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Alternative Definitions of Generalized Functions

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Alternative Definitions of Generalized Functions Appendix Alternative Definitions of Generalized Functions Distributions can be defined in three essentially different ways: as func­ tionals, sequences or improper derivatives. These three different approaches are described in §1- 3. The next three sections describe three more generaliza­ tion methods, which yield objects that are not equivalent to distributions: Mikusinski's operators, hyperfunctions and nonstandard functions. A more detailed account of the different approaches can be found in [Temple 1953, 1955], [Naas and Schmid 1961], [Slowikowski 1955] and [Beltrami 1963] and in the references below. 1. Functionals. A sequence (fJv E C,;"'(Iffi") is said to converge to 0 if all the (fJvS have their supports contained in one compact set K (independent of n) and (fJv -+ 0 uniformly together with all its derivatives (this convergence can be defined by an LF topology). The space C,;'" with this notion of convergence (topology) is called ~. A continuous linear functional on q; is called a dis­ tribution. A locally integrable functionjis identified with the distribution T defined as T«(fJ) = i j. (fJ. (1) ikl" The derivative (iJ/iJxi)T of a distribution T is defined as (2) iJ~i T«(fJ) = - T(iJ~i (fJ). A sequence of distributions 'Ii is said to converge to T in q;' if 'Ii«(fJ) -+ T«(fJ) for i -+ 00 uniformly on all bounded subsets B of functions (fJ of C,;"'. This definition of a distribution was given by Schwartz [1950/51]. He similarly defined the space of tempered distributions as the dual of ff, the rapidly decreasing functions. Sobolev [1936a] also used this approach. App., §2 Alternative Definitions of Generalized Functions 167 2. Sequences. It is a fundamental theorem in the theory of distributions that any distribution is a limit in £!I' of a sequence of continuous functions. Thus sequences of functions give an alternative method for defining distributions. This method is very similar to Cantor's construction of the real numbets. Several sequence definitions have been given by mathematicians and physicists who claim that they are closer to physical intuition than the func­ tional definition. What distinguishes the different sequence approaches is the definition of a fundamental sequence, which is not a priori given, since the space q)' is not given in advance. (a) A sequence fn of continuous functions on ~n (or Lfoc (~n) functions) is said to be a fundamental sequence if (3) is convergent for all cp E C:,. Two fundamental sequences fi and g; are equivalent if lim i ficp = lim i g;cp for all cp E C:,. (4) i-+ 00 IR" i-+ 00 Ill" An equivalence class of fundamental sequences is called a distribution. This sequence definition is the one which is closest to the functional definition since it makes uses of test functions as well. It was suggested by Tolhoek in 1944 (independently of Schwartz' work), by Mikusinski [1948], Lighthill [1958] and by Courant [Courant-Hilbert 1962, p. 777] (all three depending on Schwartz' work). (b) A sequence of functions fi E C(~) is called fundamental if for every compact subset K of ~ there exists a sequence F; E C(~) and a natural number k such that F(~)(x) = fi(x) for x E K and for all i E N and F;(x) converges uniformly on K. Two fundamental sequences fi and g; are equivalent if for all compact sub­ sets K of ~ the sequences F;, G;, mentioned above, can be chosen such that fi k)} = Fl for the same k g; = Glk ) and F;= G; for all i EN An equivalence class of fundamental sequences is called a distribution. (The extension to more than one dimension is obvious.) The distributions defined in this way are also included in Schwartz' dis­ tributions, since any Schwartz distribution is locally a derivative of a con­ tinuous function. 168 Alternative Definitions of Generalized Functions App., §3 Definition 2(b) was suggested by Mikusiriski and Sikorski [1957]. A slightly different approach was presented by Korevaar [1955]. When distributions are defined as equivalence classes of sequences, the imbedding of C(lRn) in the space of distributions and the definition of a derivative are given in an obvious way. 3. Formal derivatives of continuous functions A Schwartz distribution is locally (i.e. on every compact subset of IRn) a derivative of a continuous function. Any Schwartz distribution T E ~'(lRn) can be written as a locally finite sum: CXl ( a )Pl ( a )pn T-- L -a ... -a fp" .... Pn' (5) Pl .... Pn=O Xl Xn wherefpl ..... Pn are continuousfunctions (i.e.for any compact set KeIRn all but a finite number of the fis vanish on K). These theorems gave rise to the following definitions: (a) Consider allloca1ly finite sums of pairs