An Overview of the Theory of Distributions

Transcription

1 An Overview of the Theory of Distributions Mtt Guthrie Adpted from Hlperin [1] 1

2 Chpter 1 Introduction For the rnge of time between the introduction of the opertionl clculus t the end of the 19th century nd the mid 20th century, mny formuls were in use which hd not been dequtely clrified from the mthemticl point of view. For instnce, consider the Heviside function H(x) = { 0 for x < 0 1 for x 0. It is sid tht the derivtive of this function is the Dirc delt function δ(x), which hs the mthemticlly impossible properties tht it vnishes everywhere except t the origin where its vlue is so lrge tht (1.1) δ(x) dx = 1. (1.2) This function nd its successive derivtives hve been used with considerble success, especilly in electromgnetics nd grvittion. It ws suggested by Dirc himself tht the delt function could be voided by using insted limiting procedure involving ordinry, mthemticlly possible, functions. However, the delt function cn be kept nd mde rigorous by defining it s mesure, tht is, s set function in plce of n ordinry point function. This suggests tht the notion of point function be enlrged to include new entities 1 nd tht the notion of the derivtive be correspondingly generlized so tht within the new system of entities, every point function should hve rigorously defined derivtive. This is done in the theory of distributions. The new system of entities, clled distributions, includes ll continuous functions, ll Lebesgue loclly summble functions, nd new objects of which simple exmple is the Dirc delt function mentioned bove. The more generl but rigorous process of differentition ssigns to every distribution derivtive which is gin distribution, nd so every distribution, including every loclly summble point function, hs derivtives of ll orders. The derivtive of loclly summble point function is lwys distribution lthough not, in generl, point function. However, it coincides with the clssicl derivtive when the ltter exists nd is loclly summble. 1 Just s the notion of rtionl number ws enlrged by Dedekind to include ll rel numbers. 2

3 This theory of distributions gives rigorous content nd vlidity to the formuls of opertionl clculus mentioned bove. It cn be developed not only for functions of one vrible, but lso for functions of severl vribles nd it provides simple but more complete theory of such topics s Fourier series nd integrls, convolutions, nd prtil differentil equtions. A systemtic exposition of the theory of distributions is given in Grubb s recent Distributions nd Opertors [2]. There s lso the recommended reference work by Strichrtz, A Guide to Distribution Theory nd Fourier Trnsforms [3]. The comprehensive tretise on the subject, lthough quite old now, is Gel fnd, nd Shilov s Generlized functions [4]. A very good (though pretty dvnced) source tht s now vilble from Dover is Topologicl Vector Spces, Distributions nd Kernels by Trves [5]. Tht book is one of the clssic texts on functionl nlysis nd if you re n nlyst or spire to be, there s no reson not to hve it now. Of course, if you red French, you relly should go bck nd red Schwrtz s originl tretise [6]. However, the following pges my serve s useful introduction to some of the bsic ides. 3

4 Chpter 2 Point Functions s Functionls The following discussion will led to the precise definition of distributions given lter in this chpter. Let (, b) be finite closed intervl. A continuous function f(x) cn be considered point function 1 but it cn lso be considered in nother wy: 2 it defines the functionl F (φ) s F (φ) = f(x)φ(x) dx, (2.1) where F (φ) is number 3 defined for every continuous φ(x). This functionl is liner, tht is F (c 1 φ 1 + c 2 φ 2 ) = c 1 F (φ 1 ) + c 2 F (φ 2 ). (2.2) Since there re liner functionls which cnnot be expressed in this wy, in terms of ny continuous (or even Lebesgue summble [7] f(x)), our first suggestion is tht the distributions be defined s the rbitrry liner functionls F (φ) over some suitble set S of continuous φ(x). The φ(x) chosen will be clled testing functions. If f(x) is bsolutely continuous nd hs derivtive f (x), the derivtive will lso define liner functionl, nmely f (x)φ(x) dx. This new functionl cn be expressed in terms of the originl F (φ), for some φ, by use of integrtion by prts: f (x)φ(x) dx = f(x)φ (x) dx = F (φ ), (2.3) 1 Point functions need be defined lmost everywhere nd two functions re identified if they re equl lmost everywhere. Thus, we sy tht f(x) is identiclly zero if it hs the vlue zero lmost everywhere, tht f(x) is continuous if it cn be identified with continuous function; the upper bound of f(x) will men the essentil upper bound, the derivtive of f(x) will men the derivtive of tht function which cn be identified with f(x) nd which hs derivtive if such function exists, nd so on. 2 Just s every rtionl number ws considered by Dedekind in new wy: s defining cut in the set of ll rtionls. 3 Numbers my be tken to men rel numbers or complex numbers. 4

5 which is vlid for ll φ(x) tht vnish t nd b nd hve continuous first derivtive. This suggests tht we llow S to include only such φ(x) nd not ll continuous φ(x). Eqution (2.3) lso suggests tht for every liner F (φ), whether it comes from n f(x) possessing derivtive f (x) or not, F (φ), derivtive of F which is liner functionl on S, be defined by F (φ) = F (φ ). (2.4) If F (φ) is to be defined for ll φ S, S must be restricted to include only such φ(x) s hve φ (x) lso in S. This leds to the condition tht φ(x) mush hve derivtives of ll orders nd they (s well s φ(x) itself) must vnish t nd b. Importnt exmples of such φ(x) re the functions φ n (x) = (φ c,d (x)) 1/n, obtined by choosing n N nd c < d b nd defining φ c,d (x) = { exp ( 1 x c + 1 d x) for c < x < d 0 otherwise. It is not necessry to restrict S ny further. However, in restricting S we must be creful to identify the point function f(x) nd the functionl F (φ) which it defines. Thus, we wnt S to include sufficiently mny testing functions so tht functions f(x) which re different on (, b) will be identified with different F (φ). We do not wnt f(x)φ(x) dx to vnish for ll φ S except when f(x) is identiclly zero on (, b). This requirement is stisfied if S includes ll (2.5) for (φ c,d (x)) 1/n, nd if lim n f(x) (φ c,d (x)) 1/n dx = d c f(x) dx, (2.6) d for ll c < d b, then f(x) is identiclly zero on (, b). c f(x) dx = 0 (2.7) Another point is this: the F (φ) which re defined by point functions, nd ll derivtives of such (the F (n) (φ)) hve certin continuity property nd it is desirble to require this continuity property in defining distributions. 4 4 As will be shown in Chpter 5, this gives the smllest enlrgement or completion of the system of loclly summble point functions which permits unrestricted differentition. 5

