A generalisation of Pisier homogeneous Banach algebra

TL;DR

This paper generalizes Pisier's homogeneous Banach algebra by showing that any probability measure on the unit circle defines such an algebra, expanding the class beyond the original Gaussian-based construction.

Contribution

It extends Pisier's result by demonstrating that all probability measures on the circle generate Pisier-type algebras, not just Gaussian measures, and provides conditions for boundedness of related random Fourier series.

Findings

Any probability measure on the circle defines a Pisier-type algebra.

The paper provides a sufficient condition for the boundedness of random Fourier series with dependent variables.

Pisier's algebra is part of a larger class of similar algebras.

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 $\mathscr{P}$ 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 $\sum_{n\in \mathbb{Z}}\hat f(n) \,\xi_n \exp(2\pi i n t)$ in the general setting of dependent random variables $(\xi_n)$.