RESEARCH


 

My research is in harmonic analysis and approximation theory.  I recently started working also on combinatorics and number theory.  Currently,  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

 

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.

6.     Primality tests,  properties of the divisor functions, and the Goldbach conjecture

 

·        With my  Ph.D. students  Oleg Asipchuk and Luis Rodriguez, 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 and with Ismael Morel (a new graduate student), I am studying primality tests, the Goldbach conjecture  and other problems in number theory

Oleg’s papers

1.      Examples of exponential bases on union of  intervals  (with V. Drezels)- 2022.  Canadian Math Bulletin, Volume 66 , Issue 4 ,   pp. 1296 – 1312 (2023)

2.      Additive stability of frames  (with L. Rodriguez  and J. Glidwell)-  Submitted (2023)

3.      Existence and non-existence of ground state solutions for magnetic NLS  (with C. Leonard, S. Zheng). Submitted  (2024)

4.      Concerning the stability of exponential systems and Fourier matrices (with L. De Carli and W. Li).  Submitted (2024)

5.     Exponential bases on modified domains Submitted (2024)

Andrew’s papers

1.     Approximating divisor functions  (with L. De Carli and M. Laporta) The Journal of Analysis.  https://doi.org/10.1007/s41478-024-00747-y (2024)

2.     Generalized Vieta formulas and complete homogeneous symmetric polynomials (with L. De Carli and M. Laporta)  Submitted (2024)

3.     Egyptian fraction  and the Sierpinski  triangle, (with L. De Carli  and I. Morel) Preprint (2024)

Luis’ papers

1.     Additive stability of frames  (with O. Asipchuk  and J. Glidwell)-  Submitted (2023)

2.      Frequency-dependent stability for Gabor frames (with L. De Carli, P. Vellucci).  Submitted (2024)

 

 

Here is a list of my publications and preprints.

 

Papers in Professional Journals and preprints

[46] Frequency-dependent stability for Gabor frames (with L. Rodriguez, P. Vellucci).  Submitted (2024)

 

[45 ] A. Echezabal, L. De Carli, M. Laporta, Generalized Vieta formulas and complete homogeneous symmetric polynomials  Submitted   (2024)

 

[44]  A. Asipchuk, L. De Carli and W. Li, Concerning the stability of exponential systems and  Fourier matrices,  (2024)- Submitted

 

[43] A. Echezabal, L. De Carli, M. Laporta,  Approximating divisor functions   The Journal of Analysis.  https://doi.org/10.1007/s41478-024-00747-y (2024)

[42] L. De Carli, E. Liflyand, Lp simulation of measures Eur. J. Math. 9, no. 3, Paper No. 83  (18 pp) (2023) 

[41] P. Casazza, L. De Carli and T. Tran, Piecewise scalable frames  Linear algebra and its applications 694 (2024), 262–282. 

[40] P. Casazza, L. De Carli and T. Tran, Remark on scalable frames   (Operators and Matrices Volume 17, Number 2 (2023), 327–342

[39] L. De Carli, P. Vellucci, Applications of Lax-Milgram theorem to problems in frame theory,  in:   “Sampling Theory in Signal and Information Processing” (topical collection in data sciences, approximation, and harmonic analysis) (2023)

[38] L. De Carli and J. Edward,  Riesz bases by replacement  Sampling Theory, Signal Processing, and Data Analysis 20 no. 9, (2022).  https://doi.org/10.1007/s43670-022-00025-7

[37] L. De Carli, D. Gorbachev, and S. Tikhonov,  Weighted gradient inequalities and unique continuation problems,  Calculus of Variations and PDE's 59 (89) (2020)

[36] L. De Carli, A. Mizrahi, A. Tepper, Three problems on exponential bases  Canadian Math. Bulletin (2018) http://dx.doi.org/10.4153/CMB-2018-015-6

 [35]  L. De Carli, P. Vellucci   p-Riesz basis and  quasi-shift  invariant spaces  To appear in the Contemporary Mathematics volume “Proceedings of the AMS Special Sessions "Frames, Harmonic Analysis and Operator Theory"   edited by:   Y. Kim, S. K. Narayan, G.Picioroaga and E. Weber.

[34] L. De Carli, P. Vellucci   Stability results for Gabor frames and the p-order hold models  (short version) Linear Algebra and Its Applications 536C (2018) pp. 186—200, DOI 10.1016/j.laa.2017.09.020

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

[32] L. De Carli. Concerning exponential bases on multi-rectangles of Rd,  In: Abell, M., Iacob, E., Stokolos, A., Taylor, S., Tikhonov, S., Zhu, J. (eds) Topics in Classical and Modern Analysis. Applied and Numerical Harmonic Analysis. Chapter 3 Birkhäuser, (2019) https://doi.org/10.1007/978-3-030-12277-5_4

[31] L. De Carli    Exponential bases on multi-rectangles of Rd,  (long version, 2017)

 [30] L. De Carli and Shaikh Gohin Samad  One-parameter groups and discrete Hilbert transform, Canadian Math. Bull. 59 (2016), 497-507

[29] L. De Carli, Dmitriy Gorbachev and Sergey Tikhonov.  Pitt inequalities and restriction theorems for the Fourier transform   Revista Mat. Iberoamericana 33, (3) 2017, pp. 789–808. DOI: 10.4171/RMI/955

[28] L. De Carli and S. Pathak.  Stability of exponential bases on d- dimensional domains (2016) (preprint)

[27]   L. De Carli and Z. Hu, Parseval frames with n+1 elements in Rn,  in: Methods of Fourier analysis and approximation theory (Applied and numerical harmonic analysis) Birkhauser (2016), 23--32,

 

[26]   L. De Carli,  S. Hudson,  Split functions, Fourier transforms and multipliers.    Collect. Math. 66 (2015), no. 2, 297–309. 

