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E-raamat: Theory of Distributions

  • Formaat: PDF+DRM
  • Ilmumisaeg: 13-Jul-2015
  • Kirjastus: Springer International Publishing AG
  • Keel: eng
  • ISBN-13: 9783319195278
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Theory of DistributionsSuurem pilt
  • Formaat: PDF+DRM
  • Ilmumisaeg: 13-Jul-2015
  • Kirjastus: Springer International Publishing AG
  • Keel: eng
  • ISBN-13: 9783319195278
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This book explains many fundamental ideas on the theory of distributions. The theory of partial differential equations is one of the synthetic branches of analysis that combines ideas and methods from different fields of mathematics, ranging from functional analysis and harmonic analysis to differential geometry and topology. This presents specific difficulties to those studying this field. This book, which consists of 10 chapters, is suitable for upper undergraduate/graduate students and mathematicians seeking an accessible introduction to some aspects of the theory of distributions. It can also be used for one-semester course.

Arvustused

Uniquely, this work by Georgiev (Univ. of Sofia, Bulgaria) reads stylistically like a brief, basic, but exotic calculus text, replete with concrete sample calculations. Many researchers found distributions quite alien when they made the scene 65 years ago, but with this book, todays mathematics and physics undergraduate students may well come to regard them as quite routine. Summing Up: Recommended. Upper-division undergraduates through professionals/practitioners. (D. V. Feldman, Choice, Vol. 53 (10), June, 2016)

1 Introduction
1(26)
1.1 The Spaces l∞0 and y
1(4)
1.2 Convolution of Locally Integrable Functions
5(3)
1.3 Cones in Rn
8(2)
1.4 Exercises
10(17)
2 Generalities on Distributions
27(38)
2.1 Definition
27(4)
2.2 Order of a Distribution
31(1)
2.3 Sequences
32(2)
2.4 Support
34(2)
2.5 Singular Support
36(2)
2.6 Measures
38(1)
2.7 Multiplying Distributions by l∞Functions
39(1)
2.8 Exercises
40(25)
3 Differentiation
65(22)
3.1 Derivatives
65(3)
3.2 The Primitive of a Distribution
68(3)
3.3 Double Layers on Surfaces
71(1)
3.4 Exercises
71(16)
4 Homogeneous Distributions
87(12)
4.1 Definition and Properties
87(1)
4.2 Exercises
88(11)
5 Direct Product of Distributions
99(10)
5.1 Definition
99(2)
5.2 Properties
101(3)
5.3 Exercises
104(5)
6 Convolutions
109(42)
6.1 Definition
109(2)
6.2 Properties
111(2)
6.3 Existence
113(2)
6.4 The Convolution Algebras D(Γ +) and D(Γ)
115(1)
6.5 Regularization of Distributions
116(1)
6.6 Fractional Differentiation and Integration
117(4)
6.7 Exercises
121(30)
7 Tempered Distributions
151(10)
7.1 Definition
151(2)
7.2 Direct Product
153(1)
7.3 Convolution
154(2)
7.4 Exercises
156(5)
8 Integral Transforms
161(18)
8.1 Fourier Transform in y(Rn)
161(1)
8.2 Fourier Transform in y(Rn)
162(2)
8.3 Properties of the Fourier Transform in y(Rn)
164(1)
8.4 Fourier Transform of Distributions with Compact Support
165(1)
8.5 Fourier Transform of Convolutions
166(1)
8.6 Laplace Transform
167(3)
8.6.1 Definition
167(1)
8.6.2 Properties
168(2)
8.7 Exercises
170(9)
9 Fundamental Solutions
179(8)
9.1 Definition and Properties
179(3)
9.2 Exercises
182(5)
10 Sobolev Spaces
187(28)
10.1 Definitions
187(1)
10.2 Elementary Properties
188(3)
10.3 Approximation by Smooth Functions
191(5)
10.4 Extensions
196(3)
10.5 Traces
199(2)
10.6 Sobolev Inequalities
201(9)
10.7 The Space H--s
210(1)
10.8 Exercises
211(4)
References 215(2)
Index 217
Prof. Svetlin G. Georgiev, Assistant Professor, Department of Differential Equations, Faculty of Mathematics and Informatics, University of Sofia "St. Kliment Ohridski", Sofia, Bulgaria.
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