Distributions; General Properties
In functional analysis and its applications to physics, it is desirable to generalize the classical notion of function in the manner suggested by Dirac and carried out by Laurent Schwartz in his theory of “distributions” (called “generalized functions” by the Russian authors). In the presentation given here, in this chapter and the next few, I take the attitude that distribution theory is basically elementary and branches off from the classical development at the level of advanced calculus. The entire presentation is based on the Riemann integral, and the close relation between distributions and ordinary functions is stressed. From this point of view, L 2 spaces and their application to differential operators are also elementary, when based on distribution theory, and are so treated in Chapters 5, 6, 7, 10, and 11.
KeywordsLinear functionals test functions on ℝ and ℝn the bilinear form convergence of test functions continuous functionals real and complex distributions differentiation and integration changes of independent variables convergence of distributions mollifiers regularization of singularities
Unable to display preview. Download preview PDF.