Contents
Preface xv
Acknowledgments xvi
Chapter I. Theory of Distributions 1
Introduction to Chapters I, II, III 1
§1. Spaces of infinitely differentiable functions 2
1.1. Properties of the convolution 2
1.2. Approximation by C0
∞-functions
3
1.3. Proof of Proposition 1.1 5
1.4. Proof of property b) of the convolution 5
§2. Definition of a distribution 6
2.1. Examples of distributions 6
2.2. Regular functionals 7
2.3. Distributions in a domain 8
§3. Operations with distributions 9
3.1. Derivative of a distribution 9
3.2. Multiplication of a distribution by a
C∞-function
9
3.3. Change of variables for distributions 10
§4. Convergence of distributions 10
4.1. Delta-like sequences 12
§5. Regularizations of nonintegrable functions 14
5.1. Regularization in
R1
15
5.2. Regularization in
Rn
17
§6. Supports of distributions 20
vii
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