Pisier齐次巴拿赫代数的推广
A generalisation of Pisier homogeneous Banach algebra
摘要：
在1979年皮西尔证明了显着的一个序列的独立和同分布的标准高斯随机变量确定，通过随机傅立叶级数，齐次巴拿赫代数$\mathscr{P}$严格包含在$C(\mathbb{T})$，一类连续函数的单位圆$\mathbb{T}$和严格包含经典维纳代数$\mathbb{A}(\mathbb{T})$，即，$\mathbb{A}(\mathbb{T}) \subsetneqq \mathscr{P} \subsetneqq C(\mathbb{T}).$这改进了Zafran以前在解决Katznelson提出的一个长期存在的问题时得到的一些结果。本文推广了Pisier的结果，证明了单位圆上的任何概率测度定义了包含在$C(\mathbb{T})$中的齐次巴拿赫代数.因此，Pisier代数不是一个孤立的对象，而是一个大类的Pisier型代数的元素。考虑平稳高斯随机变量序列的谱测度的情形，得到了随机Fourier级数$\sum_{n\in \mathbb{Z}}\hat f(n) \,ξ_n \exp(2πi n t)$在一般相依随机变量$(ξ_n)$下有界的一个充分条件.
英文摘要：
In 1979 Pisier proved remarkably that a sequence of independent and identically distributed standard Gaussian random variables determines, via random Fourier series, a homogeneous Banach algebra $\mathscr{P}$ strictly contained in $C(\mathbb{T})$, the class of continuous functions on the unit circle $\mathbb{T}$ and strictly containing the classical Wiener algebra $\mathbb{A}(\mathbb{T})$, that is, $\mathbb{A}(\mathbb{T}) \subsetneqq \mathscr{P} \subsetneqq C(\mathbb{T}).$ This improved some previous results obtained by Zafran in solving a long-standing problem raised by Katznelson. In this paper we extend Pisier's result by showing that any probability measure on the unit circle defines a homogeneous Banach algebra contained in $C(\mathbb{T})$. Thus Pisier algebra is not an isolated object but rather an element in a large class of Pisier-type algebras. We consider the case of spectral measures of stationary sequences of Gaussian random variables and obtain a sufficient condition for the boundedness of the random Fourier series $\sum_{n\in \mathbb{Z}}\hat f(n) \,ξ_n \exp(2πi n t)$ in the general setting of dependent random variables $(ξ_n)$.
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