Further Evidence for Near-Tsirelson Bell-CHSH Violations in Quantum Field Theory via Haar Wavelets

David Dudal; Ken Vandermeersch

arxiv: 2410.13362 · v3 · submitted 2024-10-17 · 🧮 math-ph · hep-th· math.MP· quant-ph

Further Evidence for Near-Tsirelson Bell-CHSH Violations in Quantum Field Theory via Haar Wavelets

David Dudal , Ken Vandermeersch This is my paper

Pith reviewed 2026-05-23 19:28 UTC · model grok-4.3

classification 🧮 math-ph hep-thmath.MPquant-ph

keywords Bell-CHSHTsirelson boundHaar waveletsquantum field theoryvacuum stateBell inequality violationswavelet integrals

0 comments

The pith

Bell-CHSH violations in quantum field theory can approach Tsirelson's bound if the maximal eigenvalues of Haar wavelet integral matrices approach π.

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

The paper investigates a construction based on bumpified Haar wavelets to generate explicit Bell-CHSH violations in the vacuum state of a (1+1)-dimensional massless spinor quantum field theory. It reduces the claim that these violations can get arbitrarily close to Tsirelson's bound to the mathematical question of whether the largest eigenvalue of certain symmetric matrices built from integrals of wavelet products tends to π in the limit of finer resolutions. The authors establish a lower bound of 3.11052 using a specific subclass of wavelets and present further numerical evidence supporting the general conjecture. If correct, this would mean that the standard vacuum already allows near-maximal quantum correlations in simple field theories. The result matters for understanding how much nonlocality is inherent in quantum field theory without special state preparation.

Core claim

The construction using bumpified Haar wavelets reduces the question of near-Tsirelson Bell-CHSH violations in the (1+1)D QFT vacuum to the conjecture that the asymptotic maximal eigenvalue of the sequence of symmetric matrices composed of integrals of Haar wavelet products approaches π; a subclass of wavelets achieves an eigenvalue of 3.11052, with additional numerical evidence provided for the conjecture.

What carries the argument

Symmetric matrices formed from integrals of products of bumpified Haar wavelets, whose maximal eigenvalue is conjectured to approach π in the continuum limit.

If this is right

Bell-CHSH violations can get arbitrarily close to Tsirelson's bound.
The vacuum state of massless spinor fields in (1+1)D Minkowski space permits strong Bell violations.
The wavelet construction provides an explicit way to realize near-maximal quantum correlations in QFT.
Numerical computations can test the approach to the bound by increasing matrix size.

Where Pith is reading between the lines

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

The conjecture, if true, might extend to other wavelet bases or field theories in different dimensions.
Reaching only 3.11052 suggests the convergence to π may be slow, requiring higher resolutions for closer approaches.
This could connect to broader questions about the maximal Bell violation achievable in relativistic quantum theories.

Load-bearing premise

The maximal eigenvalue of the symmetric matrices from Haar wavelet product integrals approaches π in the continuum limit.

What would settle it

A computation of the maximal eigenvalue for a sequence of increasingly large matrices that fails to increase toward π, for example remaining below 3.12 at very high resolution, would disprove the conjecture.

Figures

Figures reproduced from arXiv: 2410.13362 by David Dudal, Ken Vandermeersch.

**Figure 1.** Figure 1: The mother Haar wavelet ψ = ψ0,0. is satisfied and the linear span of {ψn,k}n,k∈Z is dense in L 2 (R). [17] The following symmetry property will be very useful to us. Lemma 3.2. For all n, k ∈ Z we have ψn,k(−x) = −ψn,−k−1(x) almost everywhere. Proof. We start by noting that the property holds for the mother Haar wavelet ψ = ψ0,0. Indeed, ψ0,0(−x) = ψ(−x) =    +1 if 0 ≤ −x < 1 2 −1 if 1 2 ≤ −x < 1 0 o… view at source ↗

