| CHAPTER 1 LINEAR TOPOLOGICAL SPACES 1. Topological spaces and metric spaces 2. Linear topological spaces 3. Countably normed spaces 4. Continuous linear functionals 5. Weak and strong topologies 6. Perfect spaces 7. Linear operators 8. Inductive limits and unions of topological spaces Problems CHAPTER 2 SPACES OF GENERALIZED FUNCTIONS 1. Fundamental spaces and generalized functions 2. The spaces 3. The spaces 4. Multiplication and the derivatives of generalized functions 5. Structure of generalized functions on Problems CHAPTER 3 TI-~ORY OF DISTRIBUTIONS 1. Spaces of functions 2. Partition of unity, 3. Definition and some properties of distributions 4. Derivatives of distributions 5. Structure of distributions 6. Structure of distributions (continued) 7. Distributions having support on compact sets or on subspaces 8. Tensor product of distributions 9. Product of distributions by functions Applications to differential equations 10. Convolutions of distribution 1l. Convolutions of distributions with smooth 5mctions 12. The spaces and the structure of their generalized functions 13. Convolution equations 14. The spaces (S) and (S') 15. Fourier transforms of distributions Problems CHAPTER 4 CONVOLUTIONS AND FOURIER TRANSFORMS OF GENERALIZED FUNCTIONS 1. Fourier transforms of fundamental functions 2. Fourier transforms of generalized functions 3. Convolutions of generMized functions 4. The convolution theorems Problems CHAPTER 5 W SPACES 1. Theorems on complex analytic functions 2. Definition of W spaces 3. Operators in W spaces 4. Fourier transforms of W spaces 5. Nontriviality and richness of W spaces Problems CHAPTER 6 FOURIER TRANSFORMS OF ENTIRE FUNCTION 1. Entire functions of order ≤ and of fast decrease 2. Entire functions of order ≤1,142 3. Entire functions of order ≤1 and of slow increase 4. Entire functions of order ≤p and of slow increase 5. Entire functions of order ≤p and of mildly fast increase 6. Entire functions of order ≤p and of fast increase 7. Proof of Lemma Problems CHAPTER 7 THE CAUCHY PROBLEM FOR SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS 1. Systems of partial differential equations and the Cauchy problem …… CHAPTER 8 THE CAUCHY PROBLEM IN SEVERAL TIME VARIABLES CHAPTER 9 S SPACES CHAPTER 10 FURTHER APPLICATIONS TO PARTIAL DCIFFERENTIAL EQUATIONS CHAPTER 11 DIFFERENTLABILITY OF SOLUTIONS OF PAPTIAL DIFFERENTIAL EQUATIONS BIBLIOGRAPHICAL REMARKS BIBLIOGAPHY INDEX FOR SPACES INWDEX |
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