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

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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 and 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.
[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.

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.
[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.

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