**Figure 2.** Figure 2: The a.e. equality ψn,k(−x) = −ψn,−k−1(x), visually. that ψn,k(−x) = ( 2 n/2 ψ(2n (−x) − k) if − x ∈ In,k 0 otherwise = ( −2 n/2 ψ(2nx + k + 1) if x ∈ In,−k−1 0 otherwise = −ψn,−k−1(x) holds almost everywhere. See also [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** Figure 3: The graph of the function J from the proof of Lemma 3.4. Proof. Let us rewrite the given integral by first integrating with respect to x: A(n,k),(m,ℓ) = − Z ψm,−ℓ(y) Z ψn,−k(x) x + y dx dy. Then, for each y < 0 we can compute the inner integral as I(y) := Z ψn,−k(x) x + y dx = Z (−k+ 1 2 )2−n −k2−n +2n/2 x + y dx + Z (−k+1)2−n (−k+ 1 2 )2−n −2 n/2 x + y dx =2n/2 ln (−k + 1 2 )2−n + y −k2−n + y ! − 2 n/2… view at source ↗

**Figure 4.** Figure 4: Fig. 2 in [12]: Test function components for [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗

**Figure 5.** Figure 5: The test function components of f = (f1, f2) found in [12] satisfy the relation f2 = −cf1 quite closely. where d = (N1 − N0 + 1) × K. Then, the problem statement compactifies to (1 − c 2 ) X n,k,m,ℓ A(n,k),(m,ℓ) xn,k xm,ℓ = π √ 2η 1 + η 2 (6) −2c X n,k,m,ℓ A(n,k),(m,ℓ) xn,k xm,ℓ = − π √ 2η 1 + η 2 , (7) and the normality condition on f becomes the following normality condition on x: ∥x∥ 2 = 1/(1 + c 2 ). A… view at source ↗

**Figure 6.** Figure 6: Discrete plots for λmax A(N, 2) and λmax A(N, 5) , together with a comparison. 16 [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗

**Figure 7.** Figure 7: The terms an vanish relatively quickly. where the sequence a0, a1, a2, . . . is given by (see the proof of Lemma 3.4) an = A(0,1),(n,1) = −2 n/2 Z −2−n−1 −2−n J(y − 1) dy + 2n/2 Z 0 −2−n−1 J(y − 1) dy. The following closed form expression can be obtained: an = 2−n/2 2 n+1 − 2 n ln(2) − 2 [PITH_FULL_IMAGE:figures/full_fig_p019_7.png] view at source ↗

**Figure 8.** Figure 8: The first four functions fn(x) = 2 −n/2 (1 − x) x + 2n together with their linear upper bounds gn(x) = 2−3/2 (1 − x). Combining Lemmas 3.13 to 3.15, we find the following: Proposition 3.16. The asymptotic maximal eigenvalue approaches lim N→∞ λmax A(N, 1) = ln 1024 729 + 2α + 2(3 − 2 √ 2)X∞ n=1 ιn ≈ 3.1105202. Remark 3.17. This is already 3.11052/π = 99.01% of the way to π; but as could be expected, w… view at source ↗

**Figure 9.** Figure 9: The graphs of λmax F2(t) and λmax F4(t) . The graphs corresponding to different values of K are very similar. Let us define FK(t) as the K × K matrix valued Fourier series FK(t) = X∞ n=−∞ An(K)e int, t ∈ [0, 2π]. According to Theorem 3.19, we have the result lim N→∞ λmax A(N, K) = max t∈[0,2π] λmax FK(t) . The situation here is considerably more complex than the special case K = 1, as we now need t… view at source ↗

**Figure 10.** Figure 10: The basic Planck-taper window function s ε . 4.1 Bumpified Haar wavelets To construct smooth “bumpified” versions of the Haar wavelets (still compactly supported), following [10], the authors of [12] employ the Planck-taper window function. Definition 4.1. The basic Planck-taper window function with support on the interval [0, 1] is defined by s ε (x) =    1 + exp ε(2x − ε) x(x − ε) −1… view at source ↗

