My research is in harmonic
analysis and approximation theory. In particular, I am interested in:
·
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.
With my student
Oleg Asipchuk I am studying exponential bases on disconnected domains of Rd. With
my student Luis Rodrigues I am investigating the properties of
finite frames and Gabor frames.
Here are links to Oleg
and Luis’ papers.
1. Construction
on exponential bases on split intervals (with V. Drezels)-
2022. To appear in: Canadian Math Bulletin (2023)
2. Additive
stability of frames (with J. Glidwell and L. Rodriguez)-2023. Submitted.
Here is a list of my publications and preprints.
Papers
in Professional Journals and preprints
[43] A. Echezabal, L. De Carli,
M. Laporta, Approximating divisor
functions (2023) (submitted)
[42] L. De Carli, E.
Liflyand, Lp simulation of measures (2023) to appear in the European Journal of mathematics.
[41] P. Casazza, L. De Carli
and T. Tran, Piecewise scalable frames To appear in Linear algebra and
its applications (2023)
[40] P. Casazza, L. De Carli
and T. Tran, Remark on scalable frames (2022) To appear on Operators and Matrices (2023)
[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.
·
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.