L (Ok,}k), (6) k where the OkS are differential operators and the iks are continuous functions. Two such expressions are called equivalent if the results obtained from using formal partial integration on the integrals (7) are the same for all test functions q> E C:'(lRn). Equivalence classes of such expressions are called distributions. This definition was given by Tolhoek [1949] and Courant [Courant-Hilbert 1962, p.775]. (b) Consider locally finite power series CXl ZPl ... zPn (8) L J.Plt""pn 1 n , Pl, ... ,pn=O where thefpl ..... Pns are continuous functions. Two such series are equivalent if their term-by-term difference is the sum of terms of the form f(x)zfl ... z!'" ... z~n - g(X)Zfl ... Z~v+l ••• z~n, where f(x) = (a/axv)g. App., §5 Alternative Definitions of Generalized Functions 169 An equivalence class of locally finite power series is called a distribution. The mapping which sends the power series (8) into the functional T: T(rn) = "(_1)Pl+"'+P" J. (x) - a )Pl ... ( - a )pn cp(x)dx 't' L.. i n Pl.···.Pn (ox ox Pl •... 'Pn ~ 1 n is an isomorphism between the distributions defined here and those defined by Schwartz. Definition 3(b) was advanced by H. Konig [1953]. (c) Consider all pairs (f, n) of a continuous function f on a fixed interval [a, bJ = K and a natural number n. Two such pairs (f, n) and (g, m) are equivalent if I mf - rg (r is the n times iterated integral IX . ) (a+b)/2 is of the form m+n-l L aixi. i=O The equivalence class which contains (f, n) is denoted [f, n]. A system T = {[fK' nKJ} of equivalence classes corresponding to any compact interval K c IR is called a distribution if for all K' c K [fK" nd is a restriction of [fK' nKJ (with an obvious definition of a restriction). The extension to IRn is easy. R. Sikorski [1954J and S. e Silva [1955J invented this definition. It is clear how to imbed continuous functions in the space of distributions and how to define differentiation when the definitions in §3 are used. 4. Mikusinski's operators. Consider the ring of continuous complex-valued functions on IR+ u {o} with the compositions (f + g)(t) = f(t) + get), (f * g)(t) = Lf(U)9(t - u) duo Since this ring has no zero divisors [Titchmarsch 1926J, it can be extended to a field. Mikusiriski, who discovered this generalization of the function concept [1950, 1959J, called the elements in the' extension field operators. He generalized the operators to operators which did not need to have "support" on a positive half-line [1959J and showed that these "distribu­ tions" were not equivalent to Schwartz' distributions (see also Liitzen 1979, Ch. V). 5. Hyperfunctions. Consider complex functions which are holomorphic in IC\ IR. Two such complex functions are called equivalent if their difference is 170 Alternative Definitions of Generalized Functions App., §6 holomorphic in the whole complex plane. The space of equivalence classes H(C\IR)/H(IC) is called the space of hyperfunctions. It was shown by Bremer­ mann [1965, p. 50] that for every distribution T E ~'(IR) there eixsts a complex function, holomorphic on C\supp T such that f(x + if,) - f(x - if,) ~ T ind ~'. ' .... 0 In this way ~' can be imbedded in the space of hyperfunctions. To define hyperfunctions in more dimensions is considerably more difficult. Hyperfunctions were introduced by Sato [1959/60] but had already been anticipated by several other mathematicians (see Ch. 3, note 18). 6. Nonstandard functions. Laugwitz and Schmieden [1958] and Robinson [1961] extended the field of real numbers to a ring and a field, respectively, including infinitely large and infinitely small numbers. Functions in the extended ring or field give interesting generalized functions. For example, the quasi-standard function where w is an infinite natural number (w E N*\N), has the property charac­ teristic of the Dirac b-function: fb(x)f(x) = f(O), in the sense that the standard parts of the two sides of the equation are equal. In this way distributions can be represented by nonstandard functions, but the correspondence between quasi-standard functions (a subclass of the nonstandard functions) and distributions is not 1-1, since there are many quasi-standard functions representing each distribution. By forming suitable equivalence classes of quasi-standard functions, a 1-1 correspondence can be established. Equivalence relations of this kind have been defined in various ways by Laugwitz [1961], Luxemburg [1962] (similar to the method de­ scribed in §2(b) of this Appendix) and Robinson [1966] (similar to Schwartz' approach). It is worth noting that multiplication of distributions can not be defined since it would depend on the representatives of the equivalence classes. Thus in this respect the original nonstandard functions are easier to handle.
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