**Figure 11.** Figure 11: The mother bumpified Haar wavelet σ ε = σ ε 0,0 . Definition 4.4 (Bumpified Haar wavelets). The mother bumpified Haar wavelet with support on [0, 1] is defined by σ ε (x) =    +s ε (2x) if 0 ≤ x < 1 2 −s ε (2x − 1) if 1 2 ≤ x < 1 0 otherwise. Bumpified versions of the Haar wavelets ψn,k may then be defined by σ ε n,k(x) = ( 2 n/2σ ε (2nx − k) if x ∈ In,k 0 otherwise. Analogously as in Proposition… view at source ↗

read the original abstract

This paper investigates a recent construction using bumpified Haar wavelets to demonstrate explicit violations of the Bell-Clauser-Horne-Shimony-Holt inequality within the vacuum state in quantum field theory. The construction was tested for massless spinor fields in $(1+1)$-dimensional Minkowski spacetime and is claimed to achieve violations arbitrarily close to an upper bound known as Tsirelson's bound. We show that this claim can be reduced to a mathematical conjecture involving the maximal eigenvalue of a sequence of symmetric matrices composed of integrals of Haar wavelet products. More precisely, the asymptotic eigenvalue of this sequence should approach $\pi$. We present a formal argument using a subclass of wavelets, allowing us to reach $3.11052$. Although a complete proof remains elusive, we present further compelling numerical evidence to support it.

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

Reduces the Bell violation claim to an eigenvalue conjecture on Haar matrices, with a new subclass lower bound of 3.11052 and supporting numerics, but the key limit to π stays unproven.

read the letter

The main takeaway is that the authors have recast their wavelet construction for near-Tsirelson Bell-CHSH violations in (1+1)D QFT vacuum states as a precise conjecture: the largest eigenvalue of the sequence of Gram matrices from integrals of bumpified Haar wavelet products should approach π in the large-N limit. They give a formal argument that works for a restricted subclass of those wavelets, yielding a concrete lower bound of 3.11052, and they add numerical diagonalization that reaches the same value and appears consistent with the conjecture. This is a legitimate sharpening of the earlier work cited in the abstract. The reduction itself is useful because it turns the physical statement into an independent matrix problem rather than something that depends on fitted parameters or self-referential definitions. The subclass result is new and supplies something rigorous where only numerics existed before. The numerics look competently executed for the purpose. The central limitation is exactly what the stress-test note flags: the full asymptotic claim remains a conjecture. No analytic proof is given for the general case, and there are no rigorous error bounds or convergence rates on the numerical extrapolation. If the true limit is strictly less than π or the approach is too slow for the continuum QFT limit, the near-Tsirelson violations do not follow. This is a real gap, not a minor one, but it is openly acknowledged. The paper is aimed at readers working on explicit constructions of Bell violations in relativistic QFT or on wavelet techniques in field theory. Someone looking for a fully proven result will not find it, but the reduction and partial proof make the claim more testable and worth checking. It shows clear thinking and honest engagement with the prior literature. I would send it to peer review rather than desk reject; the formal reduction and subclass argument are substantive enough to merit referee time even if heavy revision is needed on the conjecture.

Referee Report

2 major / 2 minor

Summary. The paper reduces the construction of near-Tsirelson Bell-CHSH violations in (1+1)-dimensional QFT vacuum states via bumpified Haar wavelets to an independent mathematical conjecture: that the largest eigenvalue λ_N of the N×N Gram matrices formed from integrals of products of these wavelets satisfies lim_{N→∞} λ_N = π. It supplies a rigorous lower bound λ ≥ 3.11052 for a restricted subclass of wavelets together with numerical diagonalization evidence for the full sequence.

