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Teor. Veroyatnost. i Primenen., 1975, Volume 20, Issue 3, Pages 664–667 (Mi tvp3328)  

Short Communications

Some properties of lacunary series and Gaussian measures connected with uniform versions of properties of Egoroff and Lusin

B. S. Tsirel'son

Leningrad

Abstract: Let $A$ be a measurable subset of $[0,1]$ and $\operatorname{mes}A>0$. For any function $f$ satisfying
\begin{gather*} f(t)=\sum(a_k\cos\lambda_kt+b_k\sin\lambda_kt),\quad\lambda_1,\lambda_2,…>0,\quad\inf(\lambda_{k+1}/\lambda_k)>1,
\sum(a_k^2+b_k^2)<\infty\quadand |f(t)|\le1\quada.e. on A, \end{gather*}
we can find a sequence of sets $B_1\subset B_2\subset…\subset[0,1]$, $\operatorname{mes}B_n\to1$, and a function $F\in L_1[0,1]$ such that $\sum(a_k\cos\lambda_kt+b_k\sin\lambda_kt)$ converges uniformly on every $B_n$ and $|f(t)|\le F(t)$ a.e. on $[0,1]$. The sequence $\{B_n\}$ and the function $F$ depends on $\{\lambda_k\}$, $A$ only. The function $F$ may be chosen in such a way that $\int_0^1\exp(\alpha F(t)) dt<+\infty$ for some positive $\alpha$. It is interesting to observe an analogy between this theorem and similar results about Gaussian random variables.

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English version:
Theory of Probability and its Applications, 1976, 20:3, 652–655

Bibliographic databases:

Received: 13.03.1975

Citation: B. S. Tsirel'son, “Some properties of lacunary series and Gaussian measures connected with uniform versions of properties of Egoroff and Lusin”, Teor. Veroyatnost. i Primenen., 20:3 (1975), 664–667; Theory Probab. Appl., 20:3 (1976), 652–655

Citation in format AMSBIB
\Bibitem{Tsi75}
\by B.~S.~Tsirel'son
\paper Some properties of lacunary series and Gaussian measures connected with uniform versions of properties of Egoroff and Lusin
\jour Teor. Veroyatnost. i Primenen.
\yr 1975
\vol 20
\issue 3
\pages 664--667
\mathnet{http://mi.mathnet.ru/tvp3328}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=377402}
\zmath{https://zbmath.org/?q=an:0345.60023}
\transl
\jour Theory Probab. Appl.
\yr 1976
\vol 20
\issue 3
\pages 652--655
\crossref{https://doi.org/10.1137/1120074}


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