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Introduction to the Distribution Theory

1.1 Short History

The theory of distributions, or of generalized functions, constitutes a chapter of functional analysis that arose from the need to substantiate, in terms of mathematical concepts, formulae and rules of calculation used in physics, quantum mechanics and operational calculus that could not be justified by classical analysis. Thus, for example, in 1926 the English physicist P.A.M. Dirac [1] introduced in quantum mechanics the symbol d(x), called the Dirac delta function, by the formulae

(1.1)

By this symbol, Dirac mathematically described a material point of mass density equal to the unit, placed at the origin of the coordinate axis.

We notice immediately that d(x), called the impulse function, is a function not in the sense of mathematical analysis, as being zero everywhere except the origin, but that its integral is null and not equal to unity.

Also, the relations xd(x) = 0, dH(x)/dx = d(x) do not make sense in classical mathematical analysis, where

is the Heaviside function, introduced in 1898 by the English engineer Oliver Heaviside.

The created formalism regarding the use of the function d and others, although it was in contradiction with the concepts of mathematical analysis, allowed for the study of discontinuous phenomena and led to correct results from a physical point of view.

All these elements constituted the source of the theory of distributions or of the generalized functions, a theory designed to justify the formalism of calculation used in various fields of physics, mechanics and related techniques.

In 1936, S.L. Sobolev introduced distributions (generalized functions) in an explicit form, in connection with the study of the Cauchy problem for partial differential equations of hyperbolic type.

The next major event took place in 1950-1951, when L. Schwartz published a treatise in two volumes entitled "Théory des distributions" [2]. This book provided a unified and systematic presentation of the theory of distributions, including all previous approaches, thus justifying mathematically the calculation formalisms used in physics, mechanics and other fields.

Schwartz's monograph, which was based on linear functionals and on the theory of locally convex topological vector spaces, motivated further development of many chapters of mathematics: the theory of differential equations, operational calculus (Fourier and Laplace transforms), the theory of Fourier series and others.

Properties in the sense of distributions, such as the existence of the derivative of any order of a distribution and in particular of the continuous functions, the convergence of Fourier series and the possibility of term by term derivation of the convergent series of distributions, led to important technical applications of the theory of distributions, thus removing some restrictions of classical analysis.

The distribution theory had a significant further development as a result of the works developed by J. Mikusinski and R. Sikorski [3], M.I. Guelfand and G.E. Chilov [4, 5], L. Hörmander [6, 7], A. H. Zemanian [8], and so on.

Unlike the linear and continuous functionals method used by Schwartz to define distributions, J. Mikusinski and R. Sikorski introduced the concept of distribution by means of fundamental sequences of continuous functions.

This method corresponds to the spirit of classical analysis and thus it appears clearly that the concept of distribution is a generalization of the notion of function, which justifies the term generalized function, mainly used by the Russian school.

Other mathematicians, such as H. König, J. Korevaar, Sebastiano e Silva, and I. Halperin have defined the notion of distribution by various means (axiomatic, derivatives method, and so on).

Today the notion of distribution is generalized to the concept of a hyperfunction, introduced by M. Sato, [9, 10], in 1958. The hyperdistributions theory contains as special cases the extensions of the notion of distribution approached by C. Roumieu, H. Komatsu, J.F. Colombeu and others.

1.2 Fundamental Concepts and Formulae

For the purpose of distribution theory and its applications in various fields, we consider some function spaces endowed with a convergence structure, called fundamental spaces or spaces of test functions.

1.2.1 Normed Vector Spaces: Metric Spaces

We denote by G either the body of real numbers or the body of complex numbers and by +, +, 0 the sets + = [0,8), + = (0, 8), 0 = {0,1,2,..., n,...}.

Let E, F be sets of abstract objects. We denote by E × F the direct product (Cartesian) of those two sets; where the symbol "×" represents the direct or Cartesian product.

Definition 1.1 The set E is called a vector space with respect to G, and is denoted by (E, G), if the following two operations are defined: the sum, a mapping (x, y) x + y from E × F into E, and the product with scalars from G, the mapping (?, x) ?x from G × E into E, having the following properties:

The vector space (E, G) is real if G = and it is complex if G = . The elements of (E, G) are called points or vectors.

Let X be an upper bounded set of real numbers, hence there is M such that for all x X we have x = M. Then there exists a unique number M* = sup X, which is called the lowest upper bound of X, such that

Similarly, if Y is a lower bounded set of real numbers, that is, if there is m such that for all x Y we have x = m, then there exists a unique number m* = inf X, which is called the greatest lower bound of Y, such that

Example 1.1 The vector spaces n, n, n = 2 Let us consider the n-dimensional space n = × . × (n times). Two elements x, y n, x = (x1,..., xn), y = (y1,..., yn), are said to be equal, x = y, if xi = yi, .

Denote x + y = (x1 + y1 , x2 + y2,..., xn + yn), ax = (ax1, ax2,..., axn), a , then n is a real vector space, also called n-dimensional real arithmetic space.

The n-dimensional complex space n may be defined in a similar manner. The elements of this space are ordered systems of n complex numbers. The sum and product operations performed on complex numbers are defined similarly with those in n.

Definition 1.2 Let (X, G) be a real or complex vector space. A norm on (X, G) is a function || · || : X [0, 8) satisfying the following three axioms: