Editing Generalized function (section)

==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 [[measure (mathematics)|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 solution]]s of partial differential equations (i.e. solutions which are generalized functions, but may not be ordinary functions).<ref>{{Cite book |last1=Kolmogorov |first1=A. N. |url=https://www.worldcat.org/oclc/44675353 |title=Elements of the theory of functions and functional analysis |last2=Fomin |first2=S. V. |date=1999 |publisher=Dover |orig-date=1957 |isbn=0-486-40683-0 |location=Mineola, N.Y. |oclc=44675353}}</ref> Others proposing related theories at the time were [[Salomon Bochner]] and [[Kurt Friedrichs]]. Sobolev's work was extended by [[Laurent Schwartz]].<ref>{{cite journal | last1 = Schwartz | first1 = L | year = 1952 | title = Théorie des distributions | journal = Bull. Amer. Math. Soc. | volume = 58 | pages = 78–85 | doi = 10.1090/S0002-9904-1952-09555-0 | doi-access = free }}</ref>