Almost Everywhere Convergence of Riesz-Raikov Series

Ai Fan

Colloquium Mathematicae (1995)

  • Volume: 68, Issue: 2, page 241-248
  • ISSN: 0010-1354

Abstract

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Let T be a d×d matrix with integer entries and with eigenvalues >1 in modulus. Let f be a lipschitzian function of positive order. We prove that the series ∑ n = 1 ∞ c n f ( T n x ) converges almost everywhere with respect to Lebesgue measure provided that ∑ n = 1 ∞ | c n | 2 l o g 2 n < ∞ .

How to cite

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Fan, Ai. "Almost Everywhere Convergence of Riesz-Raikov Series." Colloquium Mathematicae 68.2 (1995): 241-248. <http://eudml.org/doc/210308>.

@article{Fan1995,
abstract = {Let T be a d×d matrix with integer entries and with eigenvalues >1 in modulus. Let f be a lipschitzian function of positive order. We prove that the series $∑_\{n=1\}^\{∞\} c_n f(T^\{n\}x)$ converges almost everywhere with respect to Lebesgue measure provided that $∑_\{n=1\}^\{∞\} |c_n|^2 log^\{2\}n < ∞$.},
author = {Fan, Ai},
journal = {Colloquium Mathematicae},
keywords = {Riesz-Raikov series; quasi-orthogonality; Bernoulli measures},
language = {eng},
number = {2},
pages = {241-248},
title = {Almost Everywhere Convergence of Riesz-Raikov Series},
url = {http://eudml.org/doc/210308},
volume = {68},
year = {1995},
}

TY - JOUR
AU - Fan, Ai
TI - Almost Everywhere Convergence of Riesz-Raikov Series
JO - Colloquium Mathematicae
PY - 1995
VL - 68
IS - 2
SP - 241
EP - 248
AB - Let T be a d×d matrix with integer entries and with eigenvalues >1 in modulus. Let f be a lipschitzian function of positive order. We prove that the series $∑_{n=1}^{∞} c_n f(T^{n}x)$ converges almost everywhere with respect to Lebesgue measure provided that $∑_{n=1}^{∞} |c_n|^2 log^{2}n < ∞$.
LA - eng
KW - Riesz-Raikov series; quasi-orthogonality; Bernoulli measures
UR - http://eudml.org/doc/210308
ER -

References

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  1. [1] G. Brown and A. H. Dooley, Odometer actions on G-measures, Ergodic Theory Dynamical Systems, 11 (1991), 279-307. Zbl0739.58032
  2. [2] M. Kac, R. Salem and A. Zygmund, A gap theorem, Trans. Amer. Math. Soc. 63 (1948), 235-243. Zbl0032.27402
  3. [3] S. Kakutani and K. Petersen, The speed of convergence in the Ergodic Theorem, Monatsh. Math. 91 (1981), 11-18. Zbl0446.28015
  4. [4] K. Petersen, Ergodic Theory, Cambridge Univ. Press, 1983. 
  5. [5] D. A. Raikov, On some arithmetical properties of summable functions, Mat. Sb. 1 (43) (1936), 377-384 (in Russian). Zbl0014.39701
  6. [6] F. Riesz, Sur la théorie ergodique, Comment. Math. Helv. 17 (1944-1945), 217-248. 
  7. [7] J. Rosenblatt, Convergence of series of translations, Math. Ann. 230 (1977), 245-272. Zbl0341.40002
  8. [8] J. Rosenblatt and A. del Junco, Counterexamples in ergodic theory and number theory, Math. Ann. 245 (1979), 185-197. Zbl0398.28021
  9. [9] A. Zygmund, Trigonometric Series, Vols. I and II, Cambridge Univ. Press, 1959. 

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