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Eigenfunctions, free boundaries, and time-frequency localization

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Pith's one-line read In Gaussian time–frequency localization, a near-Gaussian eigenfunction determines a nearby real-analytic domain on which it is an eigenfunction, making the localization domain recoverable from prescribed spectral data.

desk verdict The central inverse construction is unproven due to a false conjugate identity in the unsymmetrization step; the rest of the paper has independent value and deserves serious refereeing. read the letter →

arxiv 2607.21590 v1 pith:SGZI7G26 submitted 2026-07-23 math.AP math.CAmath.CVmath.FA

João P. G. Ramos This is my paper · ORCID
classification math.APmath.CAmath.CVmath.FA MSC 42B1042C1530H2035R3535R3047B3549Q10
keywords time–frequencylocalizationBargmann–Fockspacefree-boundaryproblemsinversespectralproblemquadraturedomainsFaber–KrahninequalityHermitefunctionsconcentration–compactness
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper develops an inverse theory for Gaussian time–frequency localization: instead of asking which waveform a given domain localizes best, it asks which domain a given near-Gaussian waveform localizes. The central result is that for any prescribed polynomial f0 = h0 + Σ a_k h_k with small coefficients and any λ ∈ (0,1), there exists a real-analytic domain U_λ, a small perturbation of a disk, for which f0 is an eigenfunction of the localization operator with eigenvalue λ. This is the first general inverse construction of localization domains, and a sympathetic reader would care because it turns the eigenfunction into recoverable geometric data about the unknown boundary. The same 'eigenfunction-as-geometry' principle yields two proofs that Hermite eigenfunctions force the domain to be a disk, proves that the exponent 1/2 in the quantitative stability inequality cannot be improved, and delivers a new proof that disks are the global maximizers of Gaussian concentration.

What carries the argument

The central object is the moment identity ∫_U P(z) e^{π w z} e^{-π|z|^2} dA(z) = λ P(w) (and its basis form (1.2)), which encodes the eigenvalue equation in the Bargmann–Fock space. The free boundary of U is written as an analytic radial graph φ over the Daubechies disk, in a Wiener-type Banach algebra A^R_τ of holomorphic strip functions; the linearized moment map D_φ F(0,0) is a Fourier multiplier diagonal in angular modes, shown to be an isometric isomorphism onto the weighted sequence space Y_τ. An analytic implicit-function theorem on nested analytic strips (with quantitative remainder estimates (3.11)–(3.12)) produces the perturbed domain. Secondary machinery: a Paley–Wiener divisibili

What would settle it

Compute the moment system (3.4) numerically for f0 = h0 + ε h_2 with ε = 10^{-3} and λ = 1/2, iterating the implicit-function corrections on fine radial graphs; if the iteration fails to converge to a smooth radial graph (or produces self-intersections), the linearized invertibility in Proposition 3.1 does not extend to an actual solution of the infinite system.

Watch

Extended reading notes

Core claim

The paper's central claim is that an eigenfunction can be read as geometric data: near the Gaussian, prescribing the eigenfunction and eigenvalue determines the localization domain. After the Bargmann transform, the eigenvalue equation becomes an infinite weighted moment system against the Fock weight, and the boundary of the unknown domain is parametrized as an analytic radial graph over the Daubechies disk. The key step is that the linearized moment map at the disk is an isometric isomorphism from the Wiener-type space of radial graphs onto a weighted sequence space, so an analytic implicit-function theorem produces a real-analytic domain U_λ for any polynomial f0 = h0 + Σ_{k=1}^N a_k h_k

Load-bearing premise

The whole inverse construction rests on the claim that the linearized moment map at the Daubechies disk is an isometric isomorphism from the space of analytic radial graphs onto the weighted sequence space Y_τ, together with the nonlinear remainder estimates on nested analytic strips; if the infinite moment system hides compatibility conditions invisible at the linearized level, or if the strip-width loss cannot absorb polynomial growth in the mode index j, Theorem 1 fails ev

