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 .     

Irregular lil behaviour of lacunary trigonometric series

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

BMO estimates for lacunary series

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

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.

     

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.     

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.     

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.     

Rate of decay of weakly lacunary series

Let function f(z) ≠ 0 be analytic in the unit disk and have sparse nonzero Taylor coefficients. Then the rate of decay of the function f as x → 1 − 0 depends on the rate of sparseness of its nonzero Taylor coefficients. In this paper, we consider the case f(z) = $ \sum\nolimits_{k = 0}^\infty {a_k z^{n_k } } $ \sum\nolimits_{k = 0}^\infty {a_k z^{n_k } } , where n _k ≥ A ₀(k + 2) ^p logb(k + 2).     

A metric discrepancy result for a lacunary sequence with small gaps

For any ${G(k) \uparrow \infty}$ , there exists a sequence {n _k } of integers with 1 ?? n _k+1 ? n _k ?? G(k) such that the discrepancies of {n _k x} obey the law of the iterated logarithm in the same way as uniform distributed i.i.d.     

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.