American Institute of Mathematical Sciences

Advanced
March  2020, 19(3): 1275-1289. doi: 10.3934/cpaa.2020062

Convergence of lacunary SU(1, 1)-valued trigonometric products

Jelena Rupčić

Faculty of Transport and Traffic Sciences, University of Zagreb, Vukelićeva 4, 10000 Zagreb, Croatia

Received  March 2019 Revised  July 2019 Published  November 2019

This note attempts to study lacunary trigonometric products with values in the matrix group $ \rm{SU}(1,1) $ in analogy with lacunary trigonometric series. The central questions are the characterization of their convergence in an appropriately defined $ \rm{L}^p $-metric and the characterization of their convergence almost everywhere. These can be interpreted as nonlinear analogues of the classical results by Zygmund and Kolmogorov.

Keywords: Nonlinear Fourier analysis, matrix product, lacunary trigonometric series, convergence in mean, convergence almost everywhere.
Mathematics Subject Classification: Primary: 42A55; Secondary: 40A20.
Citation: Jelena Rupčić. Convergence of lacunary SU(1, 1)-valued trigonometric products. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1275-1289. doi: 10.3934/cpaa.2020062
References:
[1]

M. J. Ablowitz, D. J. Kaup, A. C. Newell and H. Segur, The inverse scattering transform–Fourier analysis for nonlinear problems, Stud. Appl. Math., 53 (1974), 249–315. doi: 10.1137/1015113.  Google Scholar

[2]

L. Carleson, On convergence and growth of partial sums of Fourier series, Acta Math., 116 (1966), 135–157. doi: 10.1007/BF02392815.  Google Scholar

[3]

M. Christ and A. Kiselev, Maximal functions associated to filtrations, J. Funct. Anal., 179 (2001), 409–425. doi: 10.1006/jfan.2000.3687.  Google Scholar

[4]

M. Christ and A. Kiselev, WKB asymptotic behavior of almost all generalized eigenfunctions for one-dimensional Schrödinger operators with slowly decaying potentials, J. Funct. Anal., 179 (2001), 426–447. doi: 10.1006/jfan.2000.3688.  Google Scholar

[5]

L. Golinskii, Absolutely continuous measures on the unit circle with sparse Verblunsky coefficients, Mat. Fiz. Anal. Geom., 11 (2004), 408–420.  Google Scholar

[6]

A. Kolmogorov, Une contribution à l'étude de la convergence des séries de Fourier, Fund. Math., 5 (1924), 96–97. Google Scholar

[7]

V. Kovač, Uniform constants in Hausdorff-Young inequalities for the Cantor group model of the scattering transform, Proc. Amer. Math. Soc., 140 (2012), 915–926. doi: 10.1090/S0002-9939-2011-11078-5.  Google Scholar

[8]

V. Kovač, D. Oliveira e Silva and J. Rupčić, A sharp nonlinear Hausdorff-Young inequality for small potentials, Proc. Amer. Math. Soc., 147 (2019), 239–253. doi: 10.1090/proc/14268.  Google Scholar

[9]

C. Muscalu, T. Tao and C. Thiele, A Carleson theorem for a Cantor group model of the scattering transform, Nonlinearity, 16 (2003), 219–246. doi: 10.1088/0951-7715/16/1/314.  Google Scholar

[10]

C. Muscalu, T. Tao and C. Thiele, A counterexample to a multilinear endpoint question of Christ and Kiselev, Math. Res. Lett., 10 (2003), 237–246. doi: 10.4310/MRL.2003.v10.n2.a10.  Google Scholar

[11]

D. Oliveira e Silva, A variational nonlinear Hausdorff-Young inequality in the discrete setting, Math. Res. Lett., 25 (2018), 1993–2015. doi: 10.4310/MRL.2018.v25.n6.a15.  Google Scholar

[12]

B. Simon, Orthogonal Polynomials on the Unit Circle. Part 1. Classical Theory, American Mathematical Society Colloquium Publications 54, Part 1, AMS, Providence, RI, 2005.  Google Scholar

[13]

B. Simon, Orthogonal Polynomials on the Unit Circle. Part 2. Spectral Theory, American Mathematical Society Colloquium Publications 54, Part 2, AMS, Providence, RI, 2005. doi: 10.1090/coll/054.2/01.  Google Scholar

[14]

T. Tao and C. Thiele, Nonlinear Fourier Analysis, IAS/Park City Graduate Summer School, unpublished lecture notes, 2003, available at arXiv: 1201.5129. Google Scholar

[15]

S. Verblunsky, On positive harmonic functions Ⅱ, Proc. London Math. Soc., 40 (1935), 290–320. doi: 10.1112/plms/s2-40.1.290.  Google Scholar

[16]

