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Viser: The Analysis of Linear Partial Differential Operators I - Distribution Theory and Fourier Analysis
The Analysis of Linear Partial Differential Operators I
Distribution Theory and Fourier Analysis
Lars Hörmander
(2003)
Sprog: Engelsk
Detaljer om varen
- 2. Udgave
- Paperback
- Udgiver: Springer London, Limited (Juli 2003)
- ISBN: 9783540006626
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.