Convergence of averages of point transformations
Convergence of averages of point transformations
复制标题
点变换平均值的收敛
DOI：
10.1090/s0002-9939-1975-0360999-4
复制
发表时间：
1975
期刊：
影响因子：
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通讯作者：
'aNDA. Del Junco
中科院分区：
文献类型：
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作者：
M. Akcoglu;'aNDA. Del Junco
收敛 a.e.对于每个 E1 = L 1(X, 5f, [t)。这就提出了以下问题。矩阵 (ani) 上要使序列 fn(x) = ,ianif/(rZ x) 收敛 a.e 的必要条件是什么？对于每个 / E LE 和对于每个自同构 r？答案不得而知。然而，光谱方面的考虑表明了以下猜想。如果 (ani) 使得函数序列 pn(z) = X iani z 均匀有界且逐点收敛于单位圆 lZj = 1，则 /n 收敛 a.e。事实上，最近有人试图证明这一点是一个定理[1]。在这篇文章中，我们想观察以下简单的事实，它表明这个猜想远非正确。如果r是实数，则令[r]表示小于或等于r的最大整数。定义一个矩阵（ani），n= 1, 2,*，为
converges a.e. for each E1 = L 1(X, 5f, [t). This raises the following question. What are the necessary conditions on the matrix (ani) so that the sequence fn(x) = ,ianif/(rZ x) converges a.e. for each / E LE and for each automorphism r? The answer is not known. Spectral considerations would suggest, however, the following conjecture. If (ani) is such that the sequence of functions pn(z) = X iani z is uniformly bounded and pointwise convergent on the unit circle lZj = 1, then /n converges a.e. In fact, recently an attempt has been made to prove this as a theorem [1]. In this note we would like to observe the following simple fact which shows that this conjecture is far from being correct. If r is a real number, let [r] denote the greatest integer which is less than or equal to r. Define a matrix (ani), n= 1, 2,*, as