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.”

\[ 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 |