6 Finlly, with view to lter pplictions, we will define distributions on open intervls. The closed intervl gives mthemticlly simpler sitution nd indeed the open intervl will be discussed in terms of its closed subintervls. However, the word distribution will be reserved for the open intervl nd the terminology continuous liner functionl will be used for the closed intervl. The precise definitions re: Definition of distribution For ny closed finite intervl (, b), let S (,b) consist of ll continuous φ(x) possessing derivtives φ (n) (x) of ll orders which, long with φ(x) itself, vnish t nd b nd for x outside (, b). A liner functionl F (φ), defined for ll φ S (,b), is clled continuous liner functionl on (, b) if it hs the property tht whenever ll φ, φ m S (,b) nd the φ m (x) converge uniformly to φ x, nd for ech n, the derivtives φ (n) m (x) converge uniformly to φ (n) (x), then F (φ m ) will converge to F (φ). For n rbitrry open intervl I, finite or infinite, distribution on I is liner functionl F (φ) such tht for every closed finite intervl (, b) contined in I, F (φ) is defined for φ S (,b) nd, when restricted to these φ, defines continuous liner functionl on (, b). Identifiction of the distribution nd point function A distribution F (φ) on n intervl I is to be identified with point function f(x) if, for every closed finite intervl (, b) I, f(x) is summble on (, b) nd F (φ) = f(x)φ(x) dx (2.8) for ll φ S (,b). Sometimes the nottion f(φ) is used to denote the distribution identified with the point function f(x). Identifiction of the distribution nd mesure function A distribution F (φ) on n intervl I is to be identified with the Stieltjes mesure 5 dψ(x) if for every closed finite intervl (, b) contined in I, ψ(x) is of bounded vrition on (, b) nd for ll φ S (,b). F (φ) = φ(x) dψ(x) (2.9) Definition of the derivtive of distribution For ny distribution F (φ) on I, the derivtive distribution F (φ) is defined by F (φ) = F (φ ). (2.10) 5 For more informtion on this, see Lebesgue-Stieltjes Mesures on the Rel Line nd Distribution Functions [8]. 6

7 It is esily verified tht this F stisfies the conditions for distribution on I (the sme formul defines the derivtive for continuous liner functionl). The distributions defined bove include ll continuous nd even ll Lebesgue (loclly) summble point functions, ll Stieltjes mesures, nd, s we will see, vriety of new mthemticl entities. Within the system of distributions ech distribution hs derivtive nd consequently, derivtives of ll orders: F (n) (φ) = ( 1) (n) F (φ (n) ). (2.11) The derivtive of point function f(x) my be point function or Stieltjes mesure, or more generl distribution. It will be point function g(x) if nd only if f(x) is bsolutely continuous on every finite closed (, b) I, then g(x) is the ordinry point derivtive f (x) which then must be defined (lmost everywhere). The derivtive of f(x) is Stieltjes mesure dψ(x) if nd only if f(x) is of bounded vrition on every finite closed (, b) I nd then ψ(x) differs from f(x) by n rbitrry dditive constnt. In prticulr, the derivtive δ of the Heviside function H(x) is now rigorously defined (I being ny open intervl which contins the origin) nd is the mesure function dh(x); this is the proper mthemticl description of the Dirc delt function, which hs no mening s point function. We cn now use the nottion δ = H. In pplictions to certin physicl problems, loclly summble f(x) my be thought of s representing distribution of mss or electric chrge on the x xis, s such, d c f(x) dx is the totl (lgebric) chrge on (c, d) nd f(x) is the chrge density t prticulr x. From this point of view, the Dirc δ function represents the concentrtion of unit chrge t single point, the origin. δ represents dipole, nd higher derivtives of δ represent more complicted multipole terms. An interesting distribution which is quite different from the Dirc δ nd its derivtives cn be investigted with f(x) = { x 1/2 for x > 0 0 for x 0. (2.12) The derivtive of f(x) exists (s distribution), lthough it is not quite Stieltjes mesure. Roughly speking, f corresponds to negtive mss continuously distributive on the positive x xis with n infinite quntity in every neighborhood of the origin, together with n infinite positive mss t the origin, in such wy tht there is finite totl lgebric mss on every finite closed intervl. This is shown by the formul, for φ S (,b) with < 0 < b: 7

8 f (φ) = = lim ɛ 0 = lim ɛ 0 f(x)φ (x) dx = ɛ x 1/2 φ (x) dx [ [x 1/2 φ(x) ] b ɛ + ɛ 0 f(x)φ (x) dx ] 1 2 x 3/2 φ(x) dx (2.13) Becuse φ vnishes t b, f (φ) = lim ɛ 0 = lim ɛ 0 [ φ(ɛ) ɛ + [ φ(0) ɛ + ɛ ɛ ( 12 ) ] x 3/2 φ(x) dx ( 12 ) ] (2.14) x 3/2 φ(x) dx nd since φ(ɛ) φ(0) ɛ = φ(ɛ) φ(0) ɛ ɛ 0, (2.15) φ (0) = 0 s ɛ 0. Although 1 2 x 3/2 φ(x) is not summble on (0, b) for rbitrry φ S (,b), yet the brcket s whole lwys hs finite limit, which hs been clled the finite prt of the divergent integrl by Hdmrd [9]. Hdmrd uses the nottion f (φ) = F. p. 0 f (x)φ(x) dx (2.16) to indicte tht f (x) is the non-summble point derivtive of f(x) for x > 0. Finite prts of divergent integrls hve been studied in gret detil by Hdmrd. Similrly, the liner functionl [( ɛ F (φ) = lim + ɛ 0 ɛ ) ] 1 φ(x) dx x (2.17) corresponds to continuously distributed mss with infinite positive mss long the positive x xis together with infinite negtive mss long the negtive x xis in such wy tht in every neighborhood of the origin there is finite totl lgebric mss. The limit of the brcket is nother cse of Hdmrd finite prt of divergent integrl, in this cse coinciding with the Cuchy principl vlue. The distribution itself is the derivtive of the point function log x. 8

9 Chpter 3 The Clculus of Distributions Multipliction of distribution by constnt nd ddition of two distributions re defined by the formuls nd (cf )(φ) = cf (φ), (3.1) (F 1 + F 2 )(φ) = F 1 (φ) + F 2 (φ). (3.2) It is esily verified tht the usul rules of ddition nd subtrction hold, even when differentition is involved. Thus, (c 1 F 1 + c 2 F 2 ) = c 1 F 1 + c 2 F 2. (3.3) The constnt point function, f(x) = k, is identified with constnt distribution k(φ) which hs the chrcteristic property k(φ) = kφ(t) dt = k φ(t) dt (3.4) for every φ S (,b). In prticulr, when k = 0 the corresponding distribution is clled the zero distribution. Agin, it is esily verified tht if F is constnt distribution, F is the zero distribution. To prove the converse, it is useful to estblish the following expnsion lemm. Lemm. Let θ(x) be ny function in S (c,d) for which For instnce, θ(x) could be d c θ(x) dx = 1. (3.5) ( d 1 φ c,d (t) dt) φ c,d (x), c 9

