Full algebra of generalized functions and non-standard asymptotic analysis

Transcription

1 Full algebra f geeralized fuctis ad -stadard asympttic aalysis Tdr D. Tdrv Has Veraeve Abstract We cstruct a algebra f geeralized fuctis edwed with a caical embeddig f the space f Schwartz distributis. We ffer a sluti t the prblem f multiplicati f Schwartz distributis similar t but differet frm Clmbeau s sluti. We shw that the set f scalars f ur algebra is a algebraically clsed field ulike its cuterpart i Clmbeau thery, which is a rig with zer divisrs. We prve a Hah Baach extesi priciple which des t hld i Clmbeau thery. We establish a cecti betwee ur thery with -stadard aalysis ad thus aswer, althugh idirectly, a questi raised by Clmbeau. This article prvides a bridge betwee Clmbeau thery f geeralized fuctis ad -stadard aalysis. Keywrds Schwartz distributis Geeralized fuctis Clmbeau algebra Multiplicati f distributis N-stadard aalysis Ifiitesimals Ultrapwer -stadard mdel Ultrafilter Maximal filter Rbis valuati field Ultra-metric Hah Baach therem

2 1 Itrducti I the early 1970s, Rbis itrduced a real clsed, -archimedea field ρ R [33] as a factr rig f -stadard umbers i R [32]. The field ρ R is kw as Rbis field f asympttic umbers (r Rbis valuati field), because it is a atural framewrk f the classical asympttic aalysis [20]. Later Luxemburg [23] established a cecti betwee ρ R ad p-adic aalysis (see als the begiig f Sect. 8 i this article). Li Bag-He [19] studied the cecti betwee ρ R ad the aalytic represetati f Schwartz distributis, ad Pestv [30] ivlved the field ρ R ad similar cstructis i the thery f Baach spaces. Mre recetly, it was shw that the field ρ R is ismrphic t a particular Hah field f geeralized pwer series [38]. The algebras ρ E(Q) f ρ-asympttic fuctis were itrduced i [28] ad studied i Tdrv [37]. It is a differetial algebra ver Rbis s field ρ C ctaiig a cpy f the Schwartz distributis D ' (Q) [41]. Applicatis f ρ E(Q) t partial differetial equatis were preseted i Oberguggeberger [27]. We smetimes refer t the mathematics assciated directly r idirectly with the fields ρ R as stadard asympttic aalysis. O the ther had, i the early 1980s, Clmbeau develped his thery f ew geeralized fuctis withut ay cecti, at least iitially, with -stadard aalysis [6 10]. This thery is kw as Clmbeau thery r -liear thery f geeralized fuctis because it slves the prblem f the multiplicati f Schwartz distributis. Hereisa summary f Clmbeau thery preseted i aximatic like fashi: Let T d dete the usual tplgy R d ad let G be a pe set f R d.a set G(G) is called a special algebra f geeralized fuctis G (f Clmbeau type) if there exists a family G =: {G(Q)} Q T d (we use =: fr equal by defiiti ) such that: 1. Each G(Q) is a cmmutative differetial rig, that is, G(Q) is a cmmutative rig supplied with partial derivatives α,α N d 0 (liear peratrs beyig the chai rule). Here N 0 ={0, 1, 2,...}. Let C dete the rig f geeralized scalars f the family G defied as the set f the fuctis i G(R d ) with zer gradiet. Each G(Q) becmes a differetial algebra ver the rig C (hece, the termilgy Clmbeau algebras, fr shrt). 2. The rig f geeralized scalars C is f the frm C = R ir, where R is a partially rdered real rig, which is a prper extesi f R. (Real rig meas a rig with the prperty that a a 1 + +a = 0 implies a 1 = a 2 = =a = 0). The frmula x + iy = x 2 + y 2 defies a abslute value C. Csequetly, C is a prper extesi f C ad bth R ad C ctai -zer ifiitesimals. I Clmbeau thery the ifiitesimal relati i C is called assciati. 3. C is spherically cmplete uder sme ultra-metric d v C which agrees with the partial rder i R i the sese that z 1 < z 2 implies d v (0, z 1 ) d v (0, z 2 ). 4. Fr every f G(Q) ad every test fucti τ D(Q) a pairig ( f τ) C is defied (with the usual liear prperties). Here D(Q) stads fr the class f C fuctis frm Q t C with cmpact supprts. Let f, g G(Q). The fuctis f ad g are called weakly equal (r equal i the sese f geeralized distributis), i symbl f = g, if ( f τ) = (g τ) fr all τ D(Q). Similarly, f ad g

