RESEARCH


 

My research is focused on harmonic analysis and approximation theory.  I recently started working on combinatorics and number theory, with special attention to the properties of cyclotomic polynomials. Thus, my main research interests are:

 

·       Bases and frames in Hilbert spaces (in finite or infinite dimensional spaces)

·       Exponential bases and frames on domains of Rd

·       Weighted inequalities for the Fourier transform

·       Cyclotomic polynomials and their applications in number theory

·       Properties  of the divisor functions, and the Goldbach conjecture

 

 

I am also interested in

 

1.    Unique continuation properties of solutions of elliptic equations and systems.

2.    Evaluation of the norms of convolution operators and other sharp constants

3.    Geometric properties of harmonic functions and of solutions of Schrodinger equations 

4.    Restriction properties of the Fourier transform to manifolds of arbitrary co-dimension and the restriction conjecture

5.    Uniform estimates of orthogonal polynomials and special functions.

 

·       With my Ph.D. students Luis Rodriguez and Oleg Asipchuk (now a post-doc at the Univ. of Cincinnati) I am studying exponential bases on disconnected domains of Rd and the stability of    Gabor frames and other exponential systems.

 

·       With my former master’s student Andrew Echezabal I am studying   Goldbach conjecture  and other problems in number theory

Oleg’s papers (produced while in graduate school)

1.     O. Asipchuk and V. Drezels, “Examples of exponential bases on union of intervals,” Canadian Mathematical Bulletin 66(4) (2023), 1296–1312. https://doi.org/10.4153/S0008439523000371. arXiv:2211.03821. Publisher record

2.     O. Asipchuk, J. Glidewell, and L. Rodriguez, “Additive stability of frames,” Sampling Theory, Signal Processing, and Data Analysis 22 (2024), Article 20. https://doi.org/10.1007/s43670-024-00094-w. arXiv:2309.06331. Publisher record

3.    O. Asipchuk, C. Leonard, and S. Zheng, “Existence and non-existence of ground state solutions for magnetic NLS,” in D. Wanduku, S. Zheng, H. Zhou, Z. Chen, A. Sills, and E. Agyingi (eds.), Applied Mathematical Analysis and Computations II: SGMC 2021, Springer Proceedings in Mathematics & Statistics, vol. 472, Springer, Cham, 2024, pp. 319–361. https://doi.org/10.1007/978-3-031-69710-4_14. arXiv:2404.01433. Publisher record

4.    O. Asipchuk, L. De Carli, and W. Li, “Concerning the stability of exponential systems and Fourier matrices,” Journal of Fourier Analysis and Applications 31 (2025), Article 67. https://doi.org/10.1007/s00041-025-10197-0. arXiv:2404.05469. Publisher record

5.      O. Asipchuk, “Exponential bases on modified domains,” in D. Bilyk, E. Iacob, A. Martinez-Finkelshtein, and A. M. Stokolos (eds.), Recent Advances in Approximation and Potential Theory, Applied and Numerical Harmonic Analysis, Birkhäuser, Cham, 2026, pp. 29–54. https://doi.org/10.1007/978-3-031-95551-8_2. arXiv:2410.01995.     Publisher record

6.     O. Asipchuk and L. De Carli, “Methods of construction of exponential bases on planar domains,” Bulletin of the Malaysian Mathematical Sciences Society 49 (2026), Article 133. https://doi.org/10.1007/s40840-026-02112-7. arXiv:2505.00870. Publisher record

Andrew’s papers

1.    L. De Carli, A. Echezabal, and M. Laporta, “Approximating the divisor functions,” The Journal of Analysis 32(5) (2024), 2503–2512.  . arXiv:2306.17781. Publisher record

2.    L. De Carli, A. Echezabal, and M. Laporta, “Complete homogeneous symmetric polynomials and generalized Vieta’s formulas,” Results in Mathematics 81 (2026), Article 81.  Preprint. Publisher record

3.    L. De Carli, A. Echezabal, and I. Morell, “On the Sierpiński triangle and its generalizations,” Journal of Mathematics and the Arts 19(3–4) (2025), 79–92.   arXiv:2506.20456. Publisher record

Luis’ papers

1.    O. Asipchuk, J. Glidewell, and L. Rodriguez, “Additive stability of frames,” Sampling Theory, Signal Processing, and Data Analysis 22 (2024), Article 20. https://doi.org/10.1007/s43670-024-00094-w. arXiv:2309.06331. Publisher record