Significance. If the eigenvalue conjecture holds, the work supplies concrete, falsifiable evidence that QFT can realize Bell-CHSH violations arbitrarily close to Tsirelson’s bound, a result of interest at the interface of quantum field theory and quantum information. The explicit mapping to an eigenvalue problem on Haar-wavelet Gram matrices, the analytic subclass bound, and the numerical checks constitute genuine strengths that render the physical claim testable without circularity or free parameters.

major comments (2)

[Main text (conjecture statement and numerical section)] The central physical claim rests on the unproven limit lim λ_N = π. While the reduction itself is cleanly executed and the subclass argument (yielding 3.11052) is formal, the manuscript supplies neither an analytic proof of the full limit nor a rigorous error bound or convergence-rate estimate on the numerical extrapolation; this gap directly limits the strength of the “arbitrarily close” assertion.
[Numerical results] The numerical evidence is reported only up to the subclass value 3.11052; the manuscript does not state the largest N for which the full matrices were diagonalized, the observed approach rate to π, or any extrapolation procedure, making it impossible to quantify how convincingly the numerics support the continuum limit.

minor comments (2)

[Preliminaries] Notation for the bumpified Haar wavelets and the precise definition of the Gram-matrix entries ∫ ψ_i(x) ψ_j(x) dx should be collected in a single preliminary section for readability.
[Numerical results] A short table comparing the subclass bound 3.11052, the numerical maxima, and π would help readers gauge the remaining gap.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the positive assessment of the reduction to the eigenvalue conjecture, the formal subclass bound, and the overall significance. We address the major comments point by point below.

read point-by-point responses

Referee: [Main text (conjecture statement and numerical section)] The central physical claim rests on the unproven limit lim λ_N = π. While the reduction itself is cleanly executed and the subclass argument (yielding 3.11052) is formal, the manuscript supplies neither an analytic proof of the full limit nor a rigorous error bound or convergence-rate estimate on the numerical extrapolation; this gap directly limits the strength of the “arbitrarily close” assertion.

Authors: We agree that the claim of violations arbitrarily close to Tsirelson’s bound is conditional on the unproven conjecture lim_{N→∞} λ_N = π. The manuscript already states explicitly that a complete analytic proof remains elusive and presents the subclass result of 3.11052 as a rigorous lower bound together with numerical evidence. We do not have an analytic proof of the full limit or rigorous error bounds on the numerical extrapolation. We will expand the numerical section with additional details on convergence behavior to better support the evidence presented. revision: partial
Referee: [Numerical results] The numerical evidence is reported only up to the subclass value 3.11052; the manuscript does not state the largest N for which the full matrices were diagonalized, the observed approach rate to π, or any extrapolation procedure, making it impossible to quantify how convincingly the numerics support the continuum limit.

Authors: We will revise the manuscript to state the largest N for which the full matrices were diagonalized, report the observed approach rate of λ_N toward π, and describe the extrapolation procedure. These additions will make the numerical support for the conjecture more transparent and quantifiable. revision: yes

standing simulated objections not resolved

Analytic proof of lim λ_N = π together with rigorous error bounds or convergence-rate estimates on the numerical extrapolation.

Circularity Check

0 steps flagged

No circularity detected; central claim is an explicit unproven conjecture on eigenvalue limits, supported by subclass proof and numerics without reduction to inputs by construction.

full rationale

The paper explicitly reduces the QFT Bell-CHSH claim to the mathematical conjecture that lim λ_N = π for the maximal eigenvalue of Gram matrices built from integrals of bumpified Haar wavelets. A formal lower bound of 3.11052 is proven for a restricted subclass, with the general limit supported by numerical diagonalization rather than any fitted parameter or self-referential definition. No step renames a known result, smuggles an ansatz via self-citation, or equates a 'prediction' to a fitted input. Self-citations to prior work (if present) are not load-bearing for the central conjecture, which is independently evidenced here. The derivation chain is transparent and self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard properties of Haar wavelets and the definition of the QFT vacuum state; no free parameters, invented entities, or ad-hoc axioms are introduced beyond the central unproven conjecture itself.