3 are weakly assciated (r simply, assciated, fr shrt), i symbl f g, if ( f τ) (g τ) fr all τ D(Q), where i the latter frmula stads fr the ifiitesimal relati i C. 5. The family G is a sheaf. That meas that G is supplied with a restricti I t a pe set (with the usual sheaf prperties, cf. [16]) such that T d 3 O Q ad f G(Q) implies f I O G(O). Csequetly, each geeralized fucti f G(Q) has a supprt supp( f ) which is a clsed subset f Q. 6. Let Q, Q ' T d ad Diff(Q ',Q) dete the set f all C -diffemrphisms frm Q ' t Q (C -bijectis with C -iverse). A cmpsiti (chage f variables) f ψ G(Q ' ) is defied fr all f G(Q) ad all ψ Diff(Q ',Q). 7. Fr every Q T d there exists a embeddig E Q : D ' (Q) G(Q) f the space f Schwartz distributis D ' (Q) it G(Q) such that: (a) E Q preserves the vectr peratis ad partial differetiati i D ' (Q); (b) E Q is sheaf-preservig, that is, E Q preserves the restricti t pe sets. Csequetly, E Q preserves the supprt f the Schwartz distributis. (c) E Q preserves the rig peratis ad partial differetiati i the class E(Q). Here E(Q) =: C (Q) stads fr the class f C -fuctis frm Q t C (where E(Q) is treated as a subspace f D ' (Q)). (d) E Q preserves the pairig betwee D ' (Q) ad the class f test fuctis D(Q). Csequetly, E Q preserves weakly the Schwartz multiplicati i D ' (Q) (multiplicati by duality). (e) E Q preserves the usual multiplicati i the class f ctiuus fuctis C(Q) up t fuctis i G(Q) that are weakly assciated t zer. (f) E Q preserves weakly the cmpsiti with diffemrphisms (chage f variables) i the sese that fr every Q, Q ' T d,every T D ' (Q) ad every ψ Diff(Q ',Q) we have (E Q (T ) ψ τ) = (E Q ' (T ψ) τ) fr all test fuctis τ D(Q ' ).Here T ψ stads fr the cmpsiti i the sese f the distributi thery [41]. 8. A special algebra is called a full algebra f geeralized fuctis (f Clmbeau type) if the embeddig E Q is caical i the sese that E Q ca be uiquely determied by prperties expressible ly i terms which are already ivlved i the defiiti f the family G =: {G(Q)} Q T d. 9. A family G ={G(Q)} Q T d f algebras f geeralized fuctis (special r full) is called diffemrphism-ivariat if E Q preserves the cmpsiti with diffemrphisms i the sese that E Q (T ) ψ = E Q ' (T ψ) fr all Q, Q ' T d,all T D ' (Q) ad all ψ Diff(Q ',Q). We shuld meti that embeddigs E Q (caical r t) f the type described abve are, i a sese, ptimal i view f the restricti impsed by the Schwartz impssibility results [35]. Fr a discussi the tpic we refer t [10, p.8]. Every family f algebras G(Q) (special r full) f the type described abve ffers a sluti t the prblem f the multiplicati f Schwartz distributis because the Schwartz distributis ca be multiplied withi a assciative ad cmmutative differetial algebra. Full algebras f geeralized fuctis were cstructed first by Clmbeau [6]. Several years later, i a attempt t simplify Clmbeau s rigial cstructi Clmbeau

