Distribution theory and applications / Abdellah El Kinani, Mohamed Oudadess
- Author
- Kinani, Abdellah El.
- Published
- Hackensack, NJ : World Scientific, [2010]
- Copyright Date
- ©2010
- Physical Description
- xvi, 202 pages ; 24 cm.
- Additional Creators
- Oudadess, Mohamed
- Series
- Contents
- 1. Spaces of Functions -- 1.1. The space C (Ω) -- 1.2. The space Em (Ω) -- 1.3. The space E(Ω) -- 1.4. The space DmK(Ω) -- 1.5. The space DK (Ω) -- 1.6. The space Dm(Ω) -- 1.7. The Topology of Dm(Ω) -- 1.8. Density results -- 2. Distributions -- 2.1. Definitions and characterizations -- 2.2. Examples -- 2.3. Embedding of Lðloc(Ω) in D'(Ω) -- 2.4. Support of a distribution -- 2.5. Pasting principle -- 2.6. Distributions with compact support -- 2.7. Operations on distributions -- 2.8. Topology in D'(Ω) -- 2.9. Continuity of differential operators -- 3. Tensor Product -- 3.1. Tensor product of functions -- 3.2. Tensor product of distributions -- 3.3. Properties of tensor product -- 4. Convolution of Distributions -- 4.1. Regular distributions case -- 4.2. Convolution of distributions -- 4.3. Properties of the convolution product -- 4.4. Convolution algebras -- 4.5. Convolution equations -- 5. Fourier Transformation in Lð (Rn) -- 5.1. Definitions and first properties -- 5.2. Inversion Theorem -- 6. Tempered Distributions -- 6.1. The space S(Rn) -- 6.2. Multiplication operators -- 6.3. Fourier transformation in S(Rn) -- 6.4. Tempered distributions -- 6.5. Convolution operators -- 6.6. Fourier transformation of tempered distributions -- 6.7. Ultradistributions -- 7. Structure of Distributions -- 7.1. Local structure -- 7.2. Global structure -- 7.3. Structure of tempered distributions -- 8. Fourier Ranges of Some Subspaces of S'(Rn) -- 8.1. Fourier range of L(̥Rn) -- 8.2. Fourier range of Lð(Rn) -- 8.3. Fourier range of OM (Rn) -- 8.4. Fourier range of D(Rn) -- 8.5. Fourier range of E'(Rn) -- 9. Laplace Transformation -- 9.1. Laplace transforms of functions of real variables -- 9.2. Laplace transforms of distributions -- 10. Introduction to the Theory of Kernels -- 10.1. The notion of a kernel -- 10.2. Regular kernels -- 10.3. Regularizing kernels -- 10.4. Very regular kernels -- 10.5. Fundamental kernels of partial differential operators -- 10.6. Hypoelliptic operators -- 11. Sobolev Spaces -- 11.1. The spaces Hm(Ω), m element of N -- 11.2. The spaces Hm0(Ω), m element of N -- 11.3. The spaces L(̥μs), s element of R -- 11.4. The spaces Hs (Rn), s element of R -- 11.5. Injection theorem of Sobolev -- 11.6. The compact injection theorem.
- Summary
- "This book is an introductory course to the very important theory of distributions, as well as its applications in the resolution of partial differential equations (PDEs). It begins with a chapter of general interest, on the fundamental spaces (or test function spaces). The book advances and concludes with a chapter on Sobolev spaces, which are known to be very important in the resolution of PDEs." "The very basic properties of distributions are examined in detail. Several formal methods have been first used, without rigorous justifications (Dirac function, principal value of Cauchy, finite parts of Hadamard). They find their natural frame in distribution theory. It is the same for Laplace transformation which is a fundamental tool in symbolic calculations." "A detailed treatment is given to the convolution product for it is a central theme in distribution theory. Another very important instrument, covered in several chapters, is the Fourier transformation which is among the most fundamental tools in different mathematical disciplines, and also in physics. Convolution algebras, which are sufficient for the treatment of classical PDEs, are used in various applications." "The general frame for the resolution of PDEs is the theory of kernels the first elements of which are sufficient to show the practicality of distribution theory in applications." "Comments are provided to clarify the settings and sustain calculations. This book may be used by mathematicians, physicists, engineers and graduate students."--BOOK JACKET.
- Subject(s)
- ISBN
- 9789814304917 (hbk.)
9814304913 (hbk.) - Bibliography Note
- Includes bibliographical references and index.
- Source of Acquisition
- Physical and Mathematical Sciences copy: Purchased with funds from the Paterno Libraries Endowment; 2010.
- Endowment Note
- Paterno Libraries Endowment
View MARC record | catkey: 6274985