The Analysis of Linear Partial Differential Operators I
Distribution Theory and Fourier Analysis
2nd Edition
Published on 24. August 1990
Book
Hardback
XI, 440 pages
978-3-540-52345-1 (ISBN)
Description
The main change in this edition is the inclusion of exercises with answers and hints. This is meant to emphasize that this volume has been written as a general course in modern analysis on a graduate student level and not only as the beginning of a specialized course in partial differen tial equations. In particular, it could also serve as an introduction to harmonic analysis. Exercises are given primarily to the sections of gen eral interest; there are none to the last two chapters. Most of the exercises are just routine problems meant to give some familiarity with standard use of the tools introduced in the text. Others are extensions of the theory presented there. As a rule rather complete though brief solutions are then given in the answers and hints. To a large extent the exercises have been taken over from courses or examinations given by Anders Melin or myself at the University of Lund. I am grateful to Anders Melin for letting me use the problems originating from him and for numerous valuable comments on this collection. As in the revised printing of Volume II, a number of minor flaws have also been corrected in this edition. Many of these have been called to my attention by the Russian translators of the first edition, and I wish to thank them for our excellent collaboration.
More details
Series
Edition
2nd ed. 2003
Language
English
Place of publication
Heidelberg
Germany
Publishing group
Springer Berlin
Target group
College/higher education
Professional and scholarly
Research
Edition type
Revised edition
Illustrations
biography
Dimensions
Height: 23.5 cm
Width: 15.5 cm
Weight
800 gr
ISBN-13
978-3-540-52345-1 (9783540523451)
DOI
10.1007/978-3-642-61497-2
Schweitzer Classification
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Lars Hörmander
The Analysis of Linear Partial Differential Operators I
Distribution Theory and Fourier Analysis
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2nd Edition
Springer
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Lars Hörmander
The Analysis of Linear Partial Differential Operators I
Distribution Theory and Fourier Analysis
Book
07/2003
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Springer
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The Analysis of Linear Partial Differential Operators I
Distribution Theory and Fourier Analysis
Book
08/1990
2nd Edition
Springer
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Previous edition
L. Hörmander
The Analysis of Linear Partial Differential Operators I
Distribution Theory and Fourier Analysis
Book
05/1983
Springer
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Content
I. Test Functions.- Summary.- 1.1. A review of Differential Calculus.- 1.2. Existence of Test Functions.- 1.3. Convolution.- 1.4. Cutoff Functions and Partitions of Unity.- Notes.- II. Definition and Basic Properties of Distributions.- Summary.- 2.1. Basic Definitions.- 2.2. Localization.- 2.3. Distributions with Compact Support.- Notes.- III. Differentiation and Multiplication by Functions.- Summary.- 3.1. Definition and Examples.- 3.2. Homogeneous Distributions.- 3.3. Some Fundamental Solutions.- 3.4. Evaluation of Some Integrals.- Notes.- IV. Convolution.- Summary.- 4.1. Convolution with a Smooth Function.- 4.2. Convolution of Distributions.- 4.3. The Theorem of Supports.- 4.4. The Role of Fundamental Solutions.- 4.5. Basic Lp Estimates for Convolutions.- Notes.- V. Distributions in Product Spaces.- Summary.- 5.1. Tensor Products.- 5.2. The Kernel Theorem.- Notes.- VI. Composition with Smooth Maps.- Summary.- 6.1. Definitions.- 6.2. Some Fundamental Solutions.- 6.3. Distributions on a Manifold.- 6.4. The Tangent and Cotangent Bundles.- Notes.- VII. The Fourier Transformation.- Summary.- 7.1. The Fourier Transformation in ? and in ?'.- 7.2. Poisson's Summation Formula and Periodic Distributions.- 7.3. The Fourier-Laplace Transformation in ?'.- 7.4. More General Fourier-Laplace Transforms.- 7.5. The Malgrange Preparation Theorem.- 7.6. Fourier Transforms of Gaussian Functions.- 7.7. The Method of Stationary Phase.- 7.8. Oscillatory Integrals.- 7.9. H(s), Lp and Hölder Estimates.- Notes.- VIII. Spectral Analysis of Singularities.- Summary.- 8.1. The Wave Front Set.- 8.2. A Review of Operations with Distributions.- 8.3. The Wave Front Set of Solutions of Partial Differential Equations.- 8.4. The Wave Front Set with Respect to CL.- 8.5. Rules of Computation for WFL.- 8.6. WFL for Solutions of Partial Differential Equations.- 8.7. Microhyperbolicity.- Notes.- IX. Hyperfunctions.- Summary.- 9.1. Analytic Functionals.- 9.2. General Hyperfunctions.- 9.3. The Analytic Wave Front Set of a Hyperfunction.- 9.4. The Analytic Cauchy Problem.- 9.5. Hyperfunction Solutions of Partial Differential Equations.- 9.6. The Analytic Wave Front Set and the Support.- Notes.- Exercises.- Answers and Hints to All the Exercises.- Index of Notation.