In mathematics, generalized functions are objects extending the notion of functions on real or complex numbers. There is more than one recognized theory, for example the theory of distributions. Generalized functions are especially useful for treating discontinuous functions more like smooth functions, and describing discrete physical phenomena such as point charges. They are applied extensively, especially in physics and engineering. Important motivations have been the technical requirements of theories of partial differential equations and group representations.
A common feature of some of the approaches is that they build on operator aspects of everyday, numerical functions. The early history is connected with some ideas on operational calculus, and some contemporary developments are closely related to Mikio Sato's algebraic analysis.
Some early history
In the mathematics of the nineteenth century, aspects of generalized function theory appeared, for example in the definition of the Green's function, in the Laplace transform, and in Riemann's theory of trigonometric series, which were not necessarily the Fourier series of an integrable function. These were disconnected aspects of mathematical analysis at the time.
The intensive use of the Laplace transform in engineering led to the heuristic use of symbolic methods, called operational calculus. Since justifications were given that used divergent series, these methods were questionable from the point of view of pure mathematics. They are typical of later application of generalized function methods. An influential book on operational calculus was Oliver Heaviside's Electromagnetic Theory of 1899.
When the Lebesgue integral was introduced, there was for the first time a notion of generalized function central to mathematics. An integrable function, in Lebesgue's theory, is equivalent to any other which is the same almost everywhere. That means its value at each point is (in a sense) not its most important feature. In functional analysis a clear formulation is given of the essential feature of an integrable function, namely the way it defines a linear functional on other functions. This allows a definition of weak derivative.
During the late 1920s and 1930s further basic steps were taken. The Dirac delta function was boldly defined by Paul Dirac (an aspect of his scientific formalism); this was to treat measures, thought of as densities (such as charge density) like genuine functions. Sergei Sobolev, working in partial differential equation theory, defined the first rigorous theory of generalized functions in order to define weak solutions of partial differential equations (i.e. solutions which are generalized functions, but may not be ordinary functions). Others proposing related theories at the time were Salomon Bochner and Kurt Friedrichs. Sobolev's work was extended by Laurent Schwartz.
Schwartz distributions
The most definitive development was the theory of distributions developed by Laurent Schwartz, systematically working out the principle of duality for topological vector spaces. Its main rival in applied mathematics is mollifier theory, which uses sequences of smooth approximations (the 'James Lighthill' explanation).
This theory was very successful and is still widely used, but suffers from the main drawback that distributions cannot usually be multiplied: unlike most classical function spaces, they do not form an algebra. For example, it is meaningless to square the Dirac delta function. Work of Schwartz from around 1954 showed this to be an intrinsic difficulty.
Algebras of generalized functions
Some solutions to the multiplication problem have been proposed. One is based on a simple definition of by Yu. V. Egorov (see also his article in Demidov's book in the book list below) that allows arbitrary operations on, and between, generalized functions.
Another solution allowing multiplication is suggested by the path integral formulation of quantum mechanics.
Since this is required to be equivalent to the Schrödinger theory of quantum mechanics which is invariant under coordinate transformations, this property must be shared by path integrals. This fixes all products of generalized functions
as shown by H. Kleinert and A. Chervyakov. The result is equivalent to what can be derived from
dimensional regularization.
Several constructions of algebras of generalized functions have been proposed, among others those by Yu. M. Shirokov
and those by E. Rosinger, Y. Egorov, and R. Robinson.
In the first case, the multiplication is determined with some regularization of generalized function. In the second case, the algebra is constructed as multiplication of distributions. Both cases are discussed below.
Non-commutative algebra of generalized functions
The algebra of generalized functions can be built-up with an appropriate procedure of projection of a function <math>F=F(x)</math> to its smooth
<math>F_{\rm smooth}</math> and its singular <math>F_{\rm singular}</math> parts. The product of generalized functions <math>F</math> and <math>G</math> appears as
Such a rule applies to both the space of main functions and the space of operators which act on the space of the main functions.
The associativity of multiplication is achieved; and the function signum is defined in such a way, that its square is unity everywhere (including the origin of coordinates). Note that the product of singular parts does not appear in the right-hand side of (); in particular, <math>\delta(x)^2=0</math>. Such a formalism includes the conventional theory of generalized functions (without their product) as a special case. However, the resulting algebra is non-commutative: generalized functions signum and delta anticommute.
Multiplication of distributions
The problem of multiplication of distributions, a limitation of the Schwartz distribution theory, becomes serious for non-linear problems.
Various approaches are used today. The simplest one is based on the definition of generalized function given by Yu. V. Egorov.