Analysis in integer and fractional dimensions.

*(English)*Zbl 1006.46001
Cambridge Studies in Advanced Mathematics. 71. Cambridge: Cambridge University Press. xvii, 556 p. (2001).

The book is a mixture of harmonic analysis, functional analysis, and probability theory centered around the Grothendieck inequality and Grothendieck’s factorization theorem for bounded bilinear functionals on \(C_0(X)\). It tries to be self-contained apart from backgrounds in measure theory, some probability theory, and some functional and Fourier analysis. The contents of the 14 chapters are as follows:

In the first chapter a mostly historical prolog is given starting from Riesz’s Representation theorem and its bi- and multi-linear pendants and their connections to Grothendieck’s inequality, Fourier analysis (Sidon sets) and fractional dimension. The second chapter is on the (equivalent) inequalities of Khintchin (i.e. the equivalence of the \(L^1\) and \(L^2\) norms on the span of the Rademacher functions), of Littlewood, and of Orlicz (i.e. the dominance of the Fréchet variation over the mixed \(\ell^1\)-\(\ell^2\)-norm). The third chapter gives three proofs of the Grothendieck inequality, all based on Khintchin’s inequality. In the next chapter the Fréchet variation is identified as the \(\ell^1\) injective tensor norm and as the supremum norm of a function with spectrum in a \(k\)-fold product of Sidon sets. Chapter 5 is on Grothendieck’s Factorization Theorem.

In the next two chapters the rudiments of multidimensional measure theory are outlined and harmonic analysis on \(\{-1,+1\}^{\mathbb N}\) is introduced. In Chapter 8 multi-linear extensions of Grothendieck’s inequality are given using harmonic analysis and projectively bounded multi-linear functionals. In the next chapter product Fréchet measures are linked to Grothendieck’s inequality. Chapter 10 and 11 is on Brownian motion and the Wiener process and the identification of stochastic processes with Fréchet measures and the importance of Grothendieck’s inequality in this setting. The next two chapters are on the 3/2-dimensional Cartesian product and products of general fractional dimension and the relationship to exponents of interdependence in harmonic analysis and probability theory. In the last chapter some applications, open questions and possible future lines of development are given.

Every chapter ends with exercises ranging from routine to unsolved problems. Some hints for solving them are also given.

In the first chapter a mostly historical prolog is given starting from Riesz’s Representation theorem and its bi- and multi-linear pendants and their connections to Grothendieck’s inequality, Fourier analysis (Sidon sets) and fractional dimension. The second chapter is on the (equivalent) inequalities of Khintchin (i.e. the equivalence of the \(L^1\) and \(L^2\) norms on the span of the Rademacher functions), of Littlewood, and of Orlicz (i.e. the dominance of the Fréchet variation over the mixed \(\ell^1\)-\(\ell^2\)-norm). The third chapter gives three proofs of the Grothendieck inequality, all based on Khintchin’s inequality. In the next chapter the Fréchet variation is identified as the \(\ell^1\) injective tensor norm and as the supremum norm of a function with spectrum in a \(k\)-fold product of Sidon sets. Chapter 5 is on Grothendieck’s Factorization Theorem.

In the next two chapters the rudiments of multidimensional measure theory are outlined and harmonic analysis on \(\{-1,+1\}^{\mathbb N}\) is introduced. In Chapter 8 multi-linear extensions of Grothendieck’s inequality are given using harmonic analysis and projectively bounded multi-linear functionals. In the next chapter product Fréchet measures are linked to Grothendieck’s inequality. Chapter 10 and 11 is on Brownian motion and the Wiener process and the identification of stochastic processes with Fréchet measures and the importance of Grothendieck’s inequality in this setting. The next two chapters are on the 3/2-dimensional Cartesian product and products of general fractional dimension and the relationship to exponents of interdependence in harmonic analysis and probability theory. In the last chapter some applications, open questions and possible future lines of development are given.

Every chapter ends with exercises ranging from routine to unsolved problems. Some hints for solving them are also given.

Reviewer: A.Kriegl (Wien)

##### MSC:

46-02 | Research exposition (monographs, survey articles) pertaining to functional analysis |

43-02 | Research exposition (monographs, survey articles) pertaining to abstract harmonic analysis |

26-02 | Research exposition (monographs, survey articles) pertaining to real functions |

42-02 | Research exposition (monographs, survey articles) pertaining to harmonic analysis on Euclidean spaces |

42B35 | Function spaces arising in harmonic analysis |

43A05 | Measures on groups and semigroups, etc. |

43A46 | Special sets (thin sets, Kronecker sets, Helson sets, Ditkin sets, Sidon sets, etc.) |

46N30 | Applications of functional analysis in probability theory and statistics |

60H05 | Stochastic integrals |

60E15 | Inequalities; stochastic orderings |