10 where φ c,d (x) is the function defined in (2.5). Let n N. Then ny φ(x) in ny S (,b) with c < d b cn be expressed in the form φ(x) = 0 θ(x) + 1 θ (x) n θ (n) (x) + ρ n+1 n (x), (3.6) where 0, 1,..., n re uniquely determined constnts nd ρ n (x) S (,b). This lemm cn be proved by induction on n if we observe tht prticulr φ(x) S (,b) cn be expressed in the form ρ (x) for some ρ(x) S (,b) if nd only if In the expnsion for generl φ(x) S (,b), 0 = φ(t) dt = 0. (3.7) φ(t) dt. (3.8) Now using this representtion with n = 1, we rgue tht if F = 0 then F (ρ 1) = F (ρ 1 ) = 0 for ll ρ 1, nd therefore where k is constnt, the vlue of F (θ). Therefore, F (φ) = F ( 0 θ + ρ 1) = 0 F (θ) + F (ρ 1) = 0 k, (3.9) F (φ) = proving tht F is indeed constnt distribution. kφ(t) dt, (3.10) We sy tht the distribution F is primitive of G if G = F. From the preceding prgrph, it follows tht two primitives of the sme F differ by constnt distribution. As is well known, in the cse of point functions there exists n infinity of different (point function) primitives, nd in order to specify one of them it is sufficient to give its vlue t prticulr point. In the cse of n rbitrry distribution F there exists gin n infinity of primitives (distributions!) nd prticulr one cn be specified by giving its vlue for ny prticulr testing function θ(x) s described bove. This follows immeditely from the reltionship G(φ) = G( 0 θ + ρ 1) = 0 G(θ) + G(ρ 1) = 0 G(θ) F (ρ 1 ), (3.11) which cn be used to define G(φ) for every φ in terms of n rbitrry (but fixed) G(θ), nd given F. The fct tht this G is distribution cn be deduced from the reltion ρ 0 (x) = x 0 ( φ(t) dt ) x φ(t) dt θ(t) dt. (3.12) 10

11 Chpter 4 Multipliction of Distributions The product F 1 F 2 is not defined for rbitrry distributions. This reflects the fct tht the product f 1 (x)f 2 (x) of two loclly summble point functions my not be loclly summble. However, F 1 F 2 is defined in certin cses. For exmple, if F 1, F 2 cn be identified with f 1 (x), f 2 (x) respectively nd the product f 1 (x)f 2 (x) is loclly summble, then F 1 F 2 is defined to be the distribution which is identified with f 1 (x)f 2 (x). This is specil cse of the following generl definition. Definition of product of two continuous liner functionls on finite closed intervl Suppose for some n tht F (n) 1 is point function f 1 (x) nd tht F 2 is the nth derivtive of point function f 2 (x): F (n) 1 = f 1, F 2 = f (n) 2. (4.1) Suppose, too, tht the product f 1 (x)f 2 (x) is summble. Then we define F 1 F 2 by the formul F 1 F 2 = F 1 f (n) 2 ( n = (F 1 f 2 ) (n) 1 + ( 1) r ( n r ) (F 1f 2 ) (n 1) + ( ) n (F (2) 1 f 2 ) (n 2) ) (F (r) 1 f 2 ) (n r) ( 1) n F (n) 1 f 2. (4.2) Ech term on the right is continuous liner functionl since F (n) 1 f 2 is f 1 (x)f 2 (x) which ws ssumed summble for r < n, F (r) 1 f 2 is the product of n bsolutely continuous (so it is bounded) point function nd f 2. Definition of the product of distributions on n open intervl I Let F (,b) denote the distribution F restricted to the testing functions in S (,b). If F 1, F 2 re distributions on I such tht F 1(,b) F 2(,b) is defined on every (, b) I, then F 1 F 2 is defined to be the distribution on I which, when restricted to ny S (,b), coincides with F 1(,b) F 2(,b). 11

12 It is not difficult to verify tht these definitions give unique F 1 F 2 whenever they define F 1 F 2 t ll nd tht our formul bove for the product F 1 F 2 is equivlent to (F 1 F 2 )(φ) = ( 1) (n) f 2 (x)(f 1 (x)φ(x)) (n) dx. (4.3) It cn lso be verified tht the product lw (F 1 F 2 ) = F 1F 2 + F 1 F 2 is vlid nd tht the three products which occur in this sttement re necessrily defined whenever one of the products on the right is defined. There is specil but importnt cse in which this product rule cn be gretly simplified. Suppose tht F 1 is continuous point function (x) possessing point function derivtives of ll orders, so tht F (n) 1 is continuous point function for ech n. Then the product rule defines F 1 F 2 for every F 2 which, when restricted to n S (,b), cn be put in the form f (n) 2 for some f 2 nd n which might depend on nd b. The rule simplifies in this cse to for every testing function φ. 1 The preceding prgrph suggests tht we could use the reltion (F 1 F 2 )(φ) = (F 2 )(φ) = F 2 (φ) (4.4) (F 2 )(φ) = F 2 (φ) (4.5) to define F 2 whenever (x) hs (ordinry) derivtives of ll orders but F 2 is n rbitrry distribution. It is noteworthy tht such definition would not give nything new since every distribution F 2, when restricted to n S (,b), cn be put in the form f (n) 2 for some point function f 2 (x) nd some n, which my depend on nd b. This theorem will be proved in the next chpter nd it is consequence of the continuity condition in the definition of distribution. It is noted tht the originl definition of cf, where c is constnt, is included in the generl definition of product of distributions if c is considered s constnt distribution. Also note tht when the Dirc δ nd its derivtives δ (n) re multiplied by n (x) with derivtives (n) (x) of ll orders, we obtin: (x)δ = (0)δ (x)δ = (δ) δ = (0)δ (0)δ, (4.6) nd in generl, (x)δ (n) = (0)δ (n) n (0)δ (n 1) + ( ) n (0)δ (n 2) ( 1) n (n) (0)δ. (4.7) 2 1 Observe tht φ is testing function long with φ. 12