axioms (2)

standard math Standard L2 properties and orthogonality relations of Haar wavelets and their bumpified versions
Invoked when defining the matrix elements from wavelet products.
domain assumption The vacuum expectation values in massless (1+1)D spinor QFT can be expressed via the wavelet operators
Required to translate the physical Bell operator into the wavelet matrix.

pith-pipeline@v0.9.0 · 5674 in / 1323 out tokens · 41549 ms · 2026-05-23T19:28:03.423152+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/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

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

Conjecture B. ... lim N,K→∞ λ_max(A(N,K)) = π
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

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

A(n,k),(m,ℓ) = −∬ (1/(x+y)) ψ_n,-k(x) ψ_m,-ℓ(y) dx dy; block-Toeplitz structure

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

17 extracted references · 17 canonical work pages

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John S. Bell. On the Einstein Podolsky Rosen paradox. Physics Physique Fizika , 1:195–200, 1964. 36

work page 1964
[2]

Clauser, Michael A

John F. Clauser, Michael A. Horne, Abner Shimony, and Richard A. Holt. Proposed Experiment to Test Local Hidden-Variable Theories.Physical Review Letters, 23:880– 884, 1969

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Tsirelson

Boris S. Tsirelson. Quantum generalizations of Bell’s inequality. Letters in Mathe- matical Physics, 4(2):93–100, 1980

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[4]

Summers and Reinhard Werner

Stephen J. Summers and Reinhard Werner. Bell’s inequalities and quantum field theory. I. General setting. Journal of Mathematical Physics , 28(10):2440–2447, 1987

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[6]

Summers and Reinhard Werner

Stephen J. Summers and Reinhard Werner. Maximal violation of Bell’s inequalities is generic in quantum field theory. Communications in Mathematical Physics, 110:247– 259, 1987

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[7]

Orthonormal Bases of Compactly Supported Wavelets

Ingrid Daubechies. Orthonormal Bases of Compactly Supported Wavelets. Commu- nications on Pure and Applied Mathematics , 41(7):909–996, 1988

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[8]

Asymptotic Spectra of Hermitian Block Toeplitz Matrices and Preconditioning Results

Michele Miranda Jr and Paolo Tilli. Asymptotic Spectra of Hermitian Block Toeplitz Matrices and Preconditioning Results. SIAM Journal on Matrix Analysis and Ap- plications, 21:867–881, 2000

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Regalia, and Jean-Pierre Delmas

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Guimaraes, Itzhak Roditi, and Sil- vio P

David Dudal, Philipe De Fabritiis, Marcelo S. Guimaraes, Itzhak Roditi, and Sil- vio P. Sorella. Maximal violation of the bell-clauser-horne-shimony-holt inequality via bumpified haar wavelets. Physical Review D, 108:L081701, 2023

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[13]

Robert M. Gray. Toeplitz and Circulant Matrices: A Review. Foundations and Trends in Communications and Information Theory. Now Publishers, 2006

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Toeplitz Forms and Their Applications

Ulf Grenander and Gabor Szeg˝ o. Toeplitz Forms and Their Applications . California monographs in mathematical sciences. University of California Press, 1958. 37

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Haar Wavelets: With Applications

¨Ulo Lepik and Helle Hein. Haar Wavelets: With Applications . Springer International Publishing, 2014. 38

work page 2014

[1] [1]

John S. Bell. On the Einstein Podolsky Rosen paradox. Physics Physique Fizika , 1:195–200, 1964. 36

work page 1964

[2] [2]

Clauser, Michael A

John F. Clauser, Michael A. Horne, Abner Shimony, and Richard A. Holt. Proposed Experiment to Test Local Hidden-Variable Theories.Physical Review Letters, 23:880– 884, 1969

work page 1969

[3] [3]

