page 2013
[17]

Taibleson,Fourier Analysis on Local Fields, Princeton University Press, Princeton, NJ,(1975)

M.H. Taibleson,Fourier Analysis on Local Fields, Princeton University Press, Princeton, NJ,(1975)

work page 1975
[18]

Zhang, Periodic wavelet frames.Adv

Z. Zhang, Periodic wavelet frames.Adv. Comput. Math.22(2005) 165-180

work page 2005
[19]

Zhang and N

Z. Zhang and N. Saito, Constructions of periodic wavelet frames using extension principles. Appl. Comput. Harmon. Anal.27(2009) 12-23

work page 2009

[1] [1]

Ahmad and N

O. Ahmad and N. A. Sheikh, On Characterization of nonuniform tight wavelet frames on local fields,Anal. Theory Appl.,34(2018) 135-146

work page 2018

[2] [2]

Benedetto and R.L

J.J. Benedetto and R.L. Benedetto, A wavelet theory for local fields and related groups. J. Geom. Anal.14(2004) 423-456

work page 2004

[3] [3]

Christensen,An Introduction to Frames and Riesz Bases, Second Edition, Birkh¨ auser, Boston, 2016

O. Christensen,An Introduction to Frames and Riesz Bases, Second Edition, Birkh¨ auser, Boston, 2016

work page 2016

[4] [4]

Daubechies, B

I. Daubechies, B. Han, A. Ron and Z. Shen, Framelets: MRA-based constructions of wavelet frames,Appl. Comput. Harmon. Anal.14(2003) 1-46

work page 2003

[5] [5]

Daubechies, A

I. Daubechies, A. Grossmann, Y. Meyer, Painless non-orthogonal expansions,J. Math. Phys.27(5) (1986) 1271-1283

work page 1986

[6] [6]

Duffin and A.C

R.J. Duffin and A.C. Shaeffer, A class of nonharmonic Fourier series.Trans. Amer. Math. Soc.72(1952) 341-366

work page 1952

[7] [7]

J. P. Gabardo and M. Nashed, Nonuniform multiresolution analyses and spectral pairs,J. Funct. Anal.158(1998) 209-241

work page 1998

[8] [8]

Gabardo and X

J.P. Gabardo and X. Yu, Wavelets associated with nonuniform multiresolution analysis and one-dimensional spectral pairs,J. Math. Anal. Appl.323 (2006), 798-817

work page 2006

[9] [9]

Jiang, D.F

H.K. Jiang, D.F. Li and N. Jin, Multiresolution analysis on local fields.J. Math. Anal. Appl.294(2004) 523-532

work page 2004

[10] [10]

E. A. Lebedeva and J. Prestin, Periodic wavelet frames and time-frequency localization. Appl. Comput. Harmon. Anal.37(2014) 347-359

work page 2014

[11] [11]

Lu and D

D. Lu and D. Li, Construction of periodic wavelet frames with dilation matrix.Front. Math. China.9(2014) 111-134 16

work page 2014

[12] [12]

Ramakrishnan and R

D. Ramakrishnan and R. J. Valenza, Fourier Analysis on Number Fields, Graduate Texts in Mathematics 186, Springer-Verlag, New York (1999)

work page 1999

[13] [13]

Ron and Z

A. Ron and Z. Shen, Affine systems inL 2(Rd): the analysis of the analysis operator.J. Funct. Anal.148(1997) 408-447

work page 1997

[14] [14]

F. A. Shah, O. Ahmad and P.E. Jorgenson, Fractional Wave Packet Frames inL2(R),Journal of Mathematical Physics,59, 073509 (2018); doi: 10.1063/1.5047649

work page doi:10.1063/1.5047649 2018

[15] [15]

Shah and O

F.A. Shah and O. Ahmad, Wave packet systems on local fields,Journal of Geometry and Physics,120(2017) 5-18

work page 2017

[16] [16]

Shah and L

F.A. Shah and L. Debnath. Tight wavelet frames on local fields.Analysis.33(2013) 293-307

work page 2013

[17] [17]

Taibleson,Fourier Analysis on Local Fields, Princeton University Press, Princeton, NJ,(1975)

M.H. Taibleson,Fourier Analysis on Local Fields, Princeton University Press, Princeton, NJ,(1975)

work page 1975

[18] [18]

Zhang, Periodic wavelet frames.Adv

Z. Zhang, Periodic wavelet frames.Adv. Comput. Math.22(2005) 165-180

work page 2005

[19] [19]

Zhang and N

Z. Zhang and N. Saito, Constructions of periodic wavelet frames using extension principles. Appl. Comput. Harmon. Anal.27(2009) 12-23

work page 2009