| Notions of Convexity Contributor(s): Hörmander, Lars (Author) |
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ISBN: 0817645845 ISBN-13: 9780817645847 Publisher: Birkhauser OUR PRICE: $132.99 Product Type: Paperback - Other Formats Published: December 2006 Annotation: The first two chapters of this book are devoted to convexity in the classical sense, for functions of one and several real variables respectively. This gives a background for the study in the following chapters of related notions which occur in the theory of linear partial differential equations and complex analysis such as (pluri-)subharmonic functions, pseudoconvex sets, and sets which are convex for supports or singular supports with respect to a differential operator. In addition, the convexity conditions which are relevant for local or global existence of holomorphic differential equations are discussed, leading up to Tr?preau's theorem on sufficiency of condition (capital Greek letter Psi) for microlocal solvability in the analytic category. At the beginning of the book, no prerequisites are assumed beyond calculus and linear algebra. Later on, basic facts from distribution theory and functional analysis are needed. In a few places, a more extensive background in differential geometry or pseudodifferential calculus is required, but these sections can be bypassed with no loss of continuity. The major part of the book should therefore be accessible to graduate students so that it can serve as an introduction to complex analysis in one and several variables. The last sections, however, are written mainly for readers familiar with microlocal analysis. |
| Additional Information |
| BISAC Categories: - Mathematics | Mathematical Analysis - Mathematics | Differential Equations - General - Mathematics | Functional Analysis |
| Dewey: 515 |
| LCCN: 2006937427 |
| Series: Modern Birkhauser Classics |
| Physical Information: 0.79" H x 6.22" W x 9.22" (1.35 lbs) 416 pages |
| Descriptions, Reviews, Etc. |
| Publisher Description: The term convexity used to describe these lectures given at the Univer- sity of Lund in 1991-92 should be understood in a wide sense. Only Chap- ters I and II are devoted to convex sets and functions in the traditional sense of convexity. The following chapters study other kinds of convexity which occur in analysis. Most prominent is the pseudo-convexity (plurisubh- monicity) in the theory of functions of several complex variables discussed in Chapter IV. It relies on the theory of subharmonic functions in R , so Chapter III is devoted to subharmonic functions in R" for any n. Existence theorems for constant coefficient partial differential operators in R' are re- lated to various kinds of convexity conditions, depending on the operator. Chapter VI gives a survey of the rather incomplete results which are known on their geometrical meaning. There are also natural classes of "convex" functions related to subgroups of the linear group, which specialize to sev- eral of the notions already mentioned. They are discussed in Chapter V. The last chapter. Chapter VII, is devoted to the conditions for solvability of microdifferential equations, which can also be considered as a branch of convexity theory. The whole chapter is an exposition of a part of the thesis of J.-M. Trepreau. |