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Convergence of lacunary SU(1, 1)-valued trigonometric products
Faculty of Transport and Traffic Sciences, University of Zagreb, Vukelićeva 4, 10000 Zagreb, Croatia |
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.
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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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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