4 ad Le Rux [8] (ad ther authrs [2]) defied the s called simple algebras f geeralized fuctis. Later Oberguggeberger [26, Chap.III, Sect. 9] prved that the simple algebras are, actually, special algebras i the sese explaied abve. Diffemrphism ivariat full algebras were develped i Grsser et al. [13] ad als Grsser et al. [14]. The sets f geeralized scalars f all these algebras are rigs with zer divisrs [6, pp. 136]. The algebras f ρ-asympttic fuctis ρ E(Q) [28], metied earlier, are special algebras f Clmbeau type with set f geeralized scalars which is a algebraically clsed field. The cuterpart f the embeddig E Q i [28] is deted by I D,Q. It is certaily t caical because the existece f I D,Q is prved i [28] by the saturati priciple (i a -stadard aalysis framewrk) ad the fixed by had (see Remark 7.9). Amg ther thigs the purpse f this article is t cstruct a caical embeddig E Q i ρ E(Q). We achieve this by meas f the chice f a particular ultra-pwer -stadard mdel (Sect. 6) ad a particular chice f the psitive ifiitesimal ρ withi this mdel (Defiiti 6.1, #12). Clmbeau thery has umerus applicatis t rdiary ad partial differetial equatis, the thery f elasticity, fluid mechaics, thery f shck waves [6 10,26], t differetial gemetry ad relativity thery [14] ad, mre recetly, t quatum field thery [11]. Despite the remarkable achievemet ad prmisig applicatis the thery f Clmbeau has sme features which ca be certaily imprved. Here are sme f them: (a) The rig f geeralized scalars C ad the algebras f geeralized fuctis G(Q) i Clmbeau thery are cstructed as factr rigs withi the ultrapwers C I ad E(Q) I, respectively, fr a particular idex set I. The rigs f ets such as C I ad E(Q) I hwever (as well their subrigs) lack geeral theretical priciples similar t the axims f R ad C, fr example. Neither C I, r E(Q) I are edwed with priciples such as the trasfer priciple r iteral defiiti priciple i -stadard aalysis. Fr that reas Clmbeau thery has t bee able s far t get rid f the idex set I eve after the factrizati which trasfrms C I ad E(Q) I it C ad G(Q), respectively. As a result Clmbeau thery remais verly cstructive: there are t may techical parameters (with rigi i the idex set I ) ad t may quatifiers i the defiitis ad therems. (b) I a recet article Oberguggeberger ad Veraeve [29] defied the ccept f iteral sets f C ad G(Q) ad shwed that theretical priciples similar t rder cmpleteess, uderflw ad verflw priciples ad saturati priciple fr iteral sets f C ad G(Q) hld i Clmbeau thery as well althugh i mre restrictive sese cmpared with -stadard aalysis. Hwever the sets f geeralized scalars fr R ad C are still rigs with zer divisrs ad R is ly a partially rdered (t ttally rdered) rig. These facts lead t techical cmplicatis. Fr example Hah Baach extesi priciples d t hld i Clmbeau thery [40]. I this article: (i) We cstruct a family f algebras f geeralized fuctis E Q (Q) D 0 called asympttic fuctis (Sect. 4). We shw that E Q (Q) D 0 are full algebras f

5 Clmbeau type (Sect. 5) i the sese explaied abve. Thus we ffer a sluti t the prblem f the multiplicati f Schwartz distributis similar t but differet frm Clmbeau s sluti [6]. Sice the full algebras are cmmly csidered t be mre aturally cected t the thery f Schwartz distributis tha the special algebras, we lk up E Q (Q) D 0 as a imprved alterative t the algebra f ρ-asympttic fuctis ρ E(Q) defied i [28]. (ii) We believe that ur thery is a mdified ad imprved alterative t the rigial Clmbeau thery fr the fllwig reass: (a) The set f scalars - C D 0 f the algebra E Q (Q) D 0, called here asympttic umbers, is a algebraically clsed field (Therem 4.2). Recall fr cmparis that its cuterpart i Clmbeau thery C is a rig with zer divisrs [6, pp. 136]. (b) As a csequece we shw that a Hah Baach extesi priciple hlds fr liear fuctials with values i - C D 0 (Sect. 8). This result des t have a cuterpart i Clmbeau thery [40]. (c) At this stage the cstructi f E Q (Q) D 0 is already simpler tha its cuterpart i Clmbeau [6]; ur thery has e (regularizati) parameter less. (iii) Our ext gal is t simplify ur thery eve mre by establishig a cecti with -stadard aalysis (Sect. 7). Fr this purpse we cstruct a particular ultrapwer -stadard mdel called i this article the distributial stadard mdel (Sect. 6). The we replace the rigs f ets C I ad E(Q) I i Clmbeau thery by the -stadard C ad E(Q), respectively ad the regularizati parameter ε i Clmbeau thery by a particular (caical) ifiitesimal ρ i R. We shw that the field f asympttic umbers - C D 0 (defied i Sect. 4) is ismrphic t a particular Rbis field ρ C [33]. We als prve that the algebra f asympttic fuctis E Q (Q) D 0 (defied i Sect. 4) is ismrphic t a particular algebra f ρ-asympttic fuctis ρ E(Q) itrduced i [28] i the framewrk f -stadard aalysis. (iv) Amg ther thigs this article prvides a bridge betwee Clmbeau thery f geeralized fuctis ad -stadard aalysis ad we hpe that it will be beeficial fr bth. After all Rbis s -stadard aalysis [32] is histrically at least several decades lder tha Clmbeau thery. A lt f wrk had bee already de i the -stadard settig tpics similar t thse which appear i Clmbeau thery. By establishig a cecti with -stadard aalysis we aswer, althugh idirectly, a questi raised by Clmbeau himself i e f his research prjects [10, pp. 5]. Sice the article establishes a cecti betwee tw differet fields f mathematics, it is writte mstly with tw types f readers i mid.the readers with backgrud i -stadard aalysis might fid i Sects. 2 5 ad 8 (alg with the aximatic summary f Clmbeau thery preseted abve) a shrt itrducti t the -liear thery f geeralized fuctis. Ntice hwever that i these sectis we d t preset the rigial Clmbeau thery but rather a mdified (ad imprved) versi f this thery. The reader withut backgrud i -stadard aalysis will fid i