V. E. Zakharov and A. B. Shabat, A refined theory of two dimensional self-focussing and one-dimensional self-modulation of waves in non-linear media, Zh. Eksp. Teor. Fiz., 61 (1971), 118–134.  Google Scholar

[17]

A. Zygmund, On the convergence of lacunary trigonometric series, Fund. Math., 16 (1930), 90–107. doi: 10.2307/1989363.  Google Scholar

[18]

A. Zygmund, Trigonometric Series, Volumes Ⅰ and Ⅱ, Second edition, Cambridge University Press, 1959.  Google Scholar

show all references

References:
[1]

M. J. Ablowitz, D. J. Kaup, A. C. Newell and H. Segur, The inverse scattering transform–Fourier analysis for nonlinear problems, Stud. Appl. Math., 53 (1974), 249–315. doi: 10.1137/1015113.  Google Scholar

[2]

L. Carleson, On convergence and growth of partial sums of Fourier series, Acta Math., 116 (1966), 135–157. doi: 10.1007/BF02392815.  Google Scholar

[3]

M. Christ and A. Kiselev, Maximal functions associated to filtrations, J. Funct. Anal., 179 (2001), 409–425. doi: 10.1006/jfan.2000.3687.  Google Scholar

[4]

M. Christ and A. Kiselev, WKB asymptotic behavior of almost all generalized eigenfunctions for one-dimensional Schrödinger operators with slowly decaying potentials, J. Funct. Anal., 179 (2001), 426–447. doi: 10.1006/jfan.2000.3688.  Google Scholar

[5]

L. Golinskii, Absolutely continuous measures on the unit circle with sparse Verblunsky coefficients, Mat. Fiz. Anal. Geom., 11 (2004), 408–420.  Google Scholar

[6]

A. Kolmogorov, Une contribution à l'étude de la convergence des séries de Fourier, Fund. Math., 5 (1924), 96–97. Google Scholar

[7]

V. Kovač, Uniform constants in Hausdorff-Young inequalities for the Cantor group model of the scattering transform, Proc. Amer. Math. Soc., 140 (2012), 915–926. doi: 10.1090/S0002-9939-2011-11078-5.  Google Scholar

[8]

V. Kovač, D. Oliveira e Silva and J. Rupčić, A sharp nonlinear Hausdorff-Young inequality for small potentials, Proc. Amer. Math. Soc., 147 (2019), 239–253. doi: 10.1090/proc/14268.  Google Scholar

[9]

C. Muscalu, T. Tao and C. Thiele, A Carleson theorem for a Cantor group model of the scattering transform, Nonlinearity, 16 (2003), 219–246. doi: 10.1088/0951-7715/16/1/314.  Google Scholar

[10]

C. Muscalu, T. Tao and C. Thiele, A counterexample to a multilinear endpoint question of Christ and Kiselev, Math. Res. Lett., 10 (2003), 237–246. doi: 10.4310/MRL.2003.v10.n2.a10.  Google Scholar

[11]

D. Oliveira e Silva, A variational nonlinear Hausdorff-Young inequality in the discrete setting, Math. Res. Lett., 25 (2018), 1993–2015. doi: 10.4310/MRL.2018.v25.n6.a15.  Google Scholar

[12]

B. Simon, Orthogonal Polynomials on the Unit Circle. Part 1. Classical Theory, American Mathematical Society Colloquium Publications 54, Part 1, AMS, Providence, RI, 2005.  Google Scholar

[13]

B. Simon, Orthogonal Polynomials on the Unit Circle. Part 2. Spectral Theory, American Mathematical Society Colloquium Publications 54, Part 2, AMS, Providence, RI, 2005. doi: 10.1090/coll/054.2/01.  Google Scholar

[14]

T. Tao and C. Thiele, Nonlinear Fourier Analysis, IAS/Park City Graduate Summer School, unpublished lecture notes, 2003, available at arXiv: 1201.5129. Google Scholar

[15]

S. Verblunsky, On positive harmonic functions Ⅱ, Proc. London Math. Soc., 40 (1935), 290–320. doi: 10.1112/plms/s2-40.1.290.  Google Scholar

[16]

V. E. Zakharov and A. B. Shabat, A refined theory of two dimensional self-focussing and one-dimensional self-modulation of waves in non-linear media, Zh. Eksp. Teor. Fiz., 61 (1971), 118–134.  Google Scholar

[17]

A. Zygmund, On the convergence of lacunary trigonometric series, Fund. Math., 16 (1930), 90–107. doi: 10.2307/1989363.  Google Scholar

[18]

A. Zygmund, Trigonometric Series, Volumes Ⅰ and Ⅱ, Second edition, Cambridge University Press, 1959.  Google Scholar

[1]

Michael Boshernitzan, Máté Wierdl. Almost-everywhere convergence and polynomials. Journal of Modern Dynamics, 2008, 2 (3) : 465-470. doi: 10.3934/jmd.2008.2.465
[2]