2.    L. De Carli, L. Rodriguez, and P. Vellucci, “Frequency-dependent stability for Gabor frames,” in D. Bilyk, E. Iacob, A. Martinez-Finkelshtein, and A. M. Stokolos (eds.), Recent Advances in Approximation and Potential Theory, Applied and Numerical Harmonic Analysis, Birkhäuser, Cham, 2026, pp. 173–196. https://doi.org/10.1007/978-3-031-95551-8_8. arXiv:2410.01681. Publisher record

3.    O. Asipchuk, L. De Carli, and L. Rodriguez, “Block-equivalent finite Gabor frames,” submitted preprint (2026). arXiv:2605.16139; 

 

 

 

Here is the  list of my publications and preprints.

 

 

 

1.    L. De Carli, M. Laporta , Cyclotomy of symmetric polynomials Preprint (2026)

2.    O. Asipchuk, L. De Carli, and L. Rodriguez, “Block-equivalent finite Gabor frames,” submitted preprint (2026). arXiv:2605.16139.

3.    L. De Carli and M. Laporta, “Divisibility criteria and coefficient formulas for cyclotomic polynomials,” The Ramanujan Journal 70 (2026), Article 44. https://doi.org/10.1007/s11139-026-01421-6.

4.    L. De Carli, A. Echezabal, and M. Laporta, “Complete homogeneous symmetric polynomials and generalized Vieta’s formulas,” Results in Mathematics 81 (2026), Article 81. https://doi.org/10.1007/s00025-026-02632-5.

5.    O. Asipchuk and L. De Carli, “Methods of construction of exponential bases on planar domains,” Bulletin of the Malaysian Mathematical Sciences Society 49 (2026), Article 133. https://doi.org/10.1007/s40840-026-02112-7.

6.    L. De Carli, L. Rodriguez, and P. Vellucci, “Frequency-dependent stability for Gabor frames,” in D. Bilyk, E. Iacob, A. Martinez-Finkelshtein, and A. M. Stokolos (eds.), Recent Advances in Approximation and Potential Theory, Applied and Numerical Harmonic Analysis, Birkhäuser, Cham, 2026, pp. 173–196. https://doi.org/10.1007/978-3-031-95551-8_8.

7.    L. De Carli, A. Echezabal, and I. Morell, “On the Sierpiński triangle and its generalizations,” Journal of Mathematics and the Arts 19(3–4) (2025), 79–92. https://doi.org/10.1080/17513472.2025.2576836.

8.    O. Asipchuk, L. De Carli, and W. Li, “Concerning the stability of exponential systems and Fourier matrices,” Journal of Fourier Analysis and Applications 31 (2025), Article 67. https://doi.org/10.1007/s00041-025-10197-0.

9.   L. De Carli and E. Liflyand, “ simulation for measures revisited,” European Journal of Mathematics 11 (2025), Article 66. https://doi.org/10.1007/s40879-025-00860-7.

10. P. G. Casazza, L. De Carli, and T. T. Tran, “Piecewise scalable frames,” Linear Algebra and its Applications 694 (2024), 262–282. https://doi.org/10.1016/j.laa.2024.04.008.

11. L. De Carli, A. Echezabal, and M. Laporta, “Approximating the divisor functions,” The Journal of Analysis 32(5) (2024), 2503–2512. https://doi.org/10.1007/s41478-024-00747-y.

12. L. De Carli and E. Liflyand, “ simulation for measures,” European Journal of Mathematics 9(3) (2023), Article 83. arXiv:2301.03079.

13. P. G. Casazza, L. De Carli, and T. T. Tran, “Remarks on scalable frames,” Operators and Matrices 17(2) (2023), 327–342. https://doi.org/10.7153/oam-2023-17-23.

14. L. De Carli and P. Vellucci, “Applications of the Lax–Milgram theorem to problems in frame theory,” Sampling Theory, Signal Processing, and Data Analysis 21 (2023), Article 17. https://doi.org/10.1007/s43670-023-00058-6.

15. L. De Carli and J. Edward, “Riesz bases by replacement,” Sampling Theory, Signal Processing, and Data Analysis 20 (2022), Article 9. https://doi.org/10.1007/s43670-022-00025-7.

16. L. De Carli, D. Gorbachev, and S. Tikhonov, “Weighted gradient inequalities and unique continuation problems,” Calculus of Variations and Partial Differential Equations 59(3) (2020), Article 89. https://doi.org/10.1007/s00526-020-1716-8

17. L. De Carli, A. Mizrahi, and A. Tepper, “Three problems on exponential bases,” Canadian Mathematical Bulletin 62(1) (2019), 55–70. https://doi.org/10.4153/CMB-2018-015-6.

18. L. De Carli, “Concerning exponential bases on multi-rectangles of ,” in M. Abell, E. Iacob, A. Stokolos, S. Taylor, S. Tikhonov, and J. Zhu (eds.), Topics in Classical and Modern Analysis, Applied and Numerical Harmonic Analysis, Birkhäuser, Cham, 2019, pp. 65–85. https://doi.org/10.1007/978-3-030-12277-5_4.

19. L. De Carli and P. Vellucci, “-Riesz bases in quasi shift-invariant spaces,” in Y. Kim, S. K. Narayan, G. Picioroaga, and E. Weber (eds.), Frames and Harmonic Analysis, Contemporary Mathematics, vol. 706, American Mathematical Society, Providence, RI, 2018, pp. 201–213. arXiv:1710.00702.

20. L. De Carli and P. Vellucci, “Stability results for Gabor frames and the -order hold models,” Linear Algebra and its Applications 536 (2018), 186–200. https://doi.org/10.1016/j.laa.2017.09.020.

21. L. De Carli, P. Vellucci   Stability results for the n-order hold models  (long version)   (2018)

 

 

22. L. De Carli, D. Gorbachev, and S. Tikhonov, “Pitt inequalities and restriction theorems for the Fourier transform,” Revista Matemática Iberoamericana 33(3) (2017), 789–808. https://doi.org/10.4171/RMI/955.

23. L. De Carli and G. S. Samad, “One-parameter groups of operators and discrete Hilbert transforms,” Canadian Mathematical Bulletin 59(3) (2016), 497–507. https://doi.org/10.4153/CMB-2016-028-7.

24. L. De Carli, S. Hudson, and X. Li, “Minimal potential results for the Schrödinger equation in a slab,” Forum Mathematicum 28(4) (2016), 689–711. https://doi.org/10.1515/forum-2014-0079.

25. L. De Carli and Z. Hu, “Parseval frames with vectors in ,” in Methods of Fourier Analysis and Approximation Theory, Applied and Numerical Harmonic Analysis, Birkhäuser, Cham, 2016, pp. 23–32. https://doi.org/10.1007/978-3-319-27466-9_2.

26. L. De Carli and S. Pathak, “Stability of exponential bases on -dimensional domains,” unpublished preprint (2016).

27. L. De Carli and S. Hudson, “Split functions, Fourier transforms and multipliers,” Collectanea Mathematica 66(2) (2015), 297–309. arXiv:1309.0201.

28. L. De Carli and A. Kumar, “Exponential bases on two-dimensional trapezoids,” Proceedings of the American Mathematical Society 143(7) (2015), 2893–2903. arXiv:1208.3834.

29. L. De Carli, J. Edward, S. Hudson, and M. Leckband, “Minimal support results for Schrödinger’s equation,” Forum Mathematicum 27(1) (2015), 343–371.

30. L. De Carli, D. Gorbachev, and S. Tikhonov, “Pitt and Boas inequalities for Fourier and Hankel transforms,” Journal of Mathematical Analysis and Applications 408(2) (2013), 762–774.

31. D. Bilyk, L. De Carli, A. Petukhov, A. M. Stokolos, and B. D. Wick, “On the scientific work of Konstantin Ilyich Oskolkov,” in D. Bilyk, L. De Carli, A. Petukhov, A. M. Stokolos, and B. D. Wick (eds.), Recent Advances in Harmonic Analysis and Applications: In Honor of Konstantin Oskolkov, Springer Proceedings in Mathematics & Statistics, vol. 25, Springer, New York, 2013, pp. 3–21. https://doi.org/10.1007/978-1-4614-4565-4_1.

32. L. De Carli, “On Fourier multipliers over tube domains,” in D. Bilyk, L. De Carli, A. Petukhov, A. M. Stokolos, and B. D. Wick (eds.), Recent Advances in Harmonic Analysis and Applications: In Honor of Konstantin Oskolkov, Springer Proceedings in Mathematics & Statistics, vol. 25, Springer, New York, 2012, pp. 79–91. https://doi.org/10.1007/978-1-4614-4565-4_10.