13 Chpter 5 The Order Clssifiction of Distributions The system of continuous liner functionls on closed, finite intervl includes every summble f(x) nd its derivtives. This chpter will illustrte tht there re no other continuous liner functionls. Tht is, every continuous liner functionl is either f(φ) or f (n) (φ) for some suitble summble f(x) nd some finite n. 1 Let F be continuous liner functionl on (, b). F will be sid to hve finite order on (, b) either if it cn be identified with summble point function f(x) or if, for some finite r, F is the rth order derivtive of some such f(x) : F = f (r). The smllest possible r will be clled the order of F. 2 Clerly, if F cn be identified with summble f(x), then it hs order 0 nd its derivtive F hs order either 0 or 1. If F hs order r > 0, its derivtive hs order r + 1. If F hs order r, nd s r, then for some f which depends on F nd s, The vlue of s determines the following properties of f: if s > r, f is bsolutely continuous. if s = 0, f is uniquely determined to within Lebesgue equivlence. if s > 0, F = f (s). (5.1) f is determined only to within n rbitrry dditive polynomil in x of degree s 1. It should therefore be demonstrted tht F must hve finite order. The definition of continuous liner functionl requires tht F (φ m ) converges to F (φ) whenever ll φ, φ m S (,b) nd, for ech n 0, the φ m (n) (x) converge to φ (n) (x). This implies the stronger condition (C r ) for some finite r which depends on F, 1 By using higher n we cn prove this with f(x) continuous or even bsolutely continuous. 2 In The Theory of Distributions [6], slightly different definition is used. The order of F is defined there to be the lowest r for which F = F (r) 0, where F 0 is Stieltjes mesure. 13

14 F (φ m ) converges to F (φ) whenever ll φ, φ m S (,b), nd for ll n with 0 n r, the φ (n) m (x) converge uniformly to φ (n) (x). Suppose tht C r is flse for every r. A sequence of testing functions φ m cn then be defined such tht for ech m,. φ (n) m < 2 m n m, nd b. F (φ m ) > 1. The here modified bsolute vlue nottion is being used, φ = mx[ φ(x) ; x b]. Then for every n, the φ (n) m (x) converge uniformly to the zero function, nd since F is continuous liner functionl, this should imply tht F (φ m ) 0. This contrdicts b. bove nd so C r cnnot be flse for every r, tht is C r holds for some finite r. But, for n < r, φ (n) (x) = x (x t) r n 1 (r n 1)! φ(r) (t) dt, (5.2) so tht φ (n) K φ (r) n < r, for some finite K which depends only on r,, nd b. It follows tht C r is equivlent to the following condition B r, F (φ m ) converges to F (φ) whenever ll φ, φ m S (,b) nd the φ (r) m (x) converge uniformly to φ (r) (x). The condition B r, in turn, implies the following condition A r, with F r finite constnt 3. F (φ) F r φ (r) φ S (,b), (5.3) Indeed, if A r were flse, we could define sequence φ m with F (φ m ) > m φ (r) m. Then the functions µ m (x) = φ (r) m 1 m 1 φ m (x) would be in S (,b) nd the µ (r) m (x) would converge uniformly to zero since µ (r) m = m 1. But F (µ m ) > 1, contrdicting B r. Thus, A r cnnot be flse. To summrize these results, if F is continuous liner functionl on (, b), there is finite r for which F r <, nd F (φ) F r φ (r) φ S (,b). (5.4) Now we construct new functionl L, which is defined for functions which cn be put in the form φ (r), by the formul L(φ (r) ) = F (φ). The functionl L is, by the preceding prgrph, bounded liner functionl on the liner spce of the functions of the form φ (r) with norm φ (r). The Hhn- Bnch procedure [10] cn be used to extend this functionl L to ll continuous functions on (, b) 3 Let F r denote the smllest possible such constnt. 14

15 without incresing the bound of L. Then the representtion theorem of F. Riesz [10] pplies nd shows tht L(φ (r) ) = φ (r) (x) dψ(x) (5.5) for some ψ(x) of totl vrition equl to F r nd ψ(x) cn be ssumed to stisfy ψ(x) F r. Thus, F (φ) = L(φ (r) ) = = φ (r) (x) dψ(x) ψ(x)φ (r+1) (x) dx (5.6) so tht F = f (r+1) where f = ( 1) r ψ, showing tht F is indeed of finite order. If f is replced by one of its indefinite integrls we cn write F = f (r+2) with f(x) bsolutely continuous. For use in the next chpter, we emphsize tht if F r is finite then F = f (r+1) with f(x) bounded, in fct f F r. The converse to this is flse; however F r is certinly finite if F is of order r, since then F (φ) = ( 1)r f(x)φ (r) (x) dx ( ) f(x) dx φ (r). (5.7) Thus, every continuous liner functionl F on finite closed intervl cn be expressed s suitble derivtive of summble point function, nd our system of continuous liner functionls is extensive enough to permit indefinite differentition of the summble point functions. This suggests n bstrct but equivlent formultion of the system of continuous liner functionls s the system of ll symbols f (n) using ll summble f(x) nd ll nonnegtive integers n, nd imposing obvious rules of ddition, identifiction, nd so on. If distribution F on n open intervl I is considered, then the bove considertions pply to every F (,b). We write F (,b) = f (r) on (, b) nd spek of F s hving order r on (, b). However, the order of F on (, b) my be unbounded when nd b vry. A simple exmple of this is the distribution F (φ) = φ (m) (x) (5.8) m=1 on the intervl < x <. The sum on the right hs only finite number of nonzero ddends for every testing function, nd the order of this F on n intervl (, b) is the lrgest integer n for which < n < b. 15

16 Chpter 6 Continuity nd Convergence Properties of Distributions Consider continuous liner functionls on the intervl (, b). A set of F will be clled bounded if, for ech φ S (,b), the numbers F (φ) re bounded, tht is where M is finite constnt which depends on φ, M = M(φ). F (φ) M, (6.1) We sy tht F m form convergent sequence if, for ech φ, the numbers F m (φ) form convergent sequence, nd tht the F m converge to F s limit if, gin for ech φ, the F m (φ) converge to F (φ) s limit. Obviously, if the F m converge to limit, then the F m form convergent sequence. Conversely, if the F m do form convergent sequence, we define F by F (φ) = lim m 0 F m (φ). (6.2) As we will show lter in this chpter, F is continuous liner functionl nd the F m converge to F s limit. Finlly, we sy tht the series m=1 F m is convergent nd hs sum F if N m=1 F m converges to F s N becomes infinite. We now show tht if the F re bounded there must be finite r such tht F (φ) K φ (r) (6.3) for ll, φ for some finite constnt K. Tht is, the F r re bounded. Indeed, if this were flse for every r, we could, by induction on p, select sequence of φ p nd F p such tht, for ech p,. φ (n) p < 2 p n p, b. F p (φ p ) > p p 1 m=1 M(φ m), c. F p (φ m ) < 2 m m > p. 16