 

[25]   L. De Carli, S. Hudson and X. Li,  Minimal potential results for the Schrodinger equation in a slab,   Forum Math, 28 (2016), no. 4, pp 689—712.

 

[24] L. De Carli, D. Gorbachev, and S. Tikhonov,   Pitt's and Boas' inequalities for Fourier and Hankel  Transforms, Journal of Mathematical Analysis and Applications  408,   (2013),  no. 2, 762–774

 

 [23] L. De Carli, A. Kumar , Exponential bases  on two dimensional  trapezoids,   Proc. Amer. Math. Soc. 143 (2015), no. 7, 2893–2903.

 

[22] D. Bilyk, L. De Carli, A. Petukhov, A. Stokolos and B. D. Wick, On The Scientific Work of Konstantin Ilyich Oskolkov , Recent Advances in Harmonic Analysis and Applications (In Honor of Konstantin Oskolkov), Springer Proceedings in Mathematics (2012)

 

[21]  L. De Carli,  J. Edward, S. Hudson, M. Leckband,  Minimal support results for  Schrodinger's equation,    Forum Math. 27 (2015), no. 1, 343–371

 

[20]  L. De Carli,    On Fourier multipliers over tube domains,  Recent Advances in Harmonic Analysis and Applications (In Honor of Konstantin Oskolkov), Springer Proceedings  in Mathematics (2012), 79 –92.

 

[19] L. De Carli, S. Hudson,  A Faber-Krahn inequality for solutions of Schrodinger’s equation,.  Advances in Mathematics 230 (2012), pp. 2416-2427

 

[18]   L. De Carli, S. Hudson,  A generalization of Bernoulli’s inequality,   Le Matematiche 65 (2010), n. 1 

 

[17]   L. De Carli, S. Hudson,  Geometric Remarks on the Level Curves of Harmonic Functions,  Bull. London Math. Soc. 42 (2010), n. 1, 83—95.

 

[16]   L. De Carli,  M. Ash,  Growth of  Lp Lebesgue constants for convex polyhedra and other regions,     Transaction of the American Math. Soc.  361 (2009), n. 8,   4215--4232.

 

[15]   L. De Carli, Local Lp inequalities for Gegenbauer polynomials,   in; Topics in classical analysis and applications in honor of Daniel Waterman, 73--87, World Sci. Publ., Hackensack, NJ, (2008).

 

[14]   L. De Carli,  On the  Lp--Lq  norm of the Hankel transform and  related operators,     J. Math. Anal. Appl. 348 (2008), n. 1, 366--382.  

[13] L. De Carli, S. Hudson,   Unique continuation for nonnegative solutions of Schrödinger type inequalities. J. Math. Anal. Appl. 318 (2006), no 2,   467--471.

[12]   L. De Carli,   Uniform estimates of ultraspherical polynomials of large order Canadian Math. Bulletin 48 (2005), no 3, 382—393.

               

[11]   L. De Carli and L. Grafakos,  On the restriction conjecture, Michigan Math. J. 52 (2004), no. 1, 163--180.

 

[10]   L. De Carli and T. Okaji, Unique continuation theorems for Schrodinger operators from a sphere, Houston J. Math. 27 (2001), no. 1, 219--235.

 

[9]   L. De Carli and E. Laeng, On the (p,p)  norm of monotonic Fourier multipliers, C. R. Acad.  Sci. Paris Sér. I Math. 330 (2000), no. 8, 657--662.


[8]   L. De Carli and E. Laeng, Sharp Lp estimates for the segment multiplier,  Collectanea. Math. 51 (2000), no. 3, 309—326.

[7] L. De Carli,  Unique continuation for elliptic operators with non-multiple characteristics,  Israel J. Math. 118 (2000), 15--27.

[6]   L. De Carli and T. Okaji,  Strong Unique continuation for the Dirac operator,  Publ. Res. Inst. Math. Sci. 35 (1999), no. 6, 825—846.

[5]   L. De Carli and A. Iosevich, Some sharp restriction theorems for homogeneous manifolds, J. Fourier Anal. Appl. 4 (1998), no. 1, 105--128.

[4]   L. De Carli and  M. Nacinovich, Unique continuation in abstract  pseudoconcave  CR  manifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 27 (1998), no. 1, 27--46.

[3]   L. De Carli Unique continuation for a class of  higher order elliptic operators, Pacific J. Math. 179 (1997), no. 1, 1--10.

[2]   L. De Carli and A. Iosevich, A restriction theorem for flat manifolds of codimension two, Illinois J.  Math. 39 (1995), no. 4, 576--585.

[1]   L. De Carli, Lp  estimates for the Cauchy transform of distributions with respect to convex cones, Rend. Sem. Mat. Univ. 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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