Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

On the convergence of $\sum c_{n} f(nx)$
and the Lip 1/2 class


Author: István Berkes
Journal: Trans. Amer. Math. Soc. 349 (1997), 4143-4158
MSC (1991): Primary 42A55, 42A61
DOI: https://doi.org/10.1090/S0002-9947-97-01837-0
MathSciNet review: 1401764
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We investigate the almost everywhere convergence of $\sum c_{n} f(nx)$, where $f$ is a measurable function satisfying

\begin{equation*}f(x+1) = f(x), \qquad \int _{0}^{1} f(x) \, dx =0.\end{equation*}

By a known criterion, if $f$ satisfies the above conditions and belongs to the Lip $\alpha $ class for some $\alpha > 1/2$, then $\sum c_{n} f(nx)$ is a.e. convergent provided $\sum c_{n}^{2} < +\infty $. Using probabilistic methods, we prove that the above result is best possible; in fact there exist Lip 1/2 functions $f$ and almost exponentially growing sequences $(n_{k})$ such that $\sum c_{k} f(n_{k} x)$ is a.e. divergent for some $(c_{k})$ with $\sum c_{k}^{2} < +\infty $. For functions $f$ with Fourier series having a special structure, we also give necessary and sufficient convergence criteria. Finally we prove analogous results for the law of the iterated logarithm.


References [Enhancements On Off] (What's this?)

  • 1. Berkes, I., An almost sure invariance principle for lacunary trigonometric series, Acta Math. Acad. Sci. Hung. 26 (1975), 209-220. MR 54:14031
  • 2. Berkes, I., Critical LIL behavior of the trigonometric system, Trans. Amer. Math. Soc. 338 (1993), 553-585. MR 93j:60035
  • 3. Berkes, I., An optimal condition for the LIL for trigonometric series, Trans. Amer. Math. Soc. 347 (1995), 515-530. MR 95h:42006
  • 4. Berkes, I. and Philipp, W., The size of trigonometric and Walsh series and uniform distribution mod 1, J. London Math. Soc. 50 (1994), 454-464. MR 96e:11099
  • 5. Carleson, L., On convergence and growth of partial sums of Fourier series, Acta Math. 116 (1966), 135-157. MR 33:7774
  • 6. Dhompongsa, S., Uniform laws of the iterated logarithm for Lipschitz classes of functions, Acta Sci. Math. 50 (1986), 105-124. MR 87m:26003
  • 7. Erd\H{o}s, P., On the convergence of trigonometric series, Journ. of Math. and Physics 22 (1943), 37-39. MR 4:271e
  • 8. Erd\H{o}s, P., On the strong law of large numbers, Trans. Amer. Math. Soc. 67 (1949), 51-56. MR 11:375c
  • 9. Erd\H{o}s, P., On trigonometric sums with gaps., Magyar Tud. Akad. Mat. Kut. Int. Közl. 7 (1962), 37-42. MR 26:2797
  • 10. Gaposhkin, V. F., Lacunary series and independent functions, Uspekhi Mat. Nauk 21/6 (1966), 1-82 (Russian); English translation: Russian Math. Surveys 21/6, 1-82. MR 34:6374
  • 11. Gaposhkin, V. F., On series by the system $\phi (nx)$, Mat. Sbornik 69 (1966), 328-353 (Russian); English translation: Amer. Math. Soc. Transl. (2) 86 (1970), 167-197. MR 33:6198
  • 12. Gaposhkin, V. F., On a convergence system, Mat. Sbornik 74 (1967), 93-99 (Russian); English translation: Math. USSR Sb. 3 (1967), 83-90. MR 35:7065
  • 13. Gaposhkin, V. F., On convergence and divergence systems, Mat. Zametki 4 (1968), 253-260 (Russian); English translation: Math. Notes 4 (1968). MR 38:1462
  • 14. Kac, M., Convergence of certain gap series, Ann. of Math. 44 (1943), 411-415. MR 5:4c
  • 15. Kac, M., Probability methods in some problems of analysis and number theory, Bull. Amer. Math. Soc. 55 (1949), 641-665. MR 11:161b
  • 16. Kahane, J. P., Some random series of functions, Heath, Lexington, 1968. MR 40:8095
  • 17. Kaufman, R. and Philipp, W., A uniform law of the iterated logarithm for classes of functions., Annals of Probability 6 (1978), 930-952. MR 80a:60034
  • 18. Loève, M., Probability theory, Van Nostrand, New York, 1955. MR 16:598f
  • 19. Marstrand, J. M., On Khinchin's conjecture about strong uniform distribution., Proc. London Math. Soc. 21 (1970), 540-556. MR 45:185
  • 20. Nikishin, E. M., Resonance theorems and superlinear operators, Uspehi Mat. Nauk 25/6 (1970), 129-191 (Russian); English translation: Russian Math. Surveys 25/6, 125-187. MR 45:5643
  • 21. Philipp, W., Limit theorems for lacunary series and uniform distribution mod 1, Acta Arithmetica 26 (1975), 241-251. MR 52:325
  • 22. Takahashi, S., An asymptotic property of a gap sequence, Proc. Japan Acad. 38 (1962), 101-104. MR 25:4279
  • 23. Takahashi, S., On the law of the iterated logarithm for lacunary trigonometric series, Tôhoku Math. J. 24 (1972), 319-329. MR 48:9342
  • 24. Takahashi, S., On the law of the iterated logarithm for lacunary trigonometric series II, Tôhoku Math. J. 27 (1975), 391-403. MR 55:13147
  • 25. Takahashi, S., An asymptotic behavior of $ \{ f(n_{k} t)\}$, Sci. Rep. Kanazawa Univ. 33 (1988), 27-36. MR 90m:26004
  • 26. Zygmund, A., Trigonometric series, Vol I., Cambridge University Press, 1959. MR 21:6498

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 42A55, 42A61

Retrieve articles in all journals with MSC (1991): 42A55, 42A61


Additional Information

István Berkes
Affiliation: Mathematical Institute of the Hungarian Academy of Sciences, H-1364 Budapest, P.O.B. 127, Hungary
Email: berkes@math-inst.hu

DOI: https://doi.org/10.1090/S0002-9947-97-01837-0
Keywords: Almost everywhere convergence, Lipschitz classes, lacunary series, law of the iterated logarithm
Received by editor(s): March 27, 1996
Additional Notes: Research supported by Hungarian National Foundation for Scientific Research, Grants T 16384 and T 19346
Article copyright: © Copyright 1997 American Mathematical Society

American Mathematical Society