17 Let i = 0, 1,..., p 1 nd suppose the φ i, F i hve been selected to stisfy., b., c., insofr s they re involved in these conditions. Since ech F i hs F i ri finite for some r i we will hve F i (φ p ) < 2 p i if φ p stisfies the p conditions φ (r i) p < 2 p F i 1 r i, with i = 0, 1,..., p 1. (6.4) These p conditions s well s. cn be included in single condition φ (r) p < K 1, (6.5) with suitble finite K 1 nd r = mx(r 0,..., r p 1, p). Since we re ssuming tht the F r re unbounded for every r, there must be n F p nd φ p such tht φ p stisfies this condition nd F p (φ p ) stisfies b.. Then the φ i, F i with i = 0, 1,..., p will stisfy., b., c., insofr s they re involved in these conditions. Then m=1 φ m(x) would be testing function φ(x) for which F p (φ) = F p ( m=1 φ m ) = lim N F p ( N ) φ m > p, (6.6) nd therefore, for this φ, the F (φ) would not be bounded. This contrdiction shows tht the F re bounded if nd only if F (φ) K φ (r) for some fixed K nd r, for ll nd ll testing functions. Using the result emphsized in the preceding chpter it cn be concluded tht continuous liner functionls F re bounded if nd only if they cn be expressed s F = f (r) for some common r, with f bounded. As corollry, we deduce tht if F m form convergent sequence, then F (φ) = lim m F m (φ) is continuous liner functionl nd the F m converge to F ; for the convergence of the F m implies tht they re bounded, therefore F m r K for some r nd fixed finite K; nd therefore which implies tht F is continuous liner functionl. If the F m cn be expressed s F m = f (r) m m=1 F (φ) = lim m F m(φ) K φ (r) (6.7) with the f m (x) converging uniformly, then ( ) F m (φ) = ( 1) r f m (x)φ (r) (x) dx ( 1) r lim f m(x) φ (r) (x) dx (6.8) m so tht the F m do converge nd lim m F m = {lim m f m (x)} (r). The converse lso holds. Tht is, if the F m re convergent then for some r, F m = f m (r) with f m (x) converging uniformly nd lim m F m = {lim m f m (x)} (r). Suppose therefore tht the F m re convergent sequence so tht the F m (φ) converge for every φ to limit F (φ). Then the F, F m re bounded nd therefore F m = f (r) m, F = f (r) 17

18 for some suitble f(x), f m (x) with f, f m bounded. By replcing ech of f(x), f m (x) by its integrl from to x nd using r in plce of the former r + 1 it cn be supposed tht F = f (r), F m = f m (r) with the f(x), f m (x) equicontinuous on (, b). We cn lso write F m = (f m + P m ) (r) where the f m (x) re s before nd the P m (x) re rbitrry polynomils of degree less thn r. It will be shown in seprte lemm tht these polynomils cn be chosen so s to give: f m (x) + P m (x) converges uniformly to f(x). This will prove the theorem: F m converges if nd only if F m = f m (r) for some r nd uniformly convergent continuous f m (x) nd then F = lim m F m = (lim m f m (x)) (r). It remins therefore to prove the following lemm, with g m (x) in plce of our former f m (x) f(x). Lemm. If g m (x) re equicontinuous in x on (, b) nd if for some fixed r nd every testing function φ, g m (x)φ (r) (x) dx converges to zero s m, then there re polynomils P m (x) of degree less thn r such tht g m (x) + P m (x) 0 s m. We my suppose tht the g m (x) re rel vlued by considering rel nd imginry prts seprtely if they re complex vlued. We first prove the lemm for the cse r = 0. Let ɛ be n rbitrry positive number nd δ(ɛ) be such tht for ll m, g m (x) g m (y) < ɛ whenever x y < δ(ɛ). Let (, b) be covered by finite number of intervls I1,..., I t ech of length less thn δ(ɛ). Suppose tht for some q nd some I p = (c, d) sy, g q (x) ɛ x I p. Then g q (x)φ c,d (x) dx ɛ d c φ c,d (x) dx, (6.9) where φ c,d is defined in Chpter 2. Since g q(x)φ c,d (x) dx 0 s m 0 it is impossible for this to be true for n infinite number of q for the sme I p. Since there re only finite number of such I p, there is n m 0 such tht, for m m 0, g m (x) < ɛ for t lest one x p in ech I p. But every x (, b) lies in some I p, so g m (x) = g m (x) g m (x p ) + g m (x p ), (6.10) i.e., x (, b), m m 0, g m (x) < 2ɛ. Similrly, we cn obtin g m (x) < 2ɛ nd therefore g m (x) < 2ɛ x, for ll but finite number of m. This mens tht the g m (x) converge uniformly to zero, proving the lemm for the cse r = 0. Now to prove the lemm for ll r by induction. Suppose tht θ(x) is fixed function in S (,b) with θ(x) dx = 1 nd use the expnsion lemm of Chpter 3 to write φ(x) = 0θ(x) + ρ 1(x), where 0 = φ(x) dx. Then g m (x)φ (p) (x) dx = ( g m (x) = c m φ(x) dx + 0 θ (p) (x) + ρ (p+1) 1 ) dx g m (x)ρ (p+1) 1 (x) dx (6.11) 18

19 where c m = g m (x)θ (p) (x) dx (6.12) nd depends on g m but not on φ, nd is bounded for ll m. Repeted use of integrtion by prts shows tht ( 1) p c m φ(x) dx = c m p! x p φ (p) (x) dx = b m x p φ (p) (x) dx, (6.13) where b m is constnt bounded in m depending on g m but not on φ. Thus, we hve the identity (g m (x) + b m x p )φ (p) (x) dx = g m (x)ρ (p+1) 1 (x) dx, (6.14) nd it is cler tht if the lemm holds for r = p, it lso holds for r = p + 1. Therefore, it holds for ll r. We my now consider distributions on n open intervl I. We sy tht the F re bounded if, for ech φ, the F (φ) re bounded, nd we sy F m converges to F if, for ech φ, F m (φ) converges to F (φ). Now our previous work enbles us to drw relevnt conclusions bout the F (,b), F (,b) for ech (, b) I. An importnt convergence property of distributions is if F m converges to F, then F m converges to F. This follows immeditely from the fct tht F (φ) F m(φ) = (F (φ ) F m (φ )). (6.15) Thus, if F = m=1 F m, then F = m=1 F m, so tht term by term differentition is vlid without restriction for distributions. In prticulr, if f(x) is sum of uniformly convergent functions f m (x), f(x) = f m (x), (6.16) m=1 ech of which is bsolutely continuous, then f(x) my not be bsolutely continuous, but it will be summble nd therefore hve distribution derivtive f which will be the limit of N m=1 f m(x) s N. If f(x) hppens to be bsolutely continuous, then N m=1 f m(x) will converge to f (x). Let us sy tht distribution F t, depending on prmeter t, converges to F s t t 0, if, for ech φ, F t (φ) F (φ) s t t 0. Then convergence of F t to F implies tht of F t to F. Thus, strting from the function f(x) = lim t t we cn differentite twice, in the sense of distributions, to obtin 1 cos wx dw (6.17) w 2 19