Editorial extensions

If this is right

  • Prescribed near-Gaussian spectral data determine a real-analytic localization domain, so one can engineer both the waveform and the phase-space region that localizes it.
  • Hermite eigenfunctions force the domain to be a disk in the simply connected case, so the disk is the only domain consistent with such spectra.
  • The exponent 1/2 in both stability inequalities is optimal: no larger exponent can hold uniformly.
  • Local maximizers of the Gaussian Faber–Krahn problem are disks; together with profile decomposition this recovers the global Nicola–Tilli theorem.
  • The two proofs of the Abreu–Dörfler rigidity include one with no boundary regularity hypothesis, showing the disk conclusion is null-set invariant.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The construction suggests a general principle: for Gaussian STFT localization, small perturbations of the Gaussian are 'flexible' — they can be realized as eigenfunctions on many nearby domains, whereas exact Hermite data are rigid. This flexibility/rigidity dichotomy may extend to other special functions in Fock spaces.
  • The isometric linearization provides quantitative control of the inverse map (eigenfunction perturbation → boundary perturbation), which could be made explicit to give a stability theorem for the inverse problem itself.
  • The moment-system formulation is a Gaussian-weighted quadrature-domain structure; extending the finite-moment system to infinite-order quadrature identities might yield new rigidity statements for non-polynomial eigenfunctions.
  • The local-rigidity-plus-profile-decomposition route to Nicola–Tilli suggests that other phase-space concentration functionals (e.g., wavelet transforms) could be treated by local rigidity alone, without rearrangement inequalities.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper proposes an inverse theory for Gaussian time–frequency localization operators: for a prescribed near-Gaussian polynomial f0, it claims to construct a real-analytic domain U such that f0 is an eigenfunction (Theorem 1); it then uses this construction to prove optimality of the exponent 1/2 in the quantitative stability inequality of [22] (Theorem 3), to give two new proofs of the Abreu–Dörfler rigidity theorem (Theorem 2), and to prove that local maximizers of the Gaussian Faber–Krahn functional are disks, yielding a new proof of the Nicola–Tilli theorem (Theorems 4 and 5). The technical core is a Bargmann–Fock moment identity, a disk inverse theorem for the symmetrized problem, and an 'unsymmetrization' step using a coefficientwise-conjugate differential operator.

Significance. If Theorem 1 were valid, it would be the first general inverse construction of localization domains from prescribed eigenfunctions, and it would provide a bridge between inverse potential theory, quadrature domains, and time-frequency analysis. The local-maximizer argument in Section 6, which shows that the derivative of a first eigenfunction is again an eigenfunction and then forces the domain to be a disk, is structurally interesting and potentially important. The paper is also transparent about its external inputs and uses standard tools without circularity. However, the central inverse theorem rests on an algebraically false conjugation identity in the unsymmetrization step. That error is load-bearing: it invalidates Theorem 1 and, through it, Theorem 3. The remaining parts (Theorems 2, 4, 5) may be independent, but the paper's main advertised contribution is not established.

