A Generalization of Pisier Homogeneous Banach Algebra

Abstract

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 $\mathcal{P}$ strictly contained in $\mathit{C}(\mathbb{T})$, the class of continuous functions on the unit circle $\mathbb{T}$ and strictly containing the classical Wiener algebra $\mathbb{A}(\mathbb{T})$, that is, $\mathbb{A}(\mathbb{T})⫋\mathcal{P}⫋\mathit{C}(\mathbb{T})$. 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 $\mathit{C}(\mathbb{T})$. 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 ${\sum }_{\mathit{n}\in \mathbb{Z}}\hat{\mathit{f}}(\mathit{n}){\mathit{\xi }}_{\mathit{n}}exp(2\mathit{\pi }\mathit{i}\mathit{n}\mathit{t})$ in the general setting of dependent random variables $({\mathit{\xi }}_{\mathit{n}})$.