20 f = lim t t 1 cos wx dw (6.18) so tht we cn give mening to t cos wx dw s distribution lthough the integrl is not 1 convergent in the usul sense. Let us clculte the vlue of 2 0 cos 2πwx dw, which, ccording to the preceding prgrph, will hve mening s distribution. We set nd observe tht g t (x) = 2 t 0 cos 2πwx dw = sin 2πtx πx (6.19) g t (φ) = sin 2πtx φ(x) dx (6.20) πx converges to φ(0) s t becomes infinite (the well known Dirichlet integrl of Fourier series), so tht g t s distribution converges to the Dirc δ. This gives the result 2 0 cos 2πwx dw = δ. (6.21) Such formuls hve been used clssiclly in electromgnetic theory, in symbolic clcultions in wve mechnics, but often without dequte explntion. As distribution formuls, they re unmbiguous nd hve rigorous vlidity. If F t is distribution depending on prmeter t, we cn define df (not to be confused with F ) dt s the distribution F for which F (φ) = d F dt t(φ) φ, if such n F exists. Similrly, we cn define t2 t 1 F t dt to be F if t 2 F t t (φ) dt exists nd equls F (φ) φ. 20

21 Chpter 7 Nullity Sets nd Supporting Sets A distribution F, even though it is more generl thn point function, cn be sid to hve locl properties. Tht is, properties which concern prticulr, but rbitrry, x nd neighborhoods of tht x. We hve lredy seen tht for ech x there is n intervl (c, d) with c < x < d such tht F hs finite order on (c, d). We sy tht prticulr x 0 I is in the nullity set of distribution F on I if F (c,d) = 0 for some c < x 0 < d. The complement with respect to I of the nullity set of F will be clled the supporting set of F. This nullity set is open nd the supporting set is closed, reltive tot he contining intervl I. If F hppens to be identifible with continuous point function f(x), then the supporting set of F will be the closure of the set of x for which f(x) 0, the closure of n open set. The nullity set will be the interior of the set of x for which f(x) = 0, the interior of closed set. But these sttements re not vlid for rbitrry loclly summble, but continuous, f(x). The zero distribution hs nullity set which includes ll x so tht its supporting set is the empty set. Conversely, the zero distribution is the only distribution with these properties. On ny (, b) I, F = f (r) with f(x) continuous on (, b) so tht F (φ) = ( 1) r f(x)φ (r) (x) dx φ S (,b), (7.1) nd therefore for ll φ S (c,d) with c < d b. If some prticulr F (c,d) = 0, the Lemm of Chpter 6 (with ll g m of the Lemm equl to f) implies tht f(x) is polynomil of degree less thn r on (c, d). By the Heine-Borel Theorem, (, b) cn be covered by finite number of such (c, d) on ech of which f(x) is polynomil of degree less thn r. Thus, f(x) is polynomil of degree less thn r on (, b) nd F (φ) = 0 φ S (,b). Therefore F = 0, s required. This type of resoning lso shows tht for prticulr φ, F (φ) = 0 in the closure of the set x for which φ(x) 0 is contined in the nullity set of F. It follows tht the vlue of F (φ) depends only on the vlues of φ(x) in the neighborhood of the supporting set of F, tht is, on the vlues of φ(x) in ny open set contining the supporting set. The preceding sttement cnnot be shrpened to imply tht F (φ) depends only on the vlues of φ(x) on the supporting set of F. For exmple, the derivtive of the Dirc δ, nmely δ, hs the origin s the only point in its supporting set, yet δ (0) = φ (0) is not determined by the vlue 21

22 of φ(0). Since on every (, b), ech F hs the form f (r) we cn show tht for ech (, b) nd ech F there is fixed r such tht the vlue of F (φ), for φ S (,b), depends only on the vlues of φ(x), φ (x),..., φ (r) (x), on the supporting set of F. Indeed, N, the nullity set of F, is n open set nd cn be expressed s the set union of finite or countble number of disjoint open intervls c n, d n. Then F (φ) = ( 1) r f(x)φ (r) (x) dx = dn ( 1) r f(x)φ (r) (x) dx + ( 1) r n c n I N f(x)φ (r) (x) dx. (7.2) On ech c n, d n, f(x) is polynomil P n (x) of degree r 1 t most, so tht, when integrted, dn c n f(x)φ (r) (x) dx depends only on the vlues of φ(x), φ (x),..., φ (r) (x) t c n nd d n. Since ll c n, d n re in the supporting set of F nd since the vlue of the integrl of f(x)φ (r) (x) over I N, the supporting set of F, depends only on the vlues of φ (r) on the supporting set of F, the desired conclusion cn be drwn. A corollry of the previous prgrph is tht distribution hs supporting set consisting of one point only, the origin, if nd only if the distribution is finite liner combintion of the Dirc δ nd its derivtives: For ny fixed x 0, let δ x0 denote the distribution N F = c p δ (p). (7.3) p=0 δ x0 (φ) = φ(x 0 ), (7.4) so tht the δ of our previous nottion coincides with δ 0. From now on, we cll ny such distributions δ x0 Dirc delt. Then it follows, s bove, tht distribution on I hs supporting set consisting of isolted points if nd only if the distribution is finite or infinite liner combintion of Dirc delts nd their derivtives such tht for ech (, b) I, only finite number of these delts nd their derivtives re tken t points in (, b). It is esily seen tht the supporting set of F contins the supporting set of F nd n nlysis of the non-incresing fmily of supporting sets of F (n) gives some informtion bout the locl structure of F. For use lter in this chpter, we require the following observtions: If < c < d < b, there is continuous function (x) with derivtives (n) (x) of ll orders such tht (x) = 0 for x c nd (x) = 1 for x d. Such function cn be obtined by defining (x) for c < x < d s ( d (x) = c ) 1 x φ c,d (t) dt φ c,d (t) dt, (7.5) c 22