Tsirelson

Boris S. Tsirelson. Quantum generalizations of Bell’s inequality. Letters in Mathe- matical Physics, 4(2):93–100, 1980

work page 1980

[4] [4]

Summers and Reinhard Werner

Stephen J. Summers and Reinhard Werner. Bell’s inequalities and quantum field theory. I. General setting. Journal of Mathematical Physics , 28(10):2440–2447, 1987

work page 1987

[5] [5]

Summers and Reinhard Werner

Stephen J. Summers and Reinhard Werner. Bell’s inequalities and quantum field theory. II. Bell’s inequalities are maximally violated in the vacuum. Journal of Math- ematical Physics, 28(10):2448–2456, 1987

work page 1987

[6] [6]

Summers and Reinhard Werner

Stephen J. Summers and Reinhard Werner. Maximal violation of Bell’s inequalities is generic in quantum field theory. Communications in Mathematical Physics, 110:247– 259, 1987

work page 1987

[7] [7]

Orthonormal Bases of Compactly Supported Wavelets

Ingrid Daubechies. Orthonormal Bases of Compactly Supported Wavelets. Commu- nications on Pure and Applied Mathematics , 41(7):909–996, 1988

work page 1988

[8] [8]

Asymptotic Spectra of Hermitian Block Toeplitz Matrices and Preconditioning Results

Michele Miranda Jr and Paolo Tilli. Asymptotic Spectra of Hermitian Block Toeplitz Matrices and Preconditioning Results. SIAM Journal on Matrix Analysis and Ap- plications, 21:867–881, 2000

work page 2000

[9] [9]

Regalia, and Jean-Pierre Delmas

Houcem Gazzah, Philip A. Regalia, and Jean-Pierre Delmas. Asymptotic eigenvalue distribution of block Toeplitz matrices and application to blind SIMO channel iden- tification. IEEE Transactions on Information Theory , 47(3):1243–1251, 2001

work page 2001

[10] [10]

David J. A. McKechan, Craig Robinson, and Bangalore S. Sathyaprakash. A tapering window for time-domain templates and simulated signals in the detection of grav- itational waves from coalescing compact binaries. Classical and Quantum Gravity , 27:084020, 2010

work page 2010

[11] [11]

Lower bounds for the largest eigenvalue of a symmetric matrix under perturbations of rank one

Jacques B´ enass´ eni. Lower bounds for the largest eigenvalue of a symmetric matrix under perturbations of rank one. Linear and Multilinear Algebra, 2011

work page 2011

[12] [12]

Guimaraes, Itzhak Roditi, and Sil- vio P

David Dudal, Philipe De Fabritiis, Marcelo S. Guimaraes, Itzhak Roditi, and Sil- vio P. Sorella. Maximal violation of the bell-clauser-horne-shimony-holt inequality via bumpified haar wavelets. Physical Review D, 108:L081701, 2023

work page 2023

[13] [13]

Robert M. Gray. Toeplitz and Circulant Matrices: A Review. Foundations and Trends in Communications and Information Theory. Now Publishers, 2006

work page 2006

[14] [14]

Toeplitz Forms and Their Applications

Ulf Grenander and Gabor Szeg˝ o. Toeplitz Forms and Their Applications . California monographs in mathematical sciences. University of California Press, 1958. 37

work page 1958

[15] [15]

Local quantum physics: Fields, particles, algebras

Rudolf Haag. Local quantum physics: Fields, particles, algebras . Springer Science & Business Media, 2012

work page 2012

[16] [16]

A Friendly Guide to Wavelets

Gerald Kaiser. A Friendly Guide to Wavelets . Birkhauser Boston Inc., 1994

work page 1994

[17] [17]

Haar Wavelets: With Applications

¨Ulo Lepik and Helle Hein. Haar Wavelets: With Applications . Springer International Publishing, 2014. 38

work page 2014