续$\mathbf{A_2}$-分数和奇异函数

Q3 Mathematics
M. Pratsiovytyi, Y. Goncharenko, I. Lysenko, S. Ratushniak
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引用次数: 2

摘要

本文用两元字母$A_2=\{\frac12,1\}$, $a_n\in A_2$深化了无限$A_2$ -连分数$[0;a_1,a_2,...,a_n,...]$理论的度量分量,并根据它们的$A_2$ -表示:$x=[0;a_1,a_2,...,a_n,...]$建立了区段$I=[\frac12;1]$的数的正规性质。证明了几乎所有(在勒贝格测度的意义上)段$I$的数在它们的$A_2$ -表示中无限多次地使用任意长度的字母表元素的每一个元组作为该表示的连续数字。这个数字的正常性质被有效地用来证明函数$f(x=[0;a_1,a_2,...,a_n,...])=e^{\sum\limits_{n=1}^{\infty}(2a_n-1)v_n},$的奇异性,其中$v_1+v_2+...+v_n+...$是一个给定的绝对收敛级数,当函数$f$是连续的(这是只在$v_n=\frac{v_1(-1)^{n-1}}{2^{n-1}}$, $v_1\in R$的情况下)。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Continued $\mathbf{A_2}$-fractions and singular functions
In the article we deepen the metric component of theory of infinite $A_2$-continued fractions $[0;a_1,a_2,...,a_n,...]$ with a two-element alphabet $A_2=\{\frac12,1\}$, $a_n\in A_2$ and establish the normal property of numbers of the segment $I=[\frac12;1]$ in terms of their $A_2$-representations: $x=[0;a_1,a_2,...,a_n,...]$. It is proved that almost all (in the sense of the Lebesgue measure) numbers of segment $I$ in their $A_2$-representations use each of the tuples of elements of the alphabet of arbitrary length as consecutive digits of the representation infinitely many times. This normal property of the number is effectively used to prove the singularity of the function $f(x=[0;a_1,a_2,...,a_n,...])=e^{\sum\limits_{n=1}^{\infty}(2a_n-1)v_n},$where $v_1+v_2+...+v_n+...$ is a given absolutely convergent series, when function $f$ is continuous (which is the case only if $v_n=\frac{v_1(-1)^{n-1}}{2^{n-1}}$, $v_1\in R$).
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来源期刊
Matematychni Studii
Matematychni Studii Mathematics-Mathematics (all)
CiteScore
1.00
自引率
0.00%
发文量
38
期刊介绍: Journal is devoted to research in all fields of mathematics.
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