23 where φ c,d ws defined in (2.5). Therefore, every φ S (,b) cn be put in the form φ 1 + φ 2, with φ 1 S (,d) nd φ 2 S (c,b) ; set φ 1 = (1 )φ nd φ 2 = φ. Repeted pplictions of this observtion show tht if (, b) cn be covered by finite number of open intervls I p = (c p, d p ), with p = 1,..., N, then every φ S (,b) cn be expressed s sum N p=1 with φ p S (cp,d p). Finlly, if ech x 0 (, b) is covered by t lest one of fmily of open intervls I, then (, b) is covered by finite number of such I by the Heine-Borel Theorem. This leds to the prtition theorem: finite number of (c, d) cn be selected, ech n I, so tht every φ cn be expressed s sum of φ p, with ech φ p in one of the S (c,d). Moreover, this cn be done in such wy tht the condition on sequence of φ tht they nd ll their nth derivtives converge uniformly to zero is equivlent to the sme condition for their φ p p. Now suppose tht T (φ), not ssumed to be distribution, is defined for certin φ s follows: for fmily of (c, d) I, T (φ) is defined for ll φ S (c,d) nd, when restricted to S (c,d) is continuous liner functionl T (c,d) on (c, d); suppose too tht ech x 0 I is covered by the interior of t lest on (c, d). In other words, suppose tht T (φ) defines distribution loclly. Then there is one nd only one distribution F on I which extends T, tht is, for which F (c,d) = T (c,d) for ll such (c, d). For n rbitrry φ cn be expressed s the sum of finite number of φ p, with φ p in some S (c,d). Then T φ p is defined for ech p, nd F, if it exists, must stisfy F (φ) = p T (φ p). 1 On the other hnd, this ctully defines F (φ) uniquely nd this F is distribution with the properties stted bove, s follows from the preceding prgrph. The preceding observtions enble us to generlize our definition of the product of two distributions. If F 1, F 2 re two distributions on I, we sy tht their product is defined loclly t x 0 if F 1(c,d) F 2(c,d) is defined, in ccordnce with our previous definition, for some (c, d) with interior covering x 0. If the product F 1 F 2 is defined loclly t every x, there is one nd only one distribution F on I such tht F (cd ) = F 1(c,d) F 2(c,d) (c, d). We define this F to be the product F 1 F 2. In order tht the product F 1 F 2 be defined loclly t x 0, it is esily seen tht necessry (but not sufficient) condition is tht x 0 must not be n isolted point in the supporting sets of F 1 nd F 2. Thus, F 1 F 2 is not defined if there re ny common isolted points in the supporting sets. For exmple, the product of δ 0 by itself is not defined. On the other hnd, F 1 F 2 will certinly be defined nd will be the zero distribution if the supporting sets hve no points in common. 2 φ p 1 This gives simple proof tht if F is distribution with nullity set tht includes ll x, then F must be the zero distribution. 2 Tht this sufficient condition is not necessry is shown by the fct tht xδ 0 is the zero distribution. 23

24 Chpter 8 Differentil Equtions Involving Distributions Consider the differentil eqution G (m) + A m 1 G (m 1) A 0 G = F (8.1) under the ssumptions tht ech A m s is bounded function of x on closed finite intervl (, b) nd tht F is continuous function if x summble on (, b). A point function G(x) is solution of the differentil eqution if G(x),..., G (m 1) re bsolutely continuous nd the G(x),..., G (m) (x) stisfy the given eqution for lmost ll x. All such solutions cn be found by the clssicl method of successive substitutions, s follows: Let H [ 1] (x) denote x H(t) dt for ny summble H(x), so tht Let H denote H [ s] (x) = x (x t) (s 1) H(t) dt. (8.2) (s 1)! (A m 1 (x)h (m 1) (x) A 0 (x)h(x)) for ny H(x) such tht H(x),..., H (m 1) (x) exist nd re bsolutely continuous. Then set nd The series G 0 (x) = F [ m] (x) (8.3) G k+1 (x) = F [ m] (x) + ( G) [ m] (x) for k 0. (8.4) G 0 (x) + (G k+1 (x) G k (x)) k=0 will converge uniformly to sum G(x) nd the first, second,..., mth differentited series will converge uniformly to G (x),..., G (m) (x) respectively. From this, it cn be shown tht G(x) is 24

25 solution of the given differentil eqution, to be clled prticulr integrl. From the construction it lso follows tht if, for ny p, ll coefficients A (p+1) m s (x) hve A m s (x),..., A (p) m s(x) bsolutely continuous, nd A (p+1) m s (x) bounded, nd F (x) hs F (x), F (x),..., F (p) (x) bsolutely continuous, then G(x) hs G(x), G (x),..., G (m+p) (x) ll bsolutely continuous. In prticulr, if the A m s (x) nd F (x) re indefinitely differentible, then G(x) is indefinitely differentible. Now m linerly independent solutions of the homogeneous eqution (setting F = 0) cn be obtined by choosing r to be in turn 0, 1,..., m 1 nd setting nd G 0 (x) = xr r!, (8.5) G k+1 (x) = ( G k ) [ m] (x), (8.6) G(x) = G 0 (x) + (G k+1 (x) G k (x)). (8.7) k=0 Without dnger of confusion, we now dopt the nottion G p (x) for the prticulr integrl nd G 0 (x),..., G m 1 (x) for the solutions of the homogeneous eqution bove. Then for ny constnts c 0,..., c m 1, the function G(x) = G p (x) + c 0 G 0 (x) c m 1 G m 1 (x) (8.8) will be solution of the given differentil eqution, nd G p will hve differentibility properties, s explined bove, depending on those of F nd the A m s, while G 0,..., G m 1 will hve differentibility properties depending on those of the A m s only. As cn be shown, there re no other point function solutions of the given differentil eqution. We now regrd the given differentil eqution s n eqution in n unknown continuous liner functionl G. The point function solutions previously found re lso continuous liner functionls. We will show tht there re no other continuous liner functionl solutions. Indeed, if the order of G (m) were greter thn zero, it would be greter thn the orders of ech G (m 1),..., G, F, nd therefore greter thn the order of F (A m 1 G (m 1) A 0 G) which is supposed to be equl to G (m). This contrdiction shows tht the order of G (m) must be zero, so tht G (m) is summble point function, nd therefore, G is point function solution s defined bove. The bove technique of solving the differentil eqution cn be pplied if F is n rbitrrily ssigned continuous liner functionl provided tht the A m s re restricted to be bounded point functions stisfying nother condition. Let F 1 denote ny primitive of F, nd by induction, F s ny primitive of F (s 1) ; of course the F s re defined only to within n rbitrry dditive polynomil of degree s 1. Now we require tht A m s F s be defined s product of two continuous liner 25

26 functionls for s = 1,..., m. This condition is certinly stisfied if the A m s (x) hppen to be ll indefinitely differentible. Under the bove ssumptions, if the order of F is greter thn zero, we rewrite the differentil eqution s where (G F m ) (m) + A m 1 (G F m ) (m 1) A 0 (G F m ) = F 1, (8.9) F 1 = (A m 1 F 1 + A m 2 F A 0 F m ). (8.10) Now we hve n eqution in G F m in which the right side F 1 is of order less thn tht of F. By repeted reductions of the order of the right side we finlly obtin differentil eqution in which the right side is summble point function. The use of ordinry point function integrls cn then be combined with the method of successive substitutions to yield the theorem: there is continuous liner functionl G p nd m linerly independent point functions G 0,..., G m 1 such tht G = G p + c 0 G c m 1 G m 1 (8.11) is continuous liner functionl solution of the given differentil eqution for rbitrry c 0,..., c m 1, nd there re no other continuous liner functionl solutions. The order of G (m) p is precisely the order of F nd the G 0 (x),..., G m 1 (x), s point function solutions of the homogeneous eqution, hve differentibility properties on those of the A m s (x). This discussion of continuous liner functionl solutions on ech (, b) leds without difficulty to the corresponding distribution solutions of the differentil eqution on n open intervl I. The more generl eqution A m (x)g (m) + A m 1 (x)g (m 1) A 0 (x)g = F (8.12) cn be reduced to the cse A m = 1, providing tht the coefficient A m (x) cn be fctored out. In this cse, the system of solutions is s described bove. However, in the singulr cse, where A m (x) vnishes t certin points, quite different results my be obtined. Under some circumstnces, the homogeneous eqution my hve no solutions other thn the trivil one - the zero distribution. Under other circumstnces, the solutions my depend on number of rbitrry prmeters greter thn the order of the eqution. For exmple, consider the eqution xg + G = 0. (8.13) This cn be written s (xg) = 0 nd the solutions must stisfy xg = k for some constnt k. This seems to give one-prmeter fmily of solutions G = k/x. Although these re solutions in certin sense, they re not bsolutely continuous over ny intervl including the origin, nd they re not loclly summble functions nd hence cnnot be identified with distributions. From the point of view of distributions, the eqution xg = k does hve solutions, for instnce kφ(x) G(φ) = p.v. dx (8.14) x 26

