pith. sign in

arxiv: 2606.27008 · v1 · pith:YZKTO62Pnew · submitted 2026-06-25 · 🧮 math.FA

Plancherel Identities for unbounded subsets of mathbb R^d

Dorin Ervin Dutkay , Piyali Chakraborty This is my paper

Pith reviewed 2026-06-26 02:26 UTC · model grok-4.3

classification 🧮 math.FA
keywords Plancherel identityFourier transformunbounded subsetsR^dlattice invarianceisometric isomorphismharmonic analysis
0
0 comments X

The pith

Pairs of subsets of R^d invariant under translations by dual full-rank lattices have their restricted Fourier transform as an isometric isomorphism.

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

The paper introduces a class of pairs of unbounded subsets in R^d, each invariant under translations by a pair of dual full-rank lattices. For these pairs the Fourier transform restricts to an isometric isomorphism between the L2 spaces supported on each subset. This yields a Plancherel identity equating the L2 norm of a function on one subset to the L2 norm of its Fourier transform on the other. The construction extends the classical Plancherel theorem from the full space to certain proper unbounded subsets. The lattice-invariance condition supplies the structure that makes the restriction isometric.

Core claim

We present a class of pairs of subsets of R^d for which the Fourier transform, when restricted to these subsets, is an isometric isomorphism, and thus the Plancherel identity is satisfied. The sets are invariant under translations by dual full-rank lattices.

What carries the argument

Invariance of the subsets under translations by dual full-rank lattices, which supplies the structural condition enabling the restricted Fourier transform to act as an isometric isomorphism.

If this is right

  • The Plancherel identity holds exactly for functions whose support lies in one member of such a pair.
  • The Fourier transform maps the L2 space on the first subset isometrically onto the L2 space on the second.
  • The subsets may be unbounded while still satisfying the identity.
  • The result applies in every dimension d.

Where Pith is reading between the lines

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

  • The same lattice-invariance condition might be used to produce spectral sets or orthonormal bases supported on the subsets.
  • The construction could be tested numerically in low dimensions by taking explicit lattices and checking norm preservation on sample functions.
  • Analogous statements may hold for other integral transforms that interact with lattice translations.

Load-bearing premise

Invariance under translations by dual full-rank lattices is sufficient for the restricted Fourier transform to be an isometric isomorphism.

What would settle it

An explicit pair of dual-lattice-invariant subsets together with a square-integrable function on one whose L2 norm differs from the L2 norm of its Fourier transform on the other would falsify the claim.

Figures

Figures reproduced from arXiv: 2606.27008 by Dorin Ervin Dutkay, Piyali Chakraborty.

Figure 1
Figure 1. Figure 1: The nonperiodic set Ω = (0, 1)2 + Γ [PITH_FULL_IMAGE:figures/full_fig_p014_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The set FB +AZ 2 together with the lattice BZ 2 shown by the intersection points of the thin grey lines. The highlighted parallelogram is the fundamental domain FB. The dots represent the lattice AZ 2 [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The translates (AT ) −1 h − 1 2 , 1 2 i2 +(BT ) −1Z 2 . The thin grey lines represent the lattice (AT ) −1Z 2 . The dots represent the lattice (BT ) −1Z 2 . Lemma 5.5. The following statements are equivalent: (i) (BT ) −1Z d ⊆ (AT ) −1Z d . (ii) B−1A has integer entries. (iii) A = BM for some integer d × d matrix M. (iv) The lattice AZ d is contained in the lattice BZ d . In this case a fundamental domain … view at source ↗
Figure 4
Figure 4. Figure 4: R =  1 1 −1 1 , D = { 0 0  , 1 0  }, Γ = Z 2 , Γ =˜ R(Z 2 ) [PITH_FULL_IMAGE:figures/full_fig_p021_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: R =  1 −2 2 1  , D = { 0 0  , 1 0  , 2 0  , 3 0  , 4 0  }, Γ = Z × 2Z, Γ =˜ R(Z × 2Z) [PITH_FULL_IMAGE:figures/full_fig_p022_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: R =  2 1 0 2 , D = { 0 0  , 3 0  , 0 1  , 3 1  }, Γ = 3Z × Z, Γ =˜ R(3Z × Z) References [CD26] Piyali Chakraborty and Dorin Dutkay. Some Plancherel identities for unbounded subsets of R in duality. Journal of Mathematical Sciences, 2026. [CM99] Ethan M. Coven and Aaron Meyerowitz. Tiling the integers with translates of one finite set. J. Algebra, 212(1):161–174, 1999. [FMM06] Bálint Farkas, Máté Mato… view at source ↗
read the original abstract

