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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

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