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Notions of Convexity : Modern Birkhauser Classics - Lars Hormander

Notions of Convexity

Modern Birkhauser Classics

Paperback Published: 1st December 2006
ISBN: 9780817645847
Number Of Pages: 414

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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 Trepreau'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.

Prefacep. iii
Contentsp. v
Convex functions of one variablep. 1
Definitions and basic factsp. 1
Some basic inequalitiesp. 9
Conjugate convex functions (Legendre transforms)p. 16
The [Gamma] function and a difference equationp. 20
Integral representation of convex functionsp. 23
Semi-convex and quasi-convex functionsp. 26
Convexity of the minimum of a one parameter family of functionsp. 28
Convexity in a finite-dimensional vector spacep. 36
Definitions and basic factsp. 36
The Legendre transformationp. 66
Geometric inequalitiesp. 75
Smoothness of convex setsp. 94
Projective convexityp. 98
Convexity in Fourier analysisp. 111
Subharmonic functionsp. 116
Harmonic functionsp. 116
Basic facts on subharmonic functionsp. 141
Harmonic majorants and the Riesz representation formulap. 171
Exceptional setsp. 203
Plurisubharmonic functionsp. 225
Basic factsp. 225
Existence theorems in L[superscript 2] spaces with weightsp. 248
Lelong numbers of plurisubharmonic functionsp. 265
Closed positive currentsp. 271
Exceptional setsp. 285
Other convexity conditionsp. 290
Analytic functionalsp. 300
Convexity with respect to a linear groupp. 315
Smooth functions in the whole spacep. 315
General G subharmonic functionsp. 324
Convexity with respect to differential operatorsp. 328
P-convexityp. 328
An existence theorem in pseudoconvex domainsp. 332
Analytic differential equationsp. 344
Convexity and condition ([psi])p. 353
Local analytic solvability for [Partial]/[Partial]z[subscript 1]p. 353
Generalities on projections and distance functions, and a theorem of Trepreaup. 372
The symplectic point of viewp. 375
The microlocal transformation theoryp. 382
Appendixp. 391
Polynomials and multlinear formsp. 391
Commutator identitiesp. 396
Notesp. 403
Referencesp. 407
Index of notationp. 411
Indexp. 413
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9780817645847
ISBN-10: 0817645845
Series: Modern Birkhauser Classics
Audience: General
Format: Paperback
Language: English
Number Of Pages: 414
Published: 1st December 2006
Country of Publication: US
Dimensions (cm): 23.42 x 15.8  x 2.01
Weight (kg): 0.61

Earn 188 Qantas Points
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