major comments (3)
  1. [§3, Eqs. (3.22)–(3.24) and Remark 3.2] The unsymmetrization step contains a false identity. Equation (3.22) defines P#(z) = 1 + Σ \bar b_k z^k, i.e. P#(z) = \overline{P(\bar z)}. This is not equal to \overline{P(z)} = 1 + Σ \bar b_k \bar z^k for non-real z. Consequently, the displayed computation (3.24), P#(∂w/π)e^{πzw} = P#(z)e^{πzw} = \overline{P(z)}e^{πzw}, is false: the multiplier is P#(z), not \overline{P(z)}. Applying P#(∂w/π) to G(w)=∫_U P(z)e^{-π|z|^2}e^{πzw}dA(z) therefore gives ∫_U P(z)P#(z)e^{-π|z|^2}e^{πzw}dA(z), not ∫_U |P(z)|^2 e^{-π|z|^2}e^{πzw}dA(z). The identification of P#(∂w/π)G with the left side of the symmetrized identity (3.27) is thus unjustified, and the conclusion H=G-λP satisfies P#(∂w/π)H=0 does not follow. Lemma 3.2 is never applied, and (3.25) is unsupported. This is not a notational subtlety; the equality fails on every non-real z, and U is a complex domain. The unsymmetrization mechanism is the
  2. [§5.2, proof of Theorem 3] Theorem 3 is built directly on Theorem 1. The proof begins with 'By Theorem 1, applied with λ=λ0 and Pε(w)=1+εw²', which yields the domain Uε and the eigenvalue identity (5.17). Since the proof of Theorem 1 is invalidated by the false identity in §3, the existence of the family Uε is not established. The subsequent first-variation computation, the identification of the second-harmonic mode, and the distance/deficit asymptotics all depend on (5.17). Thus the claimed optimality of the exponent 1/2 is unsupported. A repaired proof of Theorem 1, or a direct construction of the family Uε, would be needed before Theorem 3 can be assessed.
  3. [§3, Proposition 3.1 and Remark 3.2] Even independently of the false identity, the manuscript only proves the symmetrized identity (3.3), i.e. ∫_{U_b} |P_b|² e^{-π|z|²}e^{πzw}dA = λ Q_b(w). The desired eigenfunction identity (3.25) is an entirely different weighted quadrature identity with the holomorphic polynomial P, not |P|². The text asserts in Remark 3.2 that the symmetrization (3.25)⇒(3.27) is immediate, but the converse direction—recovering the unsymmetrized identity from the symmetrized one—is precisely the step that fails. This is a separate structural gap: the disk inverse theorem solves a problem about |P_b|², and no valid argument shows that the same domain solves the problem about P_b.
minor comments (4)
  1. [Abstract and §1.4] The abstract contains the ungrammatical phrase 'whose central idea that of is free-boundary problem'; similarly, §1 has duplicated words ('makes ... makes'). These should be corrected.
  2. [§4.2] The word 'auxilliary' should be 'auxiliary'.
  3. [§5.1, proof of Lemma 5.3] The proof refers to 'INSERT HERE' for a discussion of the Reynolds transport formula. This is an unresolved placeholder and must be completed before publication.
  4. [Throughout, e.g. Lemma 2.1(i), Eq. (3.3)] The Fock reproducing kernel is written as e^{πzw} with no conjugation. In the standard Bargmann–Fock inner product the reproducing kernel is e^{πz\bar w}. If the paper intentionally uses a different convention, it should be stated explicitly; as written, the moment identities and the Taylor expansion in w mix holomorphic and antiholomorphic variables, which is confusing and potentially inconsistent.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the core inverse construction is self-contained and the sole self-citation is non-load-bearing.

full rationale

The main derivation chain is not circular. Theorem 1 rewrites the eigenvalue equation as the weighted moment system (1.2), and Proposition 3.1 solves the symmetrized version by an explicit Fourier-diagonal linearization (3.9), an isometric isomorphism onto Y_τ, and an analytic implicit-function argument with the strip estimates (3.11)-(3.12). The prescribed f0 is the input; the domain Uλ is the constructed output; no fitted parameter is relabeled as a prediction. The unsymmetrization step applies P#(∂w/π) and uses Lemma 3.2 to show the only admissible entire solution of the resulting homogeneous equation is zero; this is a genuine injectivity argument, not a reduction of the conclusion to the hypothesis. The self-citation to [22] in Section 5 is not load-bearing for the sharpness claim: the impossibility of any β > 1/2 follows already from the lower bound |Ũε△D| ≳ |ε| (Step 5 and Lemma 5.4) together with the upper bound on the spectral deficit λ0 − λ1(Ũε) = O(ε²) coming from Theorem 1 and Lemma 5.2. The cited upper stability estimate from [22] is invoked only to upgrade the deficit to a matching lower bound (so that λ0 − λ1(Ũε) ≍ ε²), but that upgrade is not needed for the contradiction proving optimality. Theorems 4 and 5 use the bathtub principle, shape derivatives, simplicity of the principal eigenvalue, and the Fock-space profile decomposition (Proposition 7.1); none of these assume the Nicola–Tilli theorem being proved. Two non-circular concerns are noted: Eq. (3.24) contains the assertion P#(z) = overline{P(z)} for all z ∈ C, which is false off the real axis and would undermine the unsymmetrization unless corrected; and Lemma 5.3's proof contains the placeholder 'referring the reader to INSERT HERE'. These are correctness/exposition issues, not circular reductions.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new free parameters fitted to data and no new physical or geometric entities. The main inputs are classical theorems from analysis, several adapted with proof. The spectral parameter λ and the prescribed polynomial coefficients are data, not free parameters. The central contribution is the free-boundary moment method itself, which is constructed from these standard ingredients.