We present a class of pairs of subsets of $\mathbb R^d$ for which the Fourier transform, when restricted to these subsets, is an isometric isomorphism, and thus the Plancherel identity is satisfied. The sets are invariant under translations by dual full-rank lattices.

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.

Referee Report

1 major / 0 minor

Summary. The manuscript asserts the existence of a class of pairs of unbounded subsets of R^d, invariant under translations by dual full-rank lattices, such that the Fourier transform restricted to these subsets forms an isometric isomorphism and thus satisfies the Plancherel identity.

Significance. If the claimed construction and isometry were established with explicit examples and proofs, the result would extend Plancherel theory to new families of unbounded lattice-symmetric domains, potentially relevant to harmonic analysis on periodic structures or sampling on irregular sets.

major comments (1)
  1. [Abstract] Abstract: the abstract asserts the existence of such pairs and the isometric property but supplies no derivation, explicit construction, or verification; the central claim cannot be assessed from available text.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for reviewing our manuscript. We respond to the single major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the abstract asserts the existence of such pairs and the isometric property but supplies no derivation, explicit construction, or verification; the central claim cannot be assessed from available text.

    Authors: Abstracts are designed to state the main result concisely; derivations, constructions, and verifications appear in the body of the manuscript. The full text defines an explicit class of dual-lattice-invariant subset pairs in R^d and proves that the restricted Fourier transform is an isometric isomorphism, thereby establishing the Plancherel identity on those sets. If the referee had access only to the abstract, the complete arXiv manuscript supplies the requested details. revision: no

Circularity Check

0 steps flagged

No significant circularity; result is a structural existence claim

full rationale

The paper claims existence of lattice-invariant unbounded subsets of R^d on which the restricted Fourier transform is an isometric isomorphism. The abstract and provided description present this as a new class defined by the invariance condition under dual full-rank lattices, without any fitted parameters, self-referential predictions, or load-bearing self-citations that reduce the claim to its inputs by construction. No equations are shown that equate a derived quantity to a fitted input, and the central result is not renamed from a known pattern or smuggled via prior ansatz. The derivation chain is therefore self-contained against external benchmarks and does not exhibit any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No information available from the abstract to identify free parameters, axioms, or invented entities; the result is stated at the level of existence of a class.

pith-pipeline@v0.9.1-grok · 5556 in / 1086 out tokens · 55547 ms · 2026-06-26T02:26:16.930656+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

78 extracted references · 58 canonical work pages

  1. [1]

    2024 , eprint=

    Orthogonal Fourier Analysis on Domains , author=. 2024 , eprint=

    2024

  2. [2]

    , TITLE =

    Sands, Arthur D. , TITLE =. Acta Math. Acad. Sci. Hungar. , FJOURNAL =. 1957 , PAGES =. doi:10.1007/BF02025232 , URL =

    work page doi:10.1007/bf02025232 1957

  3. [3]

    2010 , eprint=

    Tilings by translation , author=. 2010 , eprint=

    2010

  4. [4]

    and Wang, Yang , TITLE =

    Lagarias, Jeffrey C. and Wang, Yang , TITLE =. J. Fourier Anal. Appl. , FJOURNAL =. 1997 , NUMBER =. doi:10.1007/s00041-001-4051-2 , URL =

    work page doi:10.1007/s00041-001-4051-2 1997

  5. [5]

    and Meyerowitz, Aaron , TITLE =

    Coven, Ethan M. and Meyerowitz, Aaron , TITLE =. J. Algebra , FJOURNAL =. 1999 , NUMBER =. doi:10.1006/jabr.1998.7628 , URL =

    work page doi:10.1006/jabr.1998.7628 1999

  6. [6]