6 Sect. 6 a shrt itrducti t the subject. The readig f Sects. 2 5 des t require backgrud i -stadard aalysis. 2 Ultrafilter test fuctis I this secti we defie a particular ultrafilter the class f test fuctis D(R d ) clsely related t Clmbeau thery f geeralized fuctis [6]. We shall fte use the shrter tati D 0 istead f D(R d ). I what fllws we dete by R ϕ the radius f supprt f ϕ D(R d ) defied by sup{ x : x R d,ϕ(x) = 0}, ϕ = 0, R ϕ = (1) 1, ϕ = 0. Defiiti 2.1 (Directig sets). We defie the directig sequece f sets D 0, D 1, D 2... by lettig D 0 = D(R d ) ad D = ϕ D(R d ) : ϕ is real-valued, ( x R d )(ϕ( x) = ϕ(x)), R ϕ 1/, ϕ(x) dx = 1, R d α ( α N d 0 ) 1 α x ϕ(x) dx = 0, R d 1 ϕ(x) dx 1 +, R d ( α N d 0 ) α sup α ϕ(x) (R ϕ ) 2( α +d), = 1, 2,... x R d Therem 2.2 (Base fr a filter). The directig sequece (D ) is a base fr a free filter D 0 i the sese that (i) D(R d ) = D 0 D 1 D 2 D 3. (ii) D = fr all N. (iii) 0 D =. = Prf (i) Clear. (ii) Let ϕ 0 D(R) be the test fucti 1 1 c exp( 1 x 2 ), 1 x 1, ϕ 0 (x) = 0, therwise,

7 1 where c = 1 exp( 1 ) dx.welet C 1 x 2 k =: sup x R dk dx k ϕ 0 (x) fr each k N 0 ad als C α = C α1 C αd fr each multi-idex α N d 0. Fr each, m N we let B,m,d = ϕ D(R d ) : ϕ is real-valued, ϕ( x) = ϕ(x) fr all x R d, R ϕ = d, R d R d R d ϕ(x) dx = 1, α x ϕ(x) dx = 0 fr all α N d with 1 α, 3d ϕ(x) dx exp, m 1 α +d sup α ϕ(x) C α (2 d m ) fr all α N d. x R d Step 1. We shw that, if m > 2, the B,m,d =. Let first d = 1. The ϕ 0 B 0,m,1. By iducti,let ϕ 1 B 1,m,1. Defie ϕ (x) = aϕ 1 (x) + bϕ 1 (mx),fr sme cstats a, b R t be determied. The b b ϕ (x) dx = a + ad x ϕ (x) dx = a + x ϕ 1 (x) dx. m m +1 R R R b m m +1 T esure that ϕ B,m,1, we chse a + b = 1 ad a + 0 = 0. Slvig fr a, b, we fid that a = m 1 1 < 0 ad b = m m+1 1 > 0. Sice a = 0, als R ϕ = 1. Further, sice 1+x 1 x 1 + 3x exp(3x) if 0 x 1 3,wehave R s iductively, b ϕ (x) dx a + ϕ 1 (x) dx m R m = ϕ 1 (x) dx exp ϕ 1 (x) dx, m 1 m R R D ϕ (x) dx exp ϕ 0 (x) dx exp = exp. m j m j m 1 R j=1 R j=1

