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

Jelena Rupčić

doi:10.3934/cpaa.2020062

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2020, Volume 19, Issue 3: 1275-1289. Doi: 10.3934/cpaa.2020062

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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 online: March 2020

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Abstract

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.

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Mathematics Subject Classification: Primary: 42A55; Secondary: 40A20.

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[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.
[2]	L. Carleson, On convergence and growth of partial sums of Fourier series, Acta Math., 116 (1966), 135–157. doi: 10.1007/BF02392815.
[3]	M. Christ and A. Kiselev, Maximal functions associated to filtrations, J. Funct. Anal., 179 (2001), 409–425. doi: 10.1006/jfan.2000.3687.
[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.
[5]	L. Golinskii, Absolutely continuous measures on the unit circle with sparse Verblunsky coefficients, Mat. Fiz. Anal. Geom., 11 (2004), 408–420.
[6]	A. Kolmogorov, Une contribution à l'étude de la convergence des séries de Fourier, Fund. Math., 5 (1924), 96–97.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[14]	T. Tao and C. Thiele, Nonlinear Fourier Analysis, IAS/Park City Graduate Summer School, unpublished lecture notes, 2003, available at arXiv: 1201.5129.
[15]	S. Verblunsky, On positive harmonic functions Ⅱ, Proc. London Math. Soc., 40 (1935), 290–320. doi: 10.1112/plms/s2-40.1.290.
[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.
[17]	A. Zygmund, On the convergence of lacunary trigonometric series, Fund. Math., 16 (1930), 90–107. doi: 10.2307/1989363.
[18]	A. Zygmund, Trigonometric Series, Volumes Ⅰ and Ⅱ, Second edition, Cambridge University Press, 1959.