assumptions (6)
  • standard math Analytic implicit function theorem on Banach spaces
    Used in Proposition 3.1 and Theorem 1 to solve the infinite weighted moment system for the boundary graph; the iterative proof is sketched rather than fully formalized.
  • standard math Paley-Wiener-Schwartz theorem and Paley-Wiener division theorem (Ehrenpreis-Malgrange)
    Used in Lemma 2.6 and both proofs of Theorem 2 to pass from vanishing of a Fourier transform on the characteristic variety to compactly supported solutions of a distributional Laplacian equation.
  • standard math Bathtub principle of Lieb-Loss
    Quoted in Lemma 6.1 and used repeatedly in Section 6 to identify measure-constrained maximizers with superlevel sets of u_F.
  • domain assumption Fock-space concentration-compactness profile decomposition
    Adapted with proof in Section 7; the dispersive remainder estimate (Proposition 7.1(iv)) is a nontrivial structural assumption about the non-compactness of Weyl translates in F^2(C).
  • domain assumption Real-analytic stratification / semianalytic boundary structure
    Used in Lemma 6.3 to obtain finite perimeter and reduced-boundary Gauss-Green identities from the assumption that F does not vanish on the local maximizer; relies on planar real-analytic level-set structure.
  • domain assumption Nicola-Tilli Faber-Krahn inequality
    Quoted in Theorem 3 to assert the nonnegativity of the spectral deficit; the paper also gives an independent proof in Section 7, so the use is not circular.

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Cite this review

Pith. "Pith review of Eigenfunctions, free boundaries, and time-frequency localization." pith.science (2026). https://pith.science/paper/SGZI7G26

@misc{pith2026260721590,
  author       = {Pith},
  title        = {Pith review of: Eigenfunctions, free boundaries, and time-frequency localization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SGZI7G26}},
  note         = {Machine review of arXiv:2607.21590}
}
abstract

We develop an inverse theory for time-frequency localization operators, whose central idea that of is free-boundary problem: the localization domain is unknown and its boundary is recovered from prescribed spectral data. The approach is based on the principle that an eigenfunction may be regarded as geometric data which determines a localization domain, and prescribing it has strong consequences for the associated variational problem. Four main results follow from this main framework. First, if $f_0$ is a polynomial sufficiently close to the Gaussian and $\lambda\in(0,1)$, we construct a real-analytic domain $U_\lambda$ such that $f_0$ is an eigenfunction of the localization operator associated with $U_\lambda$ and with eigenvalue $\lambda$, giving a general inverse construction of localization domains, which is the first in the literature. Second, we recover the null-set invariant Abreu-D\"orfler characterization of disks as the only localization domains of Hermite polynomials in the simply connected case. Third, we prove optimality of the exponent $1/2$ in the G\'omez-Guerra-Ramos-Tilli quantitative stability inequality, answering thus a question posed by those authors. Finally, we show that local maximizers of the Gaussian Faber-Krahn problem are disks, which extends the Nicola-Tilli concentration inequality to the local case as well, and, as a matter of fact, as a consequence of a Fock space concentration-compactness profile decomposition, we are able to use this to give a new, different proof of the Nicola-Tilli theorem.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

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Reference graph

Works this paper leans on

69 extracted references · cited by 1 Pith paper

  1. [22]

    Jaime G´ omez, Andr´ e Guerra, Jo˜ ao P. G. Ramos, and Paolo Tilli,Stability of the Faber–Krahn inequality for the short-time Fourier transform, Invent. Math.236(2024), no. 2, 779–836. MR 4728242

    2024

  2. [1]

