## Random Fourier series with applications to harmonic analysis.(English)Zbl 0474.43004

Annals of Mathematics Studies, 101. Princeton, New Jersey: Princeton University Press and University of Tokyo Press. V, 150 p. hbk: $23.00; pbk:$ 9.25 (1981).
This work investigates the a.s. uniform convergence of random Fourier series on locally compact groups. In order to get a flavour of the results, we consider in this review the Abelian case. Let $$G$$ be a locally compact Abelian group, $$\Gamma$$ denote the characters of $$G$$, $$K\subset G$$ be a compact symmetric neighborhood of $$0$$ and let $$A\subset\Gamma$$ be countable. Let $$\{\xi_\gamma\}_{\gamma\in A}$$ be a Rademacher sequence, $$\{g_\gamma\}_{\gamma\in A}$$ a sequence of independent $$N(0,1)$$ random variables and $$\{\xi_\gamma\}$$ complex valued random variables satisfying $$\sup_{\gamma\in A}E|\xi_\gamma|^2<\infty$$, $$\inf_{\gamma\in A} E |\xi_\gamma|> 0$$. For a sequence of complex numbers $$\{\delta_\gamma\}$$ in $$\ell^2(A)$$, let $$\gamma\in A$$
$Z(x)=\sum_{\gamma\in A}\delta_\gamma\varepsilon_\gamma\xi_\gamma(x),\quad x\in K.\tag{$$*$$}$
The a.s. uniform convergence of this series is controlled by the pseudo metric (on $$K\oplus K$$), $$\sigma(x-y)=(\sum_{\gamma\in A}|\delta_\gamma|^2|\gamma(x)-\gamma(y)|^2)^{1/2}$$. Consider $$\sigma$$ as a function on $$K$$, let $$\tilde{\sigma}$$ denote the nondecreasing rearrangement of $$\sigma$$ and let
$I(\sigma)=\int_0^{\mu_n}\frac{\tilde{\sigma}(s)}{s\left(\log\frac{4\mu}{s}\right)^{1/2}}\,ds,$
where $$\mu$$ is the Haar measure an $$G$$ and $$\mu_n= \mu(\bigoplus_n K)$$. Then, if $$I(\sigma) < \infty$$, the series ($$*$$) converges uniformly a.s., and
$(E\|Z\|^2_\infty)^{1/2}\leq C\left(\sup_{\gamma\in A} E|\xi_\gamma|^2\right)^{1/2}(\|\{\delta_\gamma\}\|_{\ell^2(A)}+I(\sigma))$
where $$C$$ is a constant independent of $$\{\delta_\gamma\}$$ and $$\{\gamma\mid\gamma\in A\}$$. There is also a converse to this result. This material is developed in Chapter 3. The next chapter deals with ($$*$$) when viewed as a $$C(K)$$ random variable. It is shown that, if $$I(\sigma)<\infty$$, then $$Z$$ satisfies the central limit theorem and the law of the iterated logarithm. Chapter 5 deals with non-commutative versions of the above results. Chapter 6 contains several applications of random Fourier series to harmonic analysis. Let $$G$$ be a compact group, and let $$C_{\mathrm{a.s.}}(G)=\{f\in L_2(G)\,:\,\text{random Fourier series of }f\text{ is a. s. continuous}\}$$. One of the main results is that
$C_{\mathrm{a.s.}}(G)=\{f=\sum_{n=1}^\infty h_n*k_n,\;h_n\in L_2(G),\;k_n\in D\sqrt{\log L}(G),\;\sum_{n=1}^\infty\|h_n\|_2\|k_n\|<\infty\}.$
In the last chapter, the authors return to classical grounds and provide new proofs of some of the classical results of Paley-Zygmund-Salem-Kahane which motivated their work. To conclude, let me quote the authors: “We hope that these results will be of interest to probabilists who can view random Fourier series as interesting and useful examples of stochastic processes – intimately related to stationary Gaussian processes – and to harmonic analysts who may find these results and methods useful in their own work.”
Reviewer: M. Milman

### MSC:

 43A50 Convergence of Fourier series and of inverse transforms 60F05 Central limit and other weak theorems 60G15 Gaussian processes 43-02 Research exposition (monographs, survey articles) pertaining to abstract harmonic analysis

### Keywords:

random Fourier series
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