BMO estimates for lacunary series

We proveBMO andL ^p norm inequalities inR ⁿ for lacunary Walsh and generalized trigonometric series.     

A mean central limit theorem for weakly multiplicative systems and its application to lacunary trigonometric series

Summary In this paper, we discuss the rate of convergence in the mean central limit theorem for weakly multiplicative systems and apply this result to lacunary trigonometric series under probability measures on with Hölder continuous distribution function.     

On cubic lacunary Fourier series

For , we analyze the behavior, near the rational points , of , considered as a function of . We expand this series into a constant term, a term on the order of , a term linear in , a ``chirp" term on the order of , and an error term on the order of . At every such rational point, the left and right derivatives are either both finite (and equal) or both infinite, in contrast with the quadratic series, where the derivative is often finite on one side and infinite on the other. However, in the cubic series, again in contrast with the quadratic case, the chirp term generally has a different set of frequencies and amplitudes on the right and left sides. Finally, we show that almost every irrational point can be closely approximated, in a suitable Diophantine sense, by rational points where the cubic series has an infinite derivative. This implies that when , both the real and imaginary parts of the cubic series are differentiable almost nowhere.

     

Irregular discrepancy behavior of lacunary series

In 1975 Philipp showed that for any increasing sequence (n _k ) of positive integers satisfying the Hadamard gap condition n _k+1/n _k  > q > 1, k ≥ 1, the discrepancy D _N of (n _k x) mod 1 satisfies the law of the iterated logarithm $$ 1/4 \leq {\mathop {\rm lim\,sup} \limits _{N\to\infty}}\, N D_N(n_k x) (N \log \log N)^{-1/2}\leq C_q\quad \textup{a.e.}$$ Recently, Fukuyama computed the value of the lim sup for sequences of the form n _k = θ ^k , θ > 1, and in a preceding paper the author gave a Diophantine condition on (n _k ) for the value of the limsup to be equal to 1/2, the value obtained in the case of i.i.d. sequences. In this paper we utilize this number-theoretic connection to construct a lacunary sequence (n _k ) for which the lim sup in the LIL for the star-discrepancy ${D_N^*}$ is not a constant a.e. and is not equal to the lim sup in the LIL for the discrepancy D _N .     

The absolute convergence of lacunary series

A theorem is proved from which it follows that there exists a complete U-set E and a number p such that: a) if the p-lacunary trigonometric series $$\sum\nolimits_{k = 1}^\infty {a_k \sin (n_k x + \varepsilon _k ),} \frac{{\lim }}{{k \to \infty }}n_{k + 1} /n_k > p,$$ converges on E, the series of the moduli of its coefficients converges; b) if the sum of the p-lacunary trigonometric series is differentiable on E, it is continuously differentiable everywhere.     

A class of lacunary trigonometric series

It is shown that there exists a sequence of natural numbers {n_k} which does not belong to the class B₂ and which cannot be decomposed into a finite number of lacunary sequences such that: a) if the series converges on a set of positive measure, then the series consisting of the squares of the coefficients converges; b) for each set E of positive measure we can remove from the system a finite number of terms with the result that what is left is a Bessel system in L²(E); and c) if the series converges to zero on a set of positive measure, then each coefficient is zero.Translated from Matematicheskie Zametki, Vol. 14, No. 6, pp. 781–788, December, 1973.In conclusion the author wishes to thank V. F. Emel'yanov for posing the problem and for helping to solve it.     

On convergence of ultraspherical lacunary series

Summary Kolmogoroff's classical result on the convergence of lacunary Fourier trigonometric series corresponding to a function of L² class has been extended to the convergence of the Fourier Ultraspherical series possessing lacunae similar to those supposed in Kolmogoroff's theorem for the trigonometric series.     

Irregular discrepancy behavior of lacunary series II

In 1975 Philipp proved the following law of the iterated logarithm (LIL) for the discrepancy of lacunary series: let (n _k ) _{k ≥ 1} be a lacunary sequence of positive integers, i.e. a sequence satisfying the Hadamard gap condition n _k+1/n _k > q > 1. Then ${1/(4 \sqrt{2}) \leq \limsup_{N \to \infty} N D_N(n_k x) (2 N \log \log N)^{-1/2}\leq C_q}$ for almost all ${x \in (0,1)}$ in the sense of Lebesgue measure. The same result holds, if the “extremal discrepancy” D _N is replaced by the “star discrepancy” ${D_N^*}$ . It has been a long standing open problem whether the value of the limsup in the LIL has to be a constant almost everywhere or not. In a preceding paper we constructed a lacunary sequence of integers, for which the value of the limsup in the LIL for the star discrepancy is not a constant a.e. Now, using a refined version of our methods from this preceding paper, we finally construct a sequence for which also the value of the limsup in the LIL for the extremal discrepancy is not a constant a.e.     

Irregular lil behaviour of lacunary trigonometric series

Research supported by Hungarian National Foundation for Scientific Research, grant no. 1808.     

The central limit theorem for lacunary series

In this paper, the central limit theorem for lacunary trigonometric series is proved. Two gap conditions by Erdos and Takahashi are extended and unified. The criterion for the Fourier character of lacunary series is also given.

     

BMO estimates for lacunary series

We proveBMO andL ^p norm inequalities inR ⁿ for lacunary Walsh and generalized trigonometric series.     

Uniqueness theorems for multiple lacunary trigonometric series

Translated from Matematicheskie Zametki, Vol. 51, No. 6, pp. 27–31, June, 1992.     

Uniqueness theorems for multiple lacunary trigonometric series

In a many-dimensional space, we study some properties of functions with lacunary Fourier series depending only on the values of these functions in a neighborhood of a certain point. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 11, pp. 1477–1481, November, 1998.