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2017 | OriginalPaper | Buchkapitel

On the Uniform Theory of Lacunary Series

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Abstract

The theory of lacunary series starts with Weierstrass’ famous example (1872) of a continuous, nondifferentiable function and now we have a wide and nearly complete theory of lacunary subsequences of classical orthogonal systems, as well as asymptotic results for thin subsequences of general function systems. However, many applications of lacunary series in harmonic analysis, orthogonal function theory, Banach space theory, etc. require uniform limit theorems for such series, i.e., theorems holding simultaneously for a class of lacunary series, and such results are much harder to prove than dealing with individual series. The purpose of this paper is to give a survey of uniformity theory of lacunary series and discuss new results in the field. In particular, we study the permutation-invariance of lacunary series and their connection with Diophantine equations, uniform limit theorems in Banach space theory, resonance phenomena for lacunary series, lacunary sequences with random gaps, and the metric discrepancy theory of lacunary sequences.
Fußnoten
1
This is meant as \(\lim _{n\rightarrow \infty }\mathbb{E}(X_{n}Y ) = 0\) for all YL q where 1∕p + 1∕q = 1. This convergence should not be confused with weak convergence of probability distributions, also called convergence in distribution.
 
Literatur
2.
Zurück zum Zitat C. Aistleitner, On the law of the iterated logarithm for the discrepancy of lacunary sequences. Trans. Am. Math. Soc. 362, 5967–5982 (2010) MathSciNetCrossRefMATH C. Aistleitner, On the law of the iterated logarithm for the discrepancy of lacunary sequences. Trans. Am. Math. Soc. 362, 5967–5982 (2010) MathSciNetCrossRefMATH
3.
Zurück zum Zitat C. Aistleitner, I. Berkes, On the central limit theorem for f( n k x). Prob. Theory Rel. Fields 146, 267–289 (2010) CrossRefMATH C. Aistleitner, I. Berkes, On the central limit theorem for f( n k x). Prob. Theory Rel. Fields 146, 267–289 (2010) CrossRefMATH
6.
Zurück zum Zitat C. Aistleitner, I. Berkes, R. Tichy, On permutations of Hardy-Littlewood-Pólya sequences. Trans. Am. Math. Soc. 363, 6219–6244 (2011) CrossRefMATH C. Aistleitner, I. Berkes, R. Tichy, On permutations of Hardy-Littlewood-Pólya sequences. Trans. Am. Math. Soc. 363, 6219–6244 (2011) CrossRefMATH
7.
Zurück zum Zitat C. Aistleitner, I. Berkes, R. Tichy, On the asymptotic behavior of weakly lacunary sequences. Proc. Am. Math. Soc. 139, 2505–2517 (2011) CrossRefMATH C. Aistleitner, I. Berkes, R. Tichy, On the asymptotic behavior of weakly lacunary sequences. Proc. Am. Math. Soc. 139, 2505–2517 (2011) CrossRefMATH
8.
Zurück zum Zitat C. Aistleitner, I. Berkes, R. Tichy, On permutations of lacunary series. RIMS Kokyuroku Bessatsu B 34, 1–25 (2012) MathSciNetMATH C. Aistleitner, I. Berkes, R. Tichy, On permutations of lacunary series. RIMS Kokyuroku Bessatsu B 34, 1–25 (2012) MathSciNetMATH
9.
Zurück zum Zitat C. Aistleitner, I. Berkes, R. Tichy, On the system f( nx) and probabilistic number theory, in Anal. Probab. Methods Number Theory, Vilnius, 2012, ed. by E. Manstavicius et al., pp. 1–18 C. Aistleitner, I. Berkes, R. Tichy, On the system f( nx) and probabilistic number theory, in Anal. Probab. Methods Number Theory, Vilnius, 2012, ed. by E. Manstavicius et al., pp. 1–18
10.
Zurück zum Zitat C. Aistleitner, I. Berkes, R. Tichy, On the law of the iterated logarithm for permuted lacunary sequences. Proc. Steklov Inst. Math. 276, 3–20 (2012) MathSciNetCrossRefMATH C. Aistleitner, I. Berkes, R. Tichy, On the law of the iterated logarithm for permuted lacunary sequences. Proc. Steklov Inst. Math. 276, 3–20 (2012) MathSciNetCrossRefMATH
11.
Zurück zum Zitat D.J. Aldous, Limit theorems for subsequences of arbitrarily-dependent sequences of random variables. Z. Wahrsch. verw. Gebiete 40, 59–82 (1977) MathSciNetCrossRefMATH D.J. Aldous, Limit theorems for subsequences of arbitrarily-dependent sequences of random variables. Z. Wahrsch. verw. Gebiete 40, 59–82 (1977) MathSciNetCrossRefMATH
12.
Zurück zum Zitat D.J. Aldous, Subspaces of L 1 via random measures. Trans. Am. Math. Soc. 267, 445–463 (1981) MathSciNetMATH D.J. Aldous, Subspaces of L 1 via random measures. Trans. Am. Math. Soc. 267, 445–463 (1981) MathSciNetMATH
14.
16.
Zurück zum Zitat I. Berkes, Nongaussian limit distributions of lacunary trigonometric series. Can. J. Math. 43, 948–959 (1991) CrossRefMATH I. Berkes, Nongaussian limit distributions of lacunary trigonometric series. Can. J. Math. 43, 948–959 (1991) CrossRefMATH
17.
Zurück zum Zitat I. Berkes, E. Péter, Exchangeable random variables and the subsequence principle. Prob. Theory Rel. Fields 73, 395–413 (1986) MathSciNetCrossRefMATH I. Berkes, E. Péter, Exchangeable random variables and the subsequence principle. Prob. Theory Rel. Fields 73, 395–413 (1986) MathSciNetCrossRefMATH
18.
Zurück zum Zitat I. Berkes, M. Raseta, On the discrepancy and empirical distribution function of { n k α}. Unif. Distr. Theory 10, 1–17 (2015) MathSciNetMATH I. Berkes, M. Raseta, On the discrepancy and empirical distribution function of { n k α}. Unif. Distr. Theory 10, 1–17 (2015) MathSciNetMATH
19.
Zurück zum Zitat I. Berkes, M. Raseta, On trigonometric sums with random frequencies, Preprint 2016 I. Berkes, M. Raseta, On trigonometric sums with random frequencies, Preprint 2016
20.
Zurück zum Zitat I. Berkes, H.P. Rosenthal, Almost exchangeable sequences of random variables. Z. Wahrsch. verw. Gebiete 70, 473–507 (1985) MathSciNetCrossRefMATH I. Berkes, H.P. Rosenthal, Almost exchangeable sequences of random variables. Z. Wahrsch. verw. Gebiete 70, 473–507 (1985) MathSciNetCrossRefMATH
22.
Zurück zum Zitat I. Berkes, R. Tichy, Lacunary series and stable distributions, in Mathematical Statistics and Limit Theorems. Festschrift for P. Deheuvels, ed. by M. Hallin, D.M. Mason, D. Pfeifer, J. Steinebach (Springer, Berlin, 2015), pp. 7–19 I. Berkes, R. Tichy, Lacunary series and stable distributions, in Mathematical Statistics and Limit Theorems. Festschrift for P. Deheuvels, ed. by M. Hallin, D.M. Mason, D. Pfeifer, J. Steinebach (Springer, Berlin, 2015), pp. 7–19
23.
Zurück zum Zitat I. Berkes, R. Tichy, The Kadec-Pełczynski theorem in L p , 1 ≤ p < 2. Proc. Am. Math. Soc. 144, 2053–2066 (2016) I. Berkes, R. Tichy, The Kadec-Pełczynski theorem in L p , 1 ≤ p < 2. Proc. Am. Math. Soc. 144, 2053–2066 (2016)
24.
Zurück zum Zitat I. Berkes, R. Tichy, Resonance theorems and the central limit theorem for lacunary series, Preprint 2016 I. Berkes, R. Tichy, Resonance theorems and the central limit theorem for lacunary series, Preprint 2016
25.
Zurück zum Zitat I. Berkes, R. Tichy, A uniform version of the subsequence principle, Preprint 2016 I. Berkes, R. Tichy, A uniform version of the subsequence principle, Preprint 2016
26.
Zurück zum Zitat I. Berkes, M. Weber, On the convergence of ∑c k f( n k x). Mem. Am. Math. Soc. 201(943), viii+72 pp. (2009) I. Berkes, M. Weber, On the convergence of ∑c k f( n k x). Mem. Am. Math. Soc. 201(943), viii+72 pp. (2009)
27.
Zurück zum Zitat S. Bobkov, F. Götze, Concentration inequalities and limit theorems for randomized sums. Probab. Theory Rel. Fields 137, 49–81 (2007) MathSciNetCrossRefMATH S. Bobkov, F. Götze, Concentration inequalities and limit theorems for randomized sums. Probab. Theory Rel. Fields 137, 49–81 (2007) MathSciNetCrossRefMATH
29.
Zurück zum Zitat S.D. Chatterji, Un principe de sous-suites dans la théorie des probabilités, in Séminaire des probabilités VI, Strasbourg. Lecture Notes in Mathematics, vol. 258 (Springer, Berlin, 1972), pp. 72–89 S.D. Chatterji, Un principe de sous-suites dans la théorie des probabilités, in Séminaire des probabilités VI, Strasbourg. Lecture Notes in Mathematics, vol. 258 (Springer, Berlin, 1972), pp. 72–89
30.
31.
Zurück zum Zitat S.D. Chatterji, A subsequence principle in probability theory II. The law of the iterated logarithm. Invent. Math. 25, 241–251 (1974) CrossRefMATH S.D. Chatterji, A subsequence principle in probability theory II. The law of the iterated logarithm. Invent. Math. 25, 241–251 (1974) CrossRefMATH
32.
Zurück zum Zitat P. Erdős, On trigonometric sums with gaps. Magyar Tud. Akad. Mat. Kut. Int. Közl. 7, 37–42 (1962) MathSciNetMATH P. Erdős, On trigonometric sums with gaps. Magyar Tud. Akad. Mat. Kut. Int. Közl. 7, 37–42 (1962) MathSciNetMATH
33.
Zurück zum Zitat P. Erdős, I.S. Gál, On the law of the iterated logarithm. Proc. Nederl. Akad. Wetensch. Ser. A 58, 65–84 (1955) CrossRefMATH P. Erdős, I.S. Gál, On the law of the iterated logarithm. Proc. Nederl. Akad. Wetensch. Ser. A 58, 65–84 (1955) CrossRefMATH
34.
Zurück zum Zitat J.-H. Evertse, R.H.-P. Schlickewei, W.M. Schmidt, Linear equations in variables which lie in a multiplicative group. Ann. Math. 155, 807–836 (2002) MathSciNetCrossRefMATH J.-H. Evertse, R.H.-P. Schlickewei, W.M. Schmidt, Linear equations in variables which lie in a multiplicative group. Ann. Math. 155, 807–836 (2002) MathSciNetCrossRefMATH
35.
36.
Zurück zum Zitat K. Fukuyama, The law of the iterated logarithm for the discrepancies of a permutation of { n k x}. Acta Math. Acad. Sci. Hung. 123, 121–125 (2009) MathSciNetCrossRefMATH K. Fukuyama, The law of the iterated logarithm for the discrepancies of a permutation of { n k x}. Acta Math. Acad. Sci. Hung. 123, 121–125 (2009) MathSciNetCrossRefMATH
37.
38.
Zurück zum Zitat K. Fukuyama, B. Petit, Le théorème limite central pour les suites de R. C. Baker. Ergodic Theory Dyn. Syst. 21, 479–492 (2001) MATH K. Fukuyama, B. Petit, Le théorème limite central pour les suites de R. C. Baker. Ergodic Theory Dyn. Syst. 21, 479–492 (2001) MATH
39.
Zurück zum Zitat F.K. Fukuyama, M. Yamashita, Metric discrepancy results for geometric progressions with large ratios. Monatsh. Math. 180, 731–742 (2016) MathSciNetCrossRefMATH F.K. Fukuyama, M. Yamashita, Metric discrepancy results for geometric progressions with large ratios. Monatsh. Math. 180, 731–742 (2016) MathSciNetCrossRefMATH
40.
Zurück zum Zitat J. Galambos, The Asymptotic Theory of Extreme Order Statistics, 2nd ed. (Robert E. Krieger Publishing Co., Melbourne, FL, 1987) MATH J. Galambos, The Asymptotic Theory of Extreme Order Statistics, 2nd ed. (Robert E. Krieger Publishing Co., Melbourne, FL, 1987) MATH
42.
Zurück zum Zitat V.F. Gaposhkin, On some systems of almost independent functions. Siberian Math. J. 9, 198–210 (1968) CrossRefMATH V.F. Gaposhkin, On some systems of almost independent functions. Siberian Math. J. 9, 198–210 (1968) CrossRefMATH
43.
Zurück zum Zitat V.F. Gaposhkin, The central limit theorem for some weakly dependent sequences. Theory Probab. Appl. 15, 649–666 (1970) CrossRefMATH V.F. Gaposhkin, The central limit theorem for some weakly dependent sequences. Theory Probab. Appl. 15, 649–666 (1970) CrossRefMATH
44.
Zurück zum Zitat A. Garsia, Existence of almost everywhere convergent rearrangements for Fourier series of L 2 functions. Ann. Math. 79, 623–629 (1964) MathSciNetCrossRefMATH A. Garsia, Existence of almost everywhere convergent rearrangements for Fourier series of L 2 functions. Ann. Math. 79, 623–629 (1964) MathSciNetCrossRefMATH
45.
Zurück zum Zitat S. Guerre, Types and suites symétriques dans L p , 1 ≤ p < + ∞. Israel J. Math. 53, 191–208 (1986) S. Guerre, Types and suites symétriques dans L p , 1 ≤ p < + . Israel J. Math. 53, 191–208 (1986)
48.
49.
Zurück zum Zitat M.I. Kadec, W. Pełczyński, Bases, lacunary sequences and complemented subspaces in the spaces L p . Studia Math. 21, 161–176 (1961/1962) M.I. Kadec, W. Pełczyński, Bases, lacunary sequences and complemented subspaces in the spaces L p . Studia Math. 21, 161–176 (1961/1962)
51.