8 k m +k+1 +1 m 1 Further, a + b m = 2m k+1 fr k 0, m > 2 ad 1. Thus we have d k sup ϕ (x) ( a + b m ) sup ϕ 1 (x) dx k dx k x R k d k x R k+1 1 k+1 ) 2m k+1 C k (2m = C k (2m ). Hece ϕ B,m,1.Nw let d N ad ϕ B,m,1 arbitrary. We have ψ(x) =: ϕ(x 1 ) ϕ(x d ) B,m,d. Step 2. Fix d N.Let 1, let M = max{1, max α C α },let ψ B,9d,d,let 1 1 ε = ad let ϕ(x) = ε d ψ(x/ε). We shw that ϕ D.If x 1/ dm(18d) d ε 1 d, the ϕ(x) = 0. Further, sice exp(x) if 0 x < 1, we have R d R d ϕ(x) dx = ψ(x) dx exp 3d 3d d 1 9d 1 3d Fially, tice that R ϕ = εr ψ = ε d. Thus fr α we have α d sup α ϕ(x) ε sup α ψ(x) ε α d C α (2 d (9d) α +d ) ε α d C α (18d) d( α +d) α d = ε α d C α (εdm) x R d x R d 2( α +d) C α M 1 (R ϕ ). Hece ϕ D as required. (iii) Suppse ( the ctrary) that there exists ϕ =1 D. That meas (amg ther thigs) that R d ϕ(x)x α dx = 0 fr all α = 0. Thus we have α ; ϕ(0) = 0 fr all α = 0, where ; ϕ detes the Furier trasfrm f ϕ. It fllws that ; ϕ = C fr sme cstat C C sice ;ϕ is a etire fucti C d by the Paley Wieer Therem [3, Therem 8.28, pp. 97]. Hece by Furier iversi, ϕ = (2π) d Cδ D(R d ), where δ stads fr the Dirac delta fucti. The latter implies C = 0, thus ϕ = 0, ctradictig the prperty R d ϕ(x)dx = 1 i the defiiti f D. 1 x I what fllws c =: card(r) ad c + stads fr the successr f c. Therem 2.3 (Existece f ultrafilter). There exists a c + -gd ultrafilter (maximal filter) U D 0 =: D(R d ) such that D U fr all N 0 (Defiiti 2.1). Prf We bserve that card(d 0 ) = c. The existece f a (free) ultrafilter ctaiig all D fllws easily by Zr s lemma sice the set F ={A P(D 0 ) : D A fr sme N 0 } is clearly a free filter D 0.Here P(D 0 ) stads fr the pwer set f D 0. Fr the existece f a c + -gd ultrafilter ctaiig F we refer the reader t [5] (fr a presetati we als meti the Appedix i Lidstrøm [21]). Let U be a c + -gd ultrafilter D 0 =: D(R d ) ctaiig all D. We shall keep U fixed t the ed f this article.

9 Fr thse readers wh are ufamiliar with the used termilgy we preset a list f the mst imprtat prperties f U. The prperties (1) (3) belw express the fact that U is a filter, the prperty (1) (4) express the fact that U is a free filter, the prperty (1) (5) meas that U is a free ultrafilter (maximal filter) ad (6) expresses the prperty f U t be c + -gd. Lemma 2.4 (List f Prperties f U). The ultrafilter U is a set f subsets f D 0 = D(R d ) such that D U fr all N 0 ad such that: 1. If A U ad B D 0, the A B implies B U. 2. U is clsed uder fiite itersectis. 3. / U. Csequetly, U has the fiite itersecti prperty. 4. U is a free filter i the sese that A U A =. 5. Let A k P(D 0 ), k = 1, 2,...,, fr sme N. The k= 1 A k U implies A k U fr at least e k. Mrever, if the sets A k are mutually disjit, the k=1 A k U implies A k U fr exactly e k. I particular, fr every set A P(D 0 ) exactly e f A U r D 0 \ A U is true. 6. U is c + -gd i the sese that fr every set f D 0, with card(f) c, ad every reversal R : P ω (f) U there exists a strict reversal S : P ω (f) U such that S(X) R(X) fr all X P ω (f).here P ω (f) detes the set f all fiite subsets f f. Recall that a fucti R : P ω (f) U is called a reversal if X Y implies R