    , TITLE =

    Tijdeman, R. , TITLE =. Number theory (. 1995 , ISBN =. doi:10.1017/CBO9780511661990.016 , URL =

    work page doi:10.1017/cbo9780511661990.016 1995

  7. [7]

    , TITLE =

    Newman, Donald J. , TITLE =. J. Number Theory , FJOURNAL =. 1977 , NUMBER =. doi:10.1016/0022-314X(77)90054-3 , URL =

    work page doi:10.1016/0022-314x(77)90054-3 1977

  8. [8]

    Fuglede, Bent , TITLE =. J. Functional Analysis , FJOURNAL =. 1974 , PAGES =. doi:10.1016/0022-1236(74)90072-x , URL =

    work page doi:10.1016/0022-1236(74)90072-x 1974

  9. [9]

    1960 , PAGES =

    Fuglede, Bent , TITLE =. 1960 , PAGES =

    1960

  10. [10]

    Li, Qian and Zhang, Min-Min , TITLE =. Rev. Un. Mat. Argentina , FJOURNAL =. 2024 , NUMBER =

    2024

  11. [11]

    Cao, Yong-Shen and Deng, Qi-Rong and Li, Ming-Tian and Wu, Sha , TITLE =. Bull. Malays. Math. Sci. Soc. , FJOURNAL =. 2024 , NUMBER =. doi:10.1007/s40840-024-01720-5 , URL =

    work page doi:10.1007/s40840-024-01720-5 2024

  12. [12]

    Li, Wenxia and Miao, Jun Jie and Wang, Zhiqiang , TITLE =. J. Funct. Anal. , FJOURNAL =. 2024 , NUMBER =. doi:10.1016/j.jfa.2024.110539 , URL =

    work page doi:10.1016/j.jfa.2024.110539 2024

  13. [13]

    Li, Wenxia and Wang, Zhiqiang , TITLE =. J. Fourier Anal. Appl. , FJOURNAL =. 2024 , NUMBER =. doi:10.1007/s00041-024-10094-y , URL =

    work page doi:10.1007/s00041-024-10094-y 2024

  14. [14]

    Chaos Solitons Fractals , FJOURNAL =

    Liu, Zong-Sheng , TITLE =. Chaos Solitons Fractals , FJOURNAL =. 2024 , PAGES =. doi:10.1016/j.chaos.2024.114926 , URL =

    work page doi:10.1016/j.chaos.2024.114926 2024

  15. [15]

    Yin, Fengli and Zhang, Minmin , TITLE =. J. Anhui Univ. Nat. Sci. , FJOURNAL =. 2024 , NUMBER =

    2024

  16. [16]

    Forum Math

    Wang, Cong and Yin, Feng-Li and Zhang, Min-Min , TITLE =. Forum Math. , FJOURNAL =. 2024 , NUMBER =. doi:10.1515/forum-2023-0114 , URL =

    work page doi:10.1515/forum-2023-0114 2024

  17. [17]

    1967 , PAGES =

    Maurin, Krzysztof , TITLE =. 1967 , PAGES =

    1967

  18. [18]

    Nonlinearity , FJOURNAL =

    Li, Wenxia and Miao, Jun Jie and Wang, Zhiqiang , TITLE =. Nonlinearity , FJOURNAL =. 2024 , NUMBER =. doi:10.1088/1361-6544/ad0d70 , URL =

    work page doi:10.1088/1361-6544/ad0d70 2024

  19. [19]

    Dutkay, Dorin Ervin and Jorgensen, Palle E. T. , TITLE =. Sampl. Theory Signal Process. Data Anal. , FJOURNAL =. 2023 , NUMBER =. doi:10.1007/s43670-023-00072-8 , URL =

    work page doi:10.1007/s43670-023-00072-8 2023

  20. [20]