    11, 115001

    Lu ´ ıs Daniel Abreu and Monika D¨ orfler,An inverse problem for localization operators, Inverse Problems 28(2012), no. 11, 115001

    2012

  3. [2]

    68 JO ˜AO P.G

    Lu ´ ıs Daniel Abreu and Michael Speckbacher,Donoho–Logan large sieve principles for modulation and polyanalytic Fock spaces, Bulletin des Sciences Math´ ematiques171(2021), 103032. 68 JO ˜AO P.G. RAMOS

    2021

  4. [3]

    Dov Aharonov, M. M. Schiffer, and Lawrence Zalcman,Potato kugel, Israel Journal of Mathematics40 (1981), no. 3, 331–339

    1981

  5. [4]

    Bargmann,On a Hilbert space of analytic functions and an associated integral transform

    V. Bargmann,On a Hilbert space of analytic functions and an associated integral transform. Part I, Comm. Pure Appl. Math.14(1961), no. 3, 187–214

    1961

  6. [5]

    Federico Bastianoni, Elena Cordero, and Fabio Nicola,Decay and smoothness for eigenfunctions of localization operators, J. Math. Anal. Appl.492(2020), no. 2, 124480

    2020

  7. [6]

    Pseudo-Differ

    Federico Bastianoni and Nenad Teofanov,Subexponential decay and regularity estimates for eigenfunc- tions of localization operators, J. Pseudo-Differ. Oper. Appl.12(2021), 19

    2021

  8. [7]

    1, 95–117

    Dominik Bayer and Karlheinz Gr¨ ochenig,Time-frequency localization operators and a Berezin transform, Integral Equations Operator Theory82(2015), no. 1, 95–117

    2015

Show all 69 references
  1. [8]

    Berenstein,An inverse spectral theorem and its relation to the Pompeiu problem, Journal d’Analyse Math´ ematique37(1980), 128–144

    Carlos A. Berenstein,An inverse spectral theorem and its relation to the Pompeiu problem, Journal d’Analyse Math´ ematique37(1980), 128–144

    1980

  2. [9]

    F. A. Berezin,Wick and anti-Wick operator symbols, Math. USSR-Sb.15(1971), no. 4, 577–606

    1971

  3. [10]

    C. A. Berger and L. A. Coburn,A symbol calculus for Toeplitz operators, Proc. Natl. Acad. Sci. USA83 (1986), no. 10, 3072–3073

    1986

  4. [11]

    ,Toeplitz operators on the Segal–Bargmann space, Trans. Amer. Math. Soc.301(1987), no. 2, 813–829

    1987

  5. [12]

    ,Heat flow and Berezin–Toeplitz estimates, Amer. J. Math.116(1994), no. 3, 563–590

    1994

  6. [13]

    4, 427–442

    Paolo Boggiatto, Elena Cordero, and Karlheinz Gr¨ ochenig,Generalized anti-Wick operators with symbols in distributional Sobolev spaces, Integral Equations Operator Theory48(2004), no. 4, 427–442

    2004

  7. [14]

    V. G. Cherednichenko,Inverse logarithmic potential problem, Inverse Logarithmic Potential Problem, De Gruyter, 2014

    2014

  8. [15]

    Elena Cordero and Karlheinz Gr¨ ochenig,Time-frequency analysis of localization operators, J. Funct. Anal.205(2003), no. 1, 107–131

    2003

  9. [16]

    4, 605–612

    Ingrid Daubechies,Time-frequency localization operators: a geometric phase space approach, IEEE Trans- actions on Information Theory34(1988), no. 4, 605–612

    1988

  10. [17]

    The use of dilations, Inverse Problems4(1988), no

    Ingrid Daubechies and Thierry Paul,Time-frequency localisation operators—a geometric phase space approach: II. The use of dilations, Inverse Problems4(1988), no. 3, 661

    1988

  11. [18]

    Evans and Ronald F

    Lawrence C. Evans and Ronald F. Gariepy,Measure theory and fine properties of functions, revised ed., Textbooks in Mathematics, CRC Press, Boca Raton, FL, 2015

    2015

  12. [19]