27 where p.v. indictes the Cuchy principl vlue. If we denote this solution by k ( p.v. 1 x), then ll other solutions differ from it by solution of the eqution xg 1 = 0. Such G 1 must hve supporting set consisting of one point, the origin, nd therefore hs the form Then G 1 = N r=0 c r δ (r) 0. (8.15) xg 1 = N ( c r r=0 xδ (r) 0 ) = N r=0 c r ( r)δ (r 1) 0 (8.16) if nd only if c r = 0 for r > 0. Thus, G 1 = c 0 δ 0 is the set of ll such solutions G 1. Therefore, the solutions of eqution (8.13) consists precisely of ll distributions ( G = k 1 p.v. 1 ) + k 2 δ 0 (8.17) x nd depends on two independent prmeters. 27

28 Chpter 9 Distributions in Severl Vribles - Exmples The preceding theory cn be extended to distributions in n vribles where n N. In this chpter, we give definitions nd some exmples. The next chpter will sketch some of the theory. Let x now denote n n-tuple of rel numbers: x = (x i ) = (x i,..., x n ) which my be thought of s point in n-dimensionl spce. x y will men x i y i i. If = ( i ), b = (b i ) re rbitrry but fixed, with i < b i i, we let R denote the closed rectngle x b. S R will denote the set of ll continuous functions φ(x) = φ(x 1,..., x n ) which possess continuous prtil derivtives of ll orders: φ (p) (x) = φ (p 1,...,p n) (x 1,..., x n ) = p p n ( x 1 ) p 1... ( xn ) pn φ(x 1,..., x n ) (9.1) nd which, together with ll φ (p) (x), vnish on R nd outside R. A liner functionl F (φ) defined φ S R will be clled continuous liner functionl on R if: whenever φ, φ m re in S R nd for ech p, φ (p) m (x) converges uniformly to φ (p) (x), then F (φ m ) converges to F (φ). If Ω is ny bounded or unbounded open rectngle in n-dimensionl spce (for exmple, ll n- dimensionl spce), we define F to be distribution on Ω if, for ech R Ω, F (φ) is defined for φ S R nd, when restricted to such φ is continuous liner functionl to be denoted by F R. For ny distribution F on Ω nd ny p, we define prtil distribution derivtive on Ω by the formul F (p) = ( 1) p p n F (φ (p) ). (9.2) Then every distribution hs prtil derivtives of ll orders nd the result of successive differentition is lwys independent of the order of differentition. A distribution F on Ω will be identified with point function f(x) if, for ech R Ω, f(x) is summble on R, nd F (φ) = f(x) dx =... f(x 1,..., x n )φ(x 1,..., x n ) dx 1... dx n (9.3) for ech φ S R, the integrtion being tken over R or, equivlently, over ll n-dimensionl spce. 28

29 Identifiction of certin distributions with Stieltjes mesures in one or more of the x i cn lso be mde. To tke one exmple, suppose ψ(x 1, x 2, x 3,..., x n ) is rel vlued function nd suppose tht on ech R Ω the following holds: for lmost ll fixed (x 3,..., x n ), the function ψ considered s function of x 1, x 2 hs finite vritions v + (x 3,..., x n ), v (x 3,..., x n ) which re summble over R. Then ψ determines distribution F on Ω by the formul F (φ) =... dx 3... dx n ( ) φ(x 1,..., x n ) d x1 x 2 ψ(x 1, x 2,..., x n ) (9.4) A set of distributions F will be sid to be bounded if, for ech fixed φ, the F (φ) re bounded. F m will be sid to converge s m if F m (φ) converges s m, for ech fixed φ; in which cse, s will be shown in the next chpter, the limiting vlue of F m (φ) will determine limit distribution of F (φ). Similrly, F t will be sid to converge to F s t t 0 if F t (φ) F (φ) for ech fixed φ. A series m=1 F m will be sid to hve sum F if h m=1 F m converges to F s h. Continuity, respectively convergence, of distributions implies tht of their distribution derivtives of ll orders so tht term by term differentition is lwys vlid. A point x 0 will be sid to be in the nullity set of F if for some R which covers x 0, F R is the identiclly zero continuous liner functionl. The complement of the nullity set of F will be clled the supporting set of F. The sum of two distributions nd constnt times distribution re defined in the obvious wy. In the next chpter we will develop some of the theory of distributions in n vribles. We conclude this section with exmples. For fixed x 0, let F be the distribution on ll n-dimensionl spce: F (φ) = φ(x 0 ). This is generliztion of the Dirc δ function. It cn be identified with the n-dimensionl Stieltjes mesure d x1...x n H(x 1,..., x n ), where H(x) is the Heviside function: H(x) = { 0 for x0 < x 1 for x 0 x. In our terminology, this Dirc δ function, δ x0, is the pth derivtive of H(x), where p = (1, 1,..., 1). In terms of physicl concepts such s mss nd chrge distribution, this δ distribution corresponds to unit of mss or chrge concentrted t one point, x 0. If we use the sme H(x) but tke the pth derivtive for ny p = (p i ) with p i = 0 or 1 for ech i, we obtin vriety of mixed δ distributions: thus, H x 1 is the distribution F (φ) =... φ(x 01, x 2,..., x n ) dx 2... dx n, (9.6) the integrtion being tken over the n 1 dimensionl qudrnt x 1 = x 01, x i x 0i for i > 1. In physicl terms, this corresponds to n n 1 dimensionl surfce distribution with unit surfce density on the qudrnt. The Heviside function defined bove hs the vlue 1 on n n-dimensionl qudrnt nd vnishes outside the qudrnt. We now generlize by replcing the qudrnt nd vnishes outside the qudrnt. We now generlize by replcing the qudrnt by rectngle or sphere or ny other 29 (9.5)