    Dutkay, Dorin Ervin and Jorgensen, Palle E. T. , TITLE =. Proc. Amer. Math. Soc. , FJOURNAL =. 2015 , NUMBER =. doi:10.1090/S0002-9939-2015-12656-1 , URL =

    work page doi:10.1090/s0002-9939-2015-12656-1 2015

  21. [21]

    2025 , eprint=

    Commuting self-adjoint extensions of the partial differential operators on disconnected sets , author=. 2025 , eprint=

    2025

  22. [22]

    Journal of Mathematical Sciences , author=

    Some. Journal of Mathematical Sciences , author=. 2026 , eprint=

    2026

  23. [23]

    and Wang, Yang , TITLE =

    Lagarias, Jeffrey C. and Wang, Yang , TITLE =. Invent. Math. , FJOURNAL =. 1996 , NUMBER =. doi:10.1007/s002220050056 , URL =

    work page doi:10.1007/s002220050056 1996

  24. [24]

    Jorgensen , keywords =

    Piyali Chakraborty and Dorin Ervin Dutkay and Palle E.T. Jorgensen , keywords =. Fuglede’s conjecture, differential operators and unitary groups of local translations , journal =. 2025 , issn =. doi:https://doi.org/10.1016/j.exmath.2025.125662 , url =

    work page doi:10.1016/j.exmath.2025.125662 2025

  25. [25]

    Jorgensen, Palle E. T. and Pedersen, Steen and Tian, Feng , TITLE =. Complex Anal. Oper. Theory , FJOURNAL =. 2013 , NUMBER =. doi:10.1007/s11785-012-0234-x , URL =

    work page doi:10.1007/s11785-012-0234-x 2013

  26. [26]

    Jorgensen, Palle and Pedersen, Steen and Tian, Feng , TITLE =. J. Math. Phys. , FJOURNAL =. 2012 , NUMBER =. doi:10.1063/1.4709770 , URL =

    work page doi:10.1063/1.4709770 2012

  27. [27]

    Dutkay, Dorin Ervin and Jorgensen, Palle E. T. , TITLE =. J. Funct. Anal. , FJOURNAL =. 2015 , NUMBER =. doi:10.1016/j.jfa.2015.01.018 , URL =

    work page doi:10.1016/j.jfa.2015.01.018 2015

  28. [28]

    Dutkay, Dorin Ervin and Jorgensen, Palle E. T. , TITLE =. Rocky Mountain J. Math. , FJOURNAL =. 2013 , NUMBER =. doi:10.1216/RMJ-2013-43-5-1497 , URL =

    work page doi:10.1216/rmj-2013-43-5-1497 2013

  29. [29]

    Dutkay, Dorin Ervin and Jorgensen, Palle E. T. , TITLE =. J. Fourier Anal. Appl. , FJOURNAL =. 2013 , NUMBER =. doi:10.1007/s00041-013-9264-7 , URL =

    work page doi:10.1007/s00041-013-9264-7 2013

  30. [30]

    Dutkay, Dorin Ervin and Han, Deguang and Jorgensen, Palle E. T. and Picioroaga, Gabriel , TITLE =. Adv. Math. , FJOURNAL =. 2013 , PAGES =. doi:10.1016/j.aim.2013.02.016 , URL =

    work page doi:10.1016/j.aim.2013.02.016 2013

  31. [31]

    Dutkay, Dorin Ervin and Jorgensen, Palle E. T. , TITLE =. Math. Comp. , FJOURNAL =. 2012 , NUMBER =. doi:10.1090/S0025-5718-2012-02589-0 , URL =

    work page doi:10.1090/s0025-5718-2012-02589-0 2012

  32. [32]

    Dutkay, Dorin Ervin and Jorgensen, Palle E. T. , TITLE =. Math. Comp. , FJOURNAL =. 2012 , NUMBER =. doi:10.1090/S0025-5718-2012-02580-4 , URL =

    work page doi:10.1090/s0025-5718-2012-02580-4 2012

  33. [33]

    Dutkay, Dorin Ervin and Han, Deguang and Jorgensen, Palle E. T. , TITLE =. J. Funct. Anal. , FJOURNAL =. 2009 , NUMBER =. doi:10.1016/j.jfa.2009.05.014 , URL =

    work page doi:10.1016/j.jfa.2009.05.014 2009

  34. [34]

    Dutkay, Dorin Ervin and Jorgensen, Palle E. T. , TITLE =. Radon transforms, geometry, and wavelets , SERIES =. 2008 , MRCLASS =. doi:10.1090/conm/464/09077 , URL =

    work page doi:10.1090/conm/464/09077 2008

  35. [35]

    Dutkay, Dorin Ervin and Jorgensen, Palle E. T. , TITLE =. J. Funct. Anal. , FJOURNAL =. 2007 , NUMBER =. doi:10.1016/j.jfa.2007.03.002 , URL =

    work page doi:10.1016/j.jfa.2007.03.002 2007

  36. [36]

    Dutkay, Dorin Ervin and Jorgensen, Palle E. T. , TITLE =. Math. Z. , FJOURNAL =. 2007 , NUMBER =. doi:10.1007/s00209-007-0104-9 , URL =

    work page doi:10.1007/s00209-007-0104-9 2007

  37. [37]

    Jorgensen, Palle E. T. , TITLE =. 2018 , PAGES =. doi:10.1090/cbms/128 , URL =

    work page doi:10.1090/cbms/128 2018

  38. [38]

    Tao, Terence , TITLE =. Math. Res. Lett. , FJOURNAL =. 2004 , NUMBER =. doi:10.4310/MRL.2004.v11.n2.a8 , URL =

    work page doi:10.4310/mrl.2004.v11.n2.a8 2004

  39. [39]

    Jorgensen, Palle E. T. , TITLE =. Adv. in Math. , FJOURNAL =. 1982 , NUMBER =. doi:10.1016/0001-8708(82)90001-9 , URL =

    work page doi:10.1016/0001-8708(82)90001-9 1982

  40. [40]

    Pedersen, Steen , TITLE =. J. Funct. Anal. , FJOURNAL =. 1987 , NUMBER =. doi:10.1016/0022-1236(87)90061-9 , URL =

    work page doi:10.1016/0022-1236(87)90061-9 1987

  41. [41]

    Jorgensen, Palle and Pedersen, Steen and Tian, Feng , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 2015 , NUMBER =. doi:10.1090/S0002-9947-2014-06296-X , URL =

    work page doi:10.1090/s0002-9947-2014-06296-x 2015

  42. [42]

    Jorgensen, Palle E. T. and Pedersen, Steen , TITLE =. J. Fourier Anal. Appl. , FJOURNAL =. 1999 , NUMBER =. doi:10.1007/BF01259371 , URL =

    work page doi:10.1007/bf01259371 1999

  43. [43]

    1980 , PAGES =

    Reed, Michael and Simon, Barry , TITLE =. 1980 , PAGES =

    1980

  44. [44]

    , TITLE =

    Conway, John B. , TITLE =. 1990 , PAGES =

    1990

  45. [45]

    Maurin, Krzysztof , TITLE =. Math. Scand. , FJOURNAL =. 1961 , PAGES =. doi:10.7146/math.scand.a-10641 , URL =

    work page doi:10.7146/math.scand.a-10641 1961

  46. [46]

    von Neumann, John , TITLE =. Ann. of Math. (2) , FJOURNAL =. 1949 , PAGES =. doi:10.2307/1969463 , URL =

    work page doi:10.2307/1969463 1949

  47. [47]

    and Fournier, John J

    Adams, Robert A. and Fournier, John J. F. , TITLE =. 2003 , PAGES =

    2003

  48. [48]

    Jorgensen, Palle E. T. and Pedersen, Steen , TITLE =. J. Anal. Math. , FJOURNAL =. 1998 , PAGES =. doi:10.1007/BF02788699 , URL =

    work page doi:10.1007/bf02788699 1998

  49. [49]