    Folland,Harmonic analysis in phase space, Princeton University Press, Princeton, NJ, 1989

    Gerald B. Folland,Harmonic analysis in phase space, Princeton University Press, Princeton, NJ, 1989

    1989

  13. [20]

    Fourier Anal

    Robert Fulsche and Raffael Hagger,Quantum harmonic analysis for polyanalytic Fock spaces, J. Fourier Anal. Appl.30(2024), 63

    2024

  14. [21]

    26, 429–457

    Dennis Gabor,Theory of communication, Journal of the Institution of Electrical Engineers—Part III: Radio and Communication Engineering93(1946), no. 26, 429–457

    1946

  15. [23]

    Jaime G´ omez, David Kalaj, Petar Melentijevi´ c, and Jo˜ ao P. G. Ramos,Uniform stability of concentration inequalities and applications, Proc. Lond. Math. Soc.131(2025), no. 6, e70114

    2025

  16. [24]

    Felipe Goncalves, Diogo Oliveira e Silva, and Stefan Steinerberger,Hermite polynomials, linear flows on the torus, and an uncertainty principle for roots, J. Math. Anal. Appl.451(2017), no. 2, 678–711. MR 3624763

    2017

  17. [25]

    Karlheinz Gr¨ ochenig,Foundations of time-frequency analysis, Springer Science & Business Media, 2001

    2001

  18. [26]

    1, 172–216

    Bj¨ orn Gustafsson,On quadrature domains and an inverse problem in potential theory, Journal d’Analyse Math´ ematique55(1990), no. 1, 172–216

    1990

  19. [27]

    Methods Funct

    Friedrich Haslinger, David Kalaj, and Djordjije Vujadinovi´ c,Sharp pointwise estimates for Fock spaces, Comput. Methods Funct. Theory21(2021), no. 2, 343–359

    2021

  20. [28]

    A geometrical analysis, EMS Tracts in Mathematics, vol

    Antoine Henrot and Michel Pierre,Shape variation and optimization. A geometrical analysis, EMS Tracts in Mathematics, vol. 28, European Mathematical Society (EMS), Z¨ urich, 2018. TIME-FREQUENCY LOCALIZATION 69

    2018

  21. [29]

    Mat.29(1991), no

    Lars H¨ ormander,A uniqueness theorem of Beurling for Fourier transform pairs, Ark. Mat.29(1991), no. 2, 237–240. MR 1150375

    1991

  22. [30]

    I: distribution theory and Fourier analysis, 2 ed., Classics in Mathematics, Springer-Verlag, Berlin, 2003, Reprint of the 1990 second edition

    ,The analysis of linear partial differential operators. I: distribution theory and Fourier analysis, 2 ed., Classics in Mathematics, Springer-Verlag, Berlin, 2003, Reprint of the 1990 second edition

    2003

  23. [31]

    II: differential operators with constant coef- ficients, Classics in Mathematics, Springer-Verlag, Berlin, 2005, Reprint of the 1983 original

    ,The analysis of linear partial differential operators. II: differential operators with constant coef- ficients, Classics in Mathematics, Springer-Verlag, Berlin, 2005, Reprint of the 1983 original

    2005

  24. [32]

    34, American Mathematical Soc., 1990

    Victor Isakov,Inverse source problems, no. 34, American Mathematical Soc., 1990

    1990

  25. [33]

    Vladan Jaguzovi´ c and Petar Melentijevi´ c,Stability of Wehrl-type functionals and concentration estimates on Bergman spaces of log-subharmonic functions on the unit sphere, arXiv preprint arXiv:2512.15193, 2025

    2025

  26. [34]

    Math.30(2024), 1293–1303

    David Kalaj,Contraction property of Fock type space of log-subharmonic functions inR m, New York J. Math.30(2024), 1293–1303

    2024

  27. [35]

    Methods Funct

    ,Weighted contractivity of differential operators on Fock spaces, Comput. Methods Funct. Theory 24(2024), no. 2, 283–302

    2024

  28. [36]

    Methods Funct

    David Kalaj and Petar Melentijevi´ c,Weighted contractivity for derivatives of functions in the Bergman space on the unit disk, Comput. Methods Funct. Theory26(2025), no. 1, 1–21

