Lectures on Hermite and Laguerre expansions. (English) Zbl 0791.41030

Mathematical Notes (Princeton). 42. Princeton, NJ: Princeton University Press. xv, 195 p. (1993).
This concise and well written monograph is based on the author’s lecture notes for a graduate course at Cornell. It gives the theory of Laguerre and Hermite expansions considered from the viewpoint of group representations, Weyl transforms and Heisenberg groups. These topics are introduced in the first chapter. The second chapter gives the \(L^ p\) mapping properties of the Riesz means and multipliers for special Hermite expansion. Chapter 3 studies the almost everywhere and mean convergence of Hermite expansions for functions in \(L^ p(\mathbb{R}^ n)\), \(n\geq 2\). Chapter 4 proves a Marcinkiewicz type multiplier theorem for the Hermite expansions. Chapter 5 gives the almost everywhere and mean convergence of the CesĂ ro means for the Hermite series on the real line. Chapter 6 treats three types of Laguerre expansions and Chapter 7 proves two transplantation theorems used in Chapter 6.


41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
41-02 Research exposition (monographs, survey articles) pertaining to approximations and expansions