# LACUNARY SERIES AND INDEPENDENT FUNCTIONS

```@article{Gaposhkin1966LACUNARYSA,
title={LACUNARY SERIES AND INDEPENDENT FUNCTIONS},
author={V F Gaposhkin},
journal={Russian Mathematical Surveys},
year={1966},
volume={21},
pages={1 - 82}
}```
• V F Gaposhkin
• Published 1966
• Mathematics, Philosophy
• Russian Mathematical Surveys
CONTENTS Introduction Chapter I. Lacunary subsystems of general systems of functions § 1. Definitions and notation § 2. Convergence § 3. Integrability § 4. Absolute convergence § 5. The central limit theorem § 6. The law of the iterated logarithm Chapter II. Lacunary subsystems of specific systems of functions § 1. The trigonometric system § 2. The Walsh system § 3. Orthogonal polynomials § 4. Systems of the form References

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CONTENTS § 1. Introduction § 2. Fourier coefficients § 3. Convergence § 4. Convergence to 0 and to +∞ § 5. Divergent series § 6. Summability § 7. The problem of the representation of functions § 8.

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1. Introduction. 1.1. The theory of lacunary Fourier series, developed by Banach, Kolmogorov, Sidon, Zygmund, and others, deals largely with extraordinary properties that such series possess. The

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Introduction. The existence of a pair of continuous functions F(t), G(t), such that the curve x F(t), y G(t) fills completely a certain square is classical. Such a curve will, as usual, be called a

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Every function f(x) which is of period 1 and Lebesgue integrable on [0, 1 ] may be expanded in a Walsh-Fourier series(3), f(x)~ ?.?=n ak\pk(x), where ak=fof(x)ypk(x)dx, k=0, 1, 2, • • • . Fine

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The purpose of this paper is to sharpen Kolmogoroff’s celebrated law of the interated logarithm in various directions and to give best results. For the convenience of the reader, we shall link the