Continued fractions and Fourier transforms
Continued fractions and Fourier transforms
复制标题
连分数和傅立叶变换
DOI：
--
复制
发表时间：
1980
期刊：
影响因子：
--
通讯作者：
R. Kaufman
中科院分区：
文献类型：
--
作者：
R. Kaufman
设FN是真实的数x的集合，其连分式展开式x = [a0; a1，a2，.，an，.]只包含元素ai = 1，2，.，N .这里N ≥ 2。相当大的努力，[1，3]，集中在测量性能的FN和某些措施进行的FN。测度为零的集合的分类，和维数理论一样古老，依赖于傅立叶-斯蒂尔吉斯变换：一个真实的数的闭集合E称为M 0 -集合，如果E带有一个概率测度λ，其变换在无穷远处为零。除了Riemann-Lebesgue引理，E的任何纯度量性质都不能保证E是M 0 -集。然而，对于集合F N，度量性质可以用来构造测度λ。
Let F N be the set of real numbers x whose continued fraction expansion x = [ a 0 ; a 1 , a 2 ,…, a n ,…] contains only elements a i = 1,2,…, N . Here N ≥ 2. Considerable effort, [1,3], has centred on metrical properties of F N and certain measures carried by F N . A classification of sets of measure zero, as venerable as the dimensional theory, depends on Fourier-Stieltjes transforms: a closed set E of real numbers is called an M 0 -set, if E carries a probability measure λ whose transform vanishes at infinity. Aside from the Riemann-Lebesgue Lemma, no purely metrical property of E can ensure that E is an M 0 -set. For the sets F N , however, metrical properties can be used to construct the measure λ.