Having defined the basic operators of importance for functions on Gaussian space, it’s useful to also develop the analogue of the Fourier expansion.
Continue reading §11.2: Hermite polynomials


Having defined the basic operators of importance for functions on Gaussian space, it’s useful to also develop the analogue of the Fourier expansion. I’m happy to announce that the book is very nearly completed. In fact, you can preorder a copy now, either directly from Cambridge University Press, or from Amazon (currently with a 10% discount). If all goes well, the book will become physically available at the end of May. Fairly soon thereafter I will also make a pdf “final draft” version freely available here at the website. In the meantime, I will continue to serialize the final chapter (Chapter 11) as usual over the next month. In fact, I had to trim and glue together the planned Chapters 11 and 12. Also, as mentioned earlier, I dropped a chapter on Additive Combinatorics that would have gone between Chapters 8 and 9, although its “highlight”, Sanders’s Theorem, is here on the blog. Oh well, I had to wrap this project up at some point; at least you’ll have an idea of what will be in the 2nd Edition. The final destination of this chapter is a proof of the following theorem due to Mossel, O’Donnell, and Oleszkiewicz [MOO05a,MOO10], first mentioned in Chapter 5.25:
Continue reading Chapter 11: Gaussian space and Invariance Principles In Chapter 8.4 we described the problem of “threshold phenomena” for monotone functions $f : \{1,1\}^n \to \{1,1\}$. In Section 3 we collected a number of consequences of the General Hypercontractivity Theorem for functions $f \in L^2(\Omega^n, \pi^{\otimes n})$. All of these had a dependence on “$\lambda$”, the least probability of an outcome under $\pi$. This can sometimes be quite expensive; for example, the KKL Theorem and its consequence Theorem 28 are trivialized when $\lambda = 1/n^{\Theta(1)}$. In this section we will collect some applications of the General Hypercontractivity Theorem, including generalizations of the facts from Section 9.5. Let’s now study hypercontractivity for general random variables. By the end of this section we will have proved the General Hypercontractivity Theorem stated at the beginning of the chapter. 

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