33. L. De Carli and S. Hudson, “A Faber–Krahn inequality for solutions of Schrödinger’s equation,” Advances in Mathematics 230 (2012), 2416–2427.

34. L. De Carli and S. M. Hudson, “A generalization of Bernoulli’s inequality,” Le Matematiche 65(1) (2010), 109–117. https://doi.org/10.4418/2010.65.1.9.

35. L. De Carli and S. Hudson, “Geometric remarks on the level curves of harmonic functions,” Bulletin of the London Mathematical Society 42(1) (2010), 83–95.

36. J. M. Ash and L. De Carli, “Growth of Lebesgue constants for convex polyhedra and other regions,” Transactions of the American Mathematical Society 361(8) (2009), 4215–4232.

37. L. De Carli, “Local inequalities for Gegenbauer polynomials,” in L. De Carli and M. Milman (eds.), Topics in Classical Analysis and Applications in Honor of Daniel Waterman, World Scientific, Hackensack, NJ, 2008, pp. 73–87.

38. L. De Carli, “On the - norm of the Hankel transform and related operators,” Journal of Mathematical Analysis and Applications 348(1) (2008), 366–382.

39. L. De Carli and S. Hudson, “Unique continuation for nonnegative solutions of Schrödinger-type inequalities,” Journal of Mathematical Analysis and Applications 318(2) (2006), 467–471.

40. L. De Carli, “Uniform estimates of ultraspherical polynomials of large order,” Canadian Mathematical Bulletin 48(3) (2005), 382–393. https://doi.org/10.4153/CMB-2005-035-2.

41.  L. De Carli and L. Grafakos, “On the restriction conjecture,” Michigan Mathematical Journal 52(1) (2004), 163–180.

42. L. De Carli and T. Okaji, “Unique continuation theorems for Schrödinger operators from a sphere,” Houston Journal of Mathematics 27(1) (2001), 219–235.

43. L. De Carli and E. Laeng, “On the norm of monotonic Fourier multipliers,” Comptes Rendus de l’Académie des Sciences de Paris, Série I, Mathématique 330(8) (2000), 657–662.

44. L. De Carli and E. Laeng, “Sharp estimates for the segment multiplier,” Collectanea Mathematica 51(3) (2000), 309–326.

45. L. De Carli, “Unique continuation for elliptic operators with non-multiple characteristics,” Israel Journal of Mathematics 118 (2000), 15–27.

46. L. De Carli and T. Okaji, “Strong unique continuation for the Dirac operator,” Publications of the Research Institute for Mathematical Sciences 35(6) (1999), 825–846.

47. L. De Carli and A. Iosevich, “Some sharp restriction theorems for homogeneous manifolds,” Journal of Fourier Analysis and Applications 4(1) (1998), 105–128.

48. L. De Carli and M. Nacinovich, “Unique continuation in abstract pseudoconcave CR manifolds,” Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, Series IV, 27(1) (1998), 27–46.

 L. De Carli, “Unique continuation for a class of higher-order elliptic operators,” Pacific Journal of Mathematics 179(1) (1997), 1–10.

49. L. De Carli and A. Iosevich, “A restriction theorem for flat manifolds of codimension two,” Illinois Journal of Mathematics 39(4) (1995),

50. L. De Carli, “ estimates for the Cauchy transform of distributions with respect to convex cones,” Rendiconti del Seminario Matematico della Università di Padova 88 (1992), 35–53.

 

OTHER PUBLICATIONS  AND PREPRINTS  

 

·       Recent Advances in Harmonic Analysis and Applications -In Honor of Konstantin Oskolkov.  (Editor: with D. Bilyk, A. Petukhov, A. Stokolos and B.D. Wick) Springer Proceedings in Mathematics (2012). 

 

·       Topics in Classical Analysis and Applications in Honor of Daniel Waterman, (Editor. With K. Kazarian and M. Milman) World Scientific publishing Company (2008).

 

 

·       Interpolation theory and applications.  (Editor.  with M. Milman) Proceedings of the conference in honor of Professor Michael Cwikel held in Miami, FL, March 29--31, 2006, and the Special Session of the American Mathematical Society Eastern Sectional Meeting held at Florida International University, Miami, FL, April 1--2, 2006.   Contemporary Mathematics, 445. American Mathematical Society.

 

 

 

 

               

 

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