Zurück zum Zitat J. Komlós, Every sequence converging to 0 weakly in L 2 contains an unconditional convergence sequence. Ark. Math. 12, 41–49 (1974) MathSciNetCrossRefMATH J. Komlós, Every sequence converging to 0 weakly in L 2 contains an unconditional convergence sequence. Ark. Math. 12, 41–49 (1974) MathSciNetCrossRefMATH
52.
Zurück zum Zitat D.E. Menshov, Sur la convergence et la sommation des séries de fonctions orthogonales. Bull. Soc. Math. France 64, 147–170 (1936) MathSciNet D.E. Menshov, Sur la convergence et la sommation des séries de fonctions orthogonales. Bull. Soc. Math. France 64, 147–170 (1936) MathSciNet
53.
Zurück zum Zitat W. Morgenthaler, A central limit theorem for uniformly bounded orthonormal systems. Trans. Am. Math. Soc. 79, 281–311 (1955) MathSciNetCrossRefMATH W. Morgenthaler, A central limit theorem for uniformly bounded orthonormal systems. Trans. Am. Math. Soc. 79, 281–311 (1955) MathSciNetCrossRefMATH
55.
Zurück zum Zitat E.M. Nikishin, Resonance theorems and superlinear operators. Russian Math. Surv. 25/6, 125–187 (1970) E.M. Nikishin, Resonance theorems and superlinear operators. Russian Math. Surv. 25/6, 125–187 (1970)
56.
Zurück zum Zitat W. Philipp, Limit theorems for lacunary series and uniform distribution mod 1. Acta Arith. 26, 241–251 (1974/1975) W. Philipp, Limit theorems for lacunary series and uniform distribution mod 1. Acta Arith. 26, 241–251 (1974/1975)
57.
Zurück zum Zitat W. Philipp, The functional law of the iterated logarithm for empirical distribution functions of weakly dependent random variables. Ann. Probab. 5, 319–350 (1977) MathSciNetCrossRefMATH W. Philipp, The functional law of the iterated logarithm for empirical distribution functions of weakly dependent random variables. Ann. Probab. 5, 319–350 (1977) MathSciNetCrossRefMATH
58.
Zurück zum Zitat R. Ranga Rao, Relations between weak and uniform convergence of measures with applications. Ann. Math. Stat. 33, 659–680 (1962) MathSciNetCrossRefMATH R. Ranga Rao, Relations between weak and uniform convergence of measures with applications. Ann. Math. Stat. 33, 659–680 (1962) MathSciNetCrossRefMATH
60.
Zurück zum Zitat G. Shorack, J. Wellner, Empirical Processes with Applications in Statistics (Wiley, New York, 1986) MATH G. Shorack, J. Wellner, Empirical Processes with Applications in Statistics (Wiley, New York, 1986) MATH
62.
63.
Zurück zum Zitat S. Takahashi, On the law of the iterated logarithm for lacunary trigonometric series. II. Tohoku Math. J. 27, 391–403 (1975) MathSciNetCrossRefMATH S. Takahashi, On the law of the iterated logarithm for lacunary trigonometric series. II. Tohoku Math. J. 27, 391–403 (1975) MathSciNetCrossRefMATH
64.
Zurück zum Zitat R. Tijdemann, On integers with many small prime factors. Compositio Math. 26, 319–330 (1973) MathSciNet R. Tijdemann, On integers with many small prime factors. Compositio Math. 26, 319–330 (1973) MathSciNet
65.
Zurück zum Zitat P. Uljanov, Solved and unsolved problems in the theory of trigonometric and orthogonal series (Russian). Uspehi Mat. Nauk 19/1, 1–69 (1964) P. Uljanov, Solved and unsolved problems in the theory of trigonometric and orthogonal series (Russian). Uspehi Mat. Nauk 19/1, 1–69 (1964)
67.
Zurück zum Zitat M. Weiss, On the law of the iterated logarithm for uniformly bounded orthonormal systems. Trans. Am. Math. Soc. 92, 531–553 (1959) MathSciNetCrossRefMATH M. Weiss, On the law of the iterated logarithm for uniformly bounded orthonormal systems. Trans. Am. Math. Soc. 92, 531–553 (1959) MathSciNetCrossRefMATH
68.
Zurück zum Zitat A. Zygmund, Trigonometric Series, vols. I, II, 3rd ed. (Cambridge University Press, Cambridge, 2002) A. Zygmund, Trigonometric Series, vols. I, II, 3rd ed. (Cambridge University Press, Cambridge, 2002)
Metadaten
Titel
On the Uniform Theory of Lacunary Series
verfasst von
István Berkes
Copyright-Jahr
2017
DOI
https://doi.org/10.1007/978-3-319-55357-3_6

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