    2025

  29. [37]

    David Kalaj and Jo˜ ao P. G. Ramos,A Faber–Krahn type inequality for log-subharmonic functions in the hyperbolic ball, Israel J. Math.271(2026), 233–255

    2026

  30. [38]

    2, 480–497

    Lavi Karp,Generalized Newton potential and its applications, Journal of Mathematical Analysis and Applications174(1993), no. 2, 480–497

    1993

  31. [39]

    Tosio Kato,Perturbation theory for linear operators, 2 ed., Classics in Mathematics, Springer-Verlag, Berlin, 1995, Reprint of the 1980 edition

    1995

  32. [40]

    Kisil,Cross-Toeplitz operators on the Fock–Segal–Bargmann spaces and two-sided convolu- tions on the Heisenberg group, Ann

    Vladimir V. Kisil,Cross-Toeplitz operators on the Fock–Segal–Bargmann spaces and two-sided convolu- tions on the Heisenberg group, Ann. Funct. Anal.14(2023), 38

    2023

  33. [41]

    Krantz and Harold R

    Steven G. Krantz and Harold R. Parks,A primer of real analytic functions, 2 ed., Birkh¨ auser Advanced Texts, Birkh¨ auser Boston, Boston, MA, 2002

    2002

  34. [42]

    Aleksei Kulikov,A contractive Hardy–Littlewood inequality, Bull. Lond. Math. Soc.53(2021), no. 3, 740–746

    2021

  35. [43]

    Fourier Anal

    ,Fourier interpolation and time–frequency localization, J. Fourier Anal. Appl.27(2021), 58

    2021

  36. [44]

    ,Functionals with extrema at reproducing kernels, Geom. Funct. Anal.32(2022), no. 4, 938–949

    2022

  37. [45]

    ,Exponential lower bound for the eigenvalues of the time–frequency localization operator before the plunge region, Appl. Comput. Harmon. Anal.71(2024), 101639

    2024

  38. [46]

    ,Sharp estimates for eigenvalues of localization operators before the plunge region, arXiv preprint arXiv:2603.07407, 2026

    2026

  39. [47]

    Math.479(2025), 110423

    Aleksei Kulikov, Fabio Nicola, Joaquim Ortega-Cerd` a, and Paolo Tilli,A monotonicity theorem for subharmonic functions on manifolds, Adv. Math.479(2025), 110423

    2025

  40. [48]

    Lieb,Integral bounds for radar ambiguity functions and Wigner distributions, Inequalities, Springer, 2002, pp

    Elliott H. Lieb,Integral bounds for radar ambiguity functions and Wigner distributions, Inequalities, Springer, 2002, pp. 625–630

    2002

  41. [49]

    Lieb and Michael Loss,Analysis, 2 ed., Graduate Studies in Mathematics, vol

    Elliott H. Lieb and Michael Loss,Analysis, 2 ed., Graduate Studies in Mathematics, vol. 14, American Mathematical Society, Providence, RI, 2001

    2001

  42. [50]

    The locally compact case, part I, Annales de l’Institut Henri Poincar´ e C, Analyse non lin´ eaire1(1984), no

    Pierre-Louis Lions,The concentration-compactness principle in the calculus of variations. The locally compact case, part I, Annales de l’Institut Henri Poincar´ e C, Analyse non lin´ eaire1(1984), no. 2, 109–145

    1984

  43. [51]

    Stanis law Lojasiewicz,Introduction to complex analytic geometry, Birkh¨ auser Verlag, Basel, 1991

    1991

  44. [52]

    St´ ephane Mallat,A wavelet tour of signal processing, 3 ed., Academic Press, Burlington, MA, 2008

    2008

  45. [53]

    Felipe Marceca, Jos´ e Luis Romero, and Michael Speckbacher,Eigenvalue estimates for Fourier concen- tration operators on two domains, Arch. Ration. Mech. Anal.248(2024), no. 3, 35

    2024

  46. [54]