G. Gentile, V. Mastropietro. Convergence of Lindstedt series for the non linear wave equation. Communications on Pure & Applied Analysis, 2004, 3 (3) : 509-514. doi: 10.3934/cpaa.2004.3.509
[3]

Yuhong Dai, Nobuo Yamashita. Convergence analysis of sparse quasi-Newton updates with positive definite matrix completion for two-dimensional functions. Numerical Algebra, Control & Optimization, 2011, 1 (1) : 61-69. doi: 10.3934/naco.2011.1.61
[4]

Jaroslav Smítal, Marta Štefánková. Omega-chaos almost everywhere. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1323-1327. doi: 10.3934/dcds.2003.9.1323
[5]

Cheng Wang. Convergence analysis of the numerical method for the primitive equations formulated in mean vorticity on a Cartesian grid. Discrete & Continuous Dynamical Systems - B, 2004, 4 (4) : 1143-1172. doi: 10.3934/dcdsb.2004.4.1143
[6]

Andriy Bondarenko, Guy Bouchitté, Luísa Mascarenhas, Rajesh Mahadevan. Rate of convergence for correctors in almost periodic homogenization. Discrete & Continuous Dynamical Systems - A, 2005, 13 (2) : 503-514. doi: 10.3934/dcds.2005.13.503
[7]

Josu Doncel, Nicolas Gast, Bruno Gaujal. Discrete mean field games: Existence of equilibria and convergence. Journal of Dynamics & Games, 2019, 6 (3) : 221-239. doi: 10.3934/jdg.2019016
[8]

Julia Brettschneider. On uniform convergence in ergodic theorems for a class of skew product transformations. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 873-891. doi: 10.3934/dcds.2011.29.873
[9]

Yang Li, Yonghong Ren, Yun Wang, Jian Gu. Convergence analysis of a nonlinear Lagrangian method for nonconvex semidefinite programming with subproblem inexactly solved. Journal of Industrial & Management Optimization, 2015, 11 (1) : 65-81. doi: 10.3934/jimo.2015.11.65
[10]

Markus Haltmeier, Antonio Leitão, Otmar Scherzer. Kaczmarz methods for regularizing nonlinear ill-posed equations I: convergence analysis. Inverse Problems & Imaging, 2007, 1 (2) : 289-298. doi: 10.3934/ipi.2007.1.289
[11]

Alexei Kaltchenko, Nina Timofeeva. Entropy estimators with almost sure convergence and an O(n-1) variance. Advances in Mathematics of Communications, 2008, 2 (1) : 1-13. doi: 10.3934/amc.2008.2.1
[12]

Dirk Aeyels, Filip De Smet, Bavo Langerock. Area contraction in the presence of first integrals and almost global convergence. Discrete & Continuous Dynamical Systems - A, 2007, 18 (1) : 135-157. doi: 10.3934/dcds.2007.18.135
[13]

Yong Duan, Jian-Guo Liu. Convergence analysis of the vortex blob method for the $b$-equation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 1995-2011. doi: 10.3934/dcds.2014.34.1995
[14]

Jean Louis Woukeng. $\sum $-convergence and reiterated homogenization of nonlinear parabolic operators. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1753-1789. doi: 10.3934/cpaa.2010.9.1753
[15]

Yun Cai, Song Li. Convergence and stability of iteratively reweighted least squares for low-rank matrix recovery. Inverse Problems & Imaging, 2017, 11 (4) : 643-661. doi: 10.3934/ipi.2017030
[16]

Jacek Banasiak, Amartya Goswami. Singularly perturbed population models with reducible migration matrix 1. Sova-Kurtz theorem and the convergence to the aggregated model. Discrete & Continuous Dynamical Systems - A, 2015, 35 (2) : 617-635. doi: 10.3934/dcds.2015.35.617
[17]

Mina Jiang, Changjiang Zhu. Convergence rates to nonlinear diffusion waves for $p$-system with nonlinear damping on quadrant. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 887-918. doi: 10.3934/dcds.2009.23.887
[18]

Jian Su, Yinnian He. The almost unconditional convergence of the Euler implicit/explicit scheme for the three dimensional nonstationary Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3421-3438. doi: 10.3934/dcdsb.2017173
[19]

Wenqing Hu, Chris Junchi Li. A convergence analysis of the perturbed compositional gradient flow: Averaging principle and normal deviations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 4951-4977. doi: 10.3934/dcds.2018216
[20]

Leong-Kwan Li, Sally Shao. Convergence analysis of the weighted state space search algorithm for recurrent neural networks. Numerical Algebra, Control & Optimization, 2014, 4 (3) : 193-207. doi: 10.3934/naco.2014.4.193

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (11)
  • HTML views (14)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences