Distributions and Fourier Transforms. Part 1 (Goncharova O.)

Distributions and Fourier Transforms. Part 1

Автор(ы):Goncharova O.
06.10.2007
Год изд.:2002
Описание: The theory of distrubutions is usually presented in close connection with functional analysis, and the prehistory of that theory adds a very important element to the history of functional analysis. The problems or theories which shaped the prehistory of the theory of distributions are: 1. Heaviside's operational calculus; 2. Generalized derivatives and generalized solutions to differential equations; 3. Generalized Fourier transforms; 4. Improper functions; the a-functions and the partie finie; 5. The Rhams's currents. Why is the theory of distributions important? For what purpose was the theory of distributions originally created? Who invented distributions and when? You could find the answers and new questions in Lecture Notes.
Оглавление:
Distributions and Fourier Transforms. Part 1 — обложка книги. Обложка книги.
1 The Concept of Distributions [5]
  1.1 Introduction [5]
  1.2 Example about thermal diffusion [11]
2 Test functions [14]
  2.1 Some function spaces [14]
  2.2 Properties of (?) [18]
  2.3 Convergence in V (Definition of Topology on V ) [20]
3 Distributions [25]
  3.1 Functional [25]
  3.2 Distributions [26]
  3.3 Examples of Distributions [26]
    3.3.1 Regular Distribution [26]
    3.3.2 Dirac Delta Function [29]
    3.3.3 Heaviside function [31]
    3.3.4 Measures as Distributions [32]
  3.4 Computing with Distributions [32]
    3.4.1 Multiplication by functions [33]
    3.4.2 Substituting the argument: Linear Substitution, Translation, Rotation, Reflection [33]
    3.4.3 The support of a distribution [35]
  3.5 Convergence in (?) [35]
    3.5.1 Example " Approximation of (?)-function" [35]
    3.5.2 Delta-approximating sequence [36]
  3.6 Derivatives in (?) [37]
    3.6.1 Differentiation in 1D. Regular distributions [37]
    3.6.2 Heaviside function [38]
    3.6.3 Signum function [39]
    3.6.4 Example for (?) [39]
    3.6.5 Delta-distribution [40]
    3.6.6 Functions with "corners" [40]
    3.6.7 Gaufi-bracket [40]
  3.7 The Cauchy-principle value [43]
    3.7.1 The function f(t) = 1/t [44]
    3.7.2 The distribution (?) [45]
  3.8 Linear ordinary differential equations [47]
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