Generalized functions in mathematical physics. Main ideas and concepts

Notation......Page 5
1. Temperature at a point? No! In volumes contracting to the point......Page 11
2. The notion of -sequence and -function......Page 14
3. Some spaces of smooth functions. Partition of unity......Page 16
4. Examples of -sequences......Page 20
5. On the Laplace equation......Page 21
6. On the heat equation......Page 29
7. The Ostrogradsky--Gauss formula. The Green formulae and the Green function......Page 38
8. The Lebesgue integral1)......Page 41
9. The spaces Lp and Llocp......Page 50
10. Functions of Lloc1 as linear functional on C0......Page 55
11. Simplest hyperbolic equations. Generalized Sobolev solutions......Page 56
12. The space D of the Sobolev derivatives......Page 69
13. The space D# of generalized functions......Page 73
14. The problem of regularization......Page 76
15. Generalized functions with a point support. The Borel theorem......Page 78
16. The space D' of generalized functions (distributions by L. Schwartz)......Page 82
17. The Fourier series and the Fourier transform. The spaces S and S'......Page 91
18. The Fourier--Laplace transform. The Paley--Wiener theorem......Page 104
19. Fundamental solutions. Convolution......Page 109
20. On spaces Hs......Page 112
21. On pseudodifferential operators (PDO)......Page 116
22. On elliptic problems......Page 121
Addendum A new approach to the theory of generalized functions (Yu.V. Egorov)......Page 137
Bibliography......Page 141
Index......Page 147