Generalized Riesz Products on the Bohr Compactification of \({\mathbb {R}}\)

详细信息    查看全文

• 作者：E. H. El Abdalaoui
• 关键词：Generalized Riesz products ; Almost periodic functions ; Bohr compactification ; Kakutani’s criterion ; Bourgain’s singularity criterion ; Flat polynomials ; 42A05 ; 42A55 ; 11L03 ; 42A61
• 刊名：Journal of Fourier Analysis and Applications
• 出版年：2016
• 出版时间：February 2016
• 年：2016
• 卷：22
• 期：1
• 页码：20-35
• 全文大小：574 KB
• 参考文献：1.Besicovitch, A.S.: Almost Periodic Functions. Cambridge University Press, Cambridge (1932)
  2.Billingsley, P.: Convergence of Probability Measures. Wiley Series in Probability and Statistics: Probability and Statistics, 2nd edn. Wiley, New York (1999)MATH CrossRef
  3.Bohr, H.: Almost Periodic Functions. Chelsea Pub. Co., New York (1947)
  4.Bourgain, J.: On the spectral type of Ornstein class one transformations. Isr. J. Math. 84, 53–63 (1993)MATH MathSciNet CrossRef
  5.Brown, G., Moran, W.: Products of random variables and Kakutani’s criterion for orthogonality of product measures. J. Lond. Math. Soc. (2) 10, 401–405 (1975)MATH MathSciNet CrossRef
  6.Choksi, J.R., Nadkarni, M.G.: The maximal spectral type of rank one transformation. Can. Math. Bull. 37(1), 29–36 (1994)MATH MathSciNet CrossRef
  7.Chung, K.L.: A Course in Probability Theory, 3rd edn. Academic Press Inc, San Diego (2001)
  8.Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and its Applications, vol. 44. Cambridge University Press, Cambridge (1992)MATH CrossRef
  9.D. Dacunha-Castelle and M. Duflo, Probabilités et statistiques. Tome 2, (French) [Probability and statistics. Vol. 2], Problèmes à temps mobile [Movable-time problems], Collection Mathématiques Appliquées pour la Maîtrise [Collection of Applied Mathematics for the Master’s Degree], Masson, Paris, 1983
  10.Dudley, R.M.: Real Analysis and Probability. The Wadsworth & Brooks/Cole Mathematics Series. Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove (1989)MATH
  11.El Abdalaoui, E.H., Lemańczyk, M., Lesigne, E., Ulcigrai, C.: Spectral disjointness in some class of rank one flows, preprint
  12.El Abdalaoui, E.H.: A new class of rank-one transformations with singular spectrum. Ergod. Theory Dynam. Syst. 27(5), 1541–1555 (2007)MATH MathSciNet CrossRef
  13.Gelfand, I.M., Raikov, D.A., Chilov, G.E.: Commutative Normed Rings. Chelsea Pub. Co., New York (1964)
  14.Guenais, M.: Étude spectrale de certains produits gauches en théorie ergodique. Thèse de doctorat, Paris 13, (1997)
  15.Host, B., Méla, J.-F., Parreau, F.: Nonsingular transformations and spectral analysis of measures. Bull. Soc. Math. France 119(1), 33–90 (1991)MATH MathSciNet
  16.Kac, M.: Statistical Independence in Probability, Analysis and Number Theory. The Carus Mathematical Monographs. No. 12 Published by the Mathematical Association of America. Wiley, New York (1959)
  17.Kakutani, S.: On equivalence of infinite product measures. Ann. of Math. (2) 49, 214–224 (1948)MATH MathSciNet CrossRef
  18.Katznelson, Y.: An Introduction to Harmonic Analysis, 3rd edn. Cambridge Mathematical Library. Cambridge University Press, Cambridge (2004)MATH CrossRef
  19.Klemes, I.: The spectral type of staircase transformations. Thohoku Math. J. 48, 247–258 (1994)MathSciNet CrossRef
  20.Klemes, I., Reinhold, K.: Rank one transformations with singular spectre type. Isr. J. Math. 98, 1–14 (1997)MATH MathSciNet CrossRef
  21.Ledrappier, F.: Des produits de Riesz comme mesures spectrales (French, English summary). Ann. Inst. Henri Poincaré Sect. B (N.S.) 6, 335–344 (1970)MATH MathSciNet
  22.Nadkarni, M.G.: Spectral Theory of Dynamical Systems. Birkhäuser, Cambridge (1998)CrossRef
  23.Peyrière, J.: Étude de quelques propriétés des produits de Riesz. Ann. Inst. Fourier (Grenoble) 25(2), 127–169 (1975)MATH MathSciNet CrossRef
  24.Prikhod’ko, A.A.: Littlewood polynomials and their applications to the spectral theory of dynamical systems. (Russian) Mat. Sb. 204 (6), 135–160 (2013); translation in Sb. Math. 204 (5–6), 910–935 (2013)
  25.Queffelec, M.: Mesures spectrales associées à certaines suites arithmétiques (French). Bull. Soc. Math. France 107(4), 385–421 (1979)MATH MathSciNet
  26.Queffelec, H., Saffari, B.: On Bernstein’s inequality and Kahane’s ultraflat polynomials. J. Fourier Anal. Appl. 2(6), 519–582 (1996)MATH MathSciNet CrossRef
  27.Queffélec, M.: Substitution Dynamical Systems-Spectral Analysis. Lecture Notes in Mathematics, vol. 1294, 2nd edn. Springer, Berlin (2010)CrossRef
  28.Riesz, F.: Über die Fourierkoeffizienten einer stetigen Funktion von beschränkter Schwankung (German). Math. Z. 2(3–4), 312–315 (1918)MATH MathSciNet CrossRef
  29.Zygmund, A.: Trigonometric Series, vol. I, 2nd edn. Cambridge University Press, Cambridge (1959)MATH
  
• 作者单位：E. H. El Abdalaoui (1)
  
  1. Normandie University, University of Rouen, Department of Mathematics, LMRS UMR 60 85 CNRS, Avenue de l’université, B.P. 12, 76801, Saint Etienne du Rouvray, France
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Fourier Analysis
  Abstract Harmonic Analysis
  Approximations and Expansions
  Partial Differential Equations
  Applications of Mathematics
  Signal,Image and Speech Processing
  
• 出版者：Birkh盲user Boston
• ISSN：1531-5851

文摘

We study a class of generalized Riesz products connected to the spectral type of some class of rank one flows on \({\mathbb {R}}\). Applying Kac’s central limit theorem, we exhibit a large class of singular generalized Riesz products on the Bohr compactification of \({\mathbb {R}}\). Keywords Generalized Riesz products Almost periodic functions Bohr compactification Kakutani’s criterion Bourgain’s singularity criterion Flat polynomials