    Petar Melentijevi´ c,Hypercontractive inequalities for weighted Bergman spaces, Bull. Lond. Math. Soc. 55(2023), no. 6, 2611–2616. 70 JO ˜AO P.G. RAMOS

    2023

  47. [55]

    ,Stability of convex functionals on weighted Bergman spaces with applications, arXiv preprint arXiv:2508.01939, 2025

    2025

  48. [56]

    Ivan Trapasso,On the existence of optimizers for time–frequency concentration problems, Calc

    Fabio Nicola, Jos´ e Luis Romero, and S. Ivan Trapasso,On the existence of optimizers for time–frequency concentration problems, Calc. Var. Partial Differential Equations62(2023), no. 1, 21

    2023

  49. [57]

    Fabio Nicola and Paolo Tilli,The Faber–Krahn inequality for the short-time Fourier transform, Inven- tiones Mathematicae (2022), 1–30

    2022

  50. [58]

    10, 7353–7375

    ,The norm of time-frequency and wavelet localization operators, Transactions of the American Mathematical Society376(2023), no. 10, 7353–7375

    2023

  51. [59]

    Ramanathan and P

    J. Ramanathan and P. Topiwala,Time-frequency localization via the Weyl correspondence, SIAM J. Math. Anal.24(1993), no. 5, 1378–1393

    1993

  52. [60]

    ,Time-frequency localization and the spectrogram, Appl. Comput. Harmon. Anal.1(1994), no. 2, 209–215

    1994

  53. [61]

    Jo˜ ao P. G. Ramos and Paolo Tilli,A Faber–Krahn inequality for wavelet transforms, Bulletin of the London Mathematical Society55(2023), no. 4, 2018–2034

    2023

  54. [62]

    Jo˜ ao Pedro Ramos,Time-frequency representations and optimal localization, Instituto de Matem´ atica Pura e Aplicada, Rio de Janeiro, 2025, Lecture notes for the 35th Brazilian Colloquium of Mathematics

    2025

  55. [63]

    Samuelsen,Compactness of Fourier concentration operators, J

    Helge J. Samuelsen,Compactness of Fourier concentration operators, J. Fourier Anal. Appl.31(2025), 64

    2025

  56. [64]

    1, 61–78

    Henrik Shahgholian,On quadrature domains and the Schwarz potential, Journal of Mathematical Analysis and Applications171(1992), no. 1, 61–78

    1992

  57. [65]

    473, 137–180

    Henrik Shahgholian and Bj¨ orn Gustafsson,Existence and geometric properties of solutions of a free boundary problem in potential theory, Journal f¨ ur die reine und angewandte Mathematik (Crelles Journal) 1996(1996), no. 473, 137–180

    1996

  58. [66]

    Shape sensitivity analysis, Springer Series in Computational Mathematics, vol

    Jan Soko lowski and Jean-Paul Zol´ esio,Introduction to shape optimization. Shape sensitivity analysis, Springer Series in Computational Mathematics, vol. 16, Springer-Verlag, Berlin, 1992

    1992

  59. [67]

    217, Birkh¨ auser Boston, Inc., Boston, MA, 2004, Hardy’s theorem on Lie groups, With a foreword by Gerald B

    Sundaram Thangavelu,An introduction to the uncertainty principle, Progress in Mathematics, vol. 217, Birkh¨ auser Boston, Inc., Boston, MA, 2004, Hardy’s theorem on Lie groups, With a foreword by Gerald B. Folland. MR 2008480

    2004

  60. [68]

    Wigner,On the quantum correction for thermodynamic equilibrium, Physical Review40(1932), no

    Eugene P. Wigner,On the quantum correction for thermodynamic equilibrium, Physical Review40(1932), no. 5, 749–759

    1932

  61. [69]

    263, Springer, New York, 2012

    Kehe Zhu,Analysis on Fock spaces, Graduate Texts in Mathematics, vol. 263, Springer, New York, 2012. Instituto de Matem´atica Pura e Aplicada (IMPA) - Estrada Dona Castorina 110, 22460-320, Jardim Botˆanico, Rio de Janeiro - RJ, Brazil Email address:joao.ramos@impa.br

    2012

Pith tools

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