    Jorgensen, Palle E. T. and Pedersen, Steen , TITLE =. Electron. Res. Announc. Amer. Math. Soc. , FJOURNAL =. 1998 , PAGES =. doi:10.1090/S1079-6762-98-00044-4 , URL =

    work page doi:10.1090/s1079-6762-98-00044-4 1998

  50. [50]

    Jorgensen, Palle E. T. and Pedersen, Steen , TITLE =. Exposition. Math. , FJOURNAL =. 1993 , NUMBER =

    1993

  51. [51]

    Jorgensen, Palle E. T. and Pedersen, Steen , TITLE =. J. Funct. Anal. , FJOURNAL =. 1992 , NUMBER =. doi:10.1016/0022-1236(92)90101-N , URL =

    work page doi:10.1016/0022-1236(92)90101-n 1992

  52. [52]

    Jorgensen, Palle E. T. and Pedersen, Steen , TITLE =. C. R. Acad. Sci. Paris S\'. 1991 , NUMBER =

    1991

  53. [53]

    Etkind, Mark Mordechai and Lev, Nir , TITLE =. Proc. Lond. Math. Soc. (3) , FJOURNAL =. 2023 , NUMBER =

    2023

  54. [54]

    , TITLE =

    Dunford, Nelson and Schwartz, Jacob T. , TITLE =. 1988 , PAGES =

    1988

  55. [55]

    Functional

    Nelson, Edward , TITLE =. Functional. 1970 , MRCLASS =

    1970

  56. [56]

    1969 , PAGES =

    Nelson, Edward , TITLE =. 1969 , PAGES =

    1969

  57. [57]

    , TITLE =

    Mackey, George W. , TITLE =. Math. Ann. , FJOURNAL =. 1961/62 , PAGES =. doi:10.1007/BF01452360 , URL =

    work page doi:10.1007/bf01452360 1961

  58. [58]

    , TITLE =

    Mackey, George W. , TITLE =. Proc. 1961 , MRCLASS =

    1961

  59. [59]

    Jorgensen, Palle E. T. , TITLE =. Internat. J. Math. , FJOURNAL =. 1991 , NUMBER =. doi:10.1142/S0129167X91000168 , URL =

    work page doi:10.1142/s0129167x91000168 1991

  60. [60]

    , TITLE =

    J rgensen, Palle T. , TITLE =. Bull. Amer. Math. Soc. , FJOURNAL =. 1976 , NUMBER =. doi:10.1090/S0002-9904-1976-14217-6 , URL =

    work page doi:10.1090/s0002-9904-1976-14217-6 1976

  61. [61]

    Farkas, B\'. On. J. Fourier Anal. Appl. , FJOURNAL =. 2006 , NUMBER =. doi:10.1007/s00041-005-5069-7 , URL =

    work page doi:10.1007/s00041-005-5069-7 2006

  62. [62]

    Lev, Nir , TITLE =. Rev. Mat. Iberoam. , FJOURNAL =. 2022 , NUMBER =. doi:10.4171/RMI/1318 , URL =

    work page doi:10.4171/rmi/1318 2022

  63. [63]

    Lev, Nir and Matolcsi, M\'. The. Acta Math. , FJOURNAL =. 2022 , NUMBER =. doi:10.4310/acta.2022.v228.n2.a3 , URL =

    work page doi:10.4310/acta.2022.v228.n2.a3 2022

  64. [64]

    Extended abstracts fall 2019---spaces of analytic functions: approximation, interpolation, sampling , SERIES =

    Debernardi, Alberto and Lev, Nir , TITLE =. Extended abstracts fall 2019---spaces of analytic functions: approximation, interpolation, sampling , SERIES =. [2021] 2021 , MRCLASS =. doi:10.1007/978-3-030-74417-5\_11 , URL =

    work page doi:10.1007/978-3-030-74417-5 2019

  65. [65]

    Greenfeld, Rachel and Lev, Nir , TITLE =. J. Anal. Math. , FJOURNAL =. 2020 , NUMBER =. doi:10.1007/s11854-020-0092-9 , URL =

    work page doi:10.1007/s11854-020-0092-9 2020

  66. [66]

    Lev, Nir and Olevskii, Alexander , TITLE =. Adv. Math. , FJOURNAL =. 2017 , PAGES =. doi:10.1016/j.aim.2017.05.015 , URL =

    work page doi:10.1016/j.aim.2017.05.015 2017

  67. [67]

    and Lev, Nir , TITLE =

    Kolountzakis, Mihail N. and Lev, Nir , TITLE =. Int. Math. Res. Not. IMRN , FJOURNAL =. 2016 , NUMBER =. doi:10.1093/imrn/rnv283 , URL =

    work page doi:10.1093/imrn/rnv283 2016

  68. [68]

    Iosevich, Alex and Mayeli, Azita and Pakianathan, Jonathan , TITLE =. Anal. PDE , FJOURNAL =. 2017 , NUMBER =. doi:10.2140/apde.2017.10.757 , URL =

    work page doi:10.2140/apde.2017.10.757 2017

  69. [69]

    , TITLE =

    Iosevich, Alex and Kolountzakis, Mihal N. , TITLE =. Anal. PDE , FJOURNAL =. 2013 , NUMBER =. doi:10.2140/apde.2013.6.819 , URL =

    work page doi:10.2140/apde.2013.6.819 2013

  70. [70]

    Iosevich, Alex and Rudnev, Mischa , TITLE =. Int. Math. Res. Not. , FJOURNAL =. 2003 , NUMBER =. doi:10.1155/S1073792803208126 , URL =

    work page doi:10.1155/s1073792803208126 2003

  71. [71]

    , TITLE =

    Evans, Lawrence C. , TITLE =. 2010 , PAGES =. doi:10.1090/gsm/019 , URL =

    work page doi:10.1090/gsm/019 2010

  72. [72]

    and Lions, J

    Deny, J. and Lions, J. L. , TITLE =. Ann. Inst. Fourier (Grenoble) , FJOURNAL =. 1945 , PAGES =

    1945

  73. [73]

    Iosevich, Alex and Katz, Nets and Pedersen, Steen , TITLE =. Math. Res. Lett. , FJOURNAL =. 1999 , NUMBER =. doi:10.4310/MRL.1999.v6.n2.a13 , URL =

    work page doi:10.4310/mrl.1999.v6.n2.a13 1999

  74. [74]

    and Jorgensen, Palle E

    Herr, John E. and Jorgensen, Palle E. T. and Weber, Eric S. , TITLE =. From classical analysis to analysis on fractals. [2023] 2023 , MRCLASS =. doi:10.1007/978-3-031-37800-3\_9 , URL =

    work page doi:10.1007/978-3-031-37800-3 2023

  75. [75]

    and Jorgensen, Palle E

    Herr, John E. and Jorgensen, Palle E. T. and Weber, Eric S. , TITLE =. Analysis, probability and mathematical physics on fractals , SERIES =. [2020] 2020 , MRCLASS =

    2020

  76. [76]

    and Jorgensen, Palle E

    Herr, John E. and Jorgensen, Palle E. T. and Weber, Eric S. , TITLE =. J. Anal. Math. , FJOURNAL =. 2019 , NUMBER =. doi:10.1007/s11854-019-0026-6 , URL =

    work page doi:10.1007/s11854-019-0026-6 2019

  77. [77]

    and Jorgensen, Palle E

    Herr, John E. and Jorgensen, Palle E. T. and Weber, Eric S. , TITLE =. Linear Algebra Appl. , FJOURNAL =. 2019 , PAGES =. doi:10.1016/j.laa.2018.02.023 , URL =

    work page doi:10.1016/j.laa.2018.02.023 2019

  78. [78]

    and Jorgensen, Palle E

    Herr, John E. and Jorgensen, Palle E. T. and Weber, Eric S. , TITLE =. Frames and harmonic analysis , SERIES =. [2018] 2018 , MRCLASS =. doi:10.1090/conm/706/14211 , URL =

    work page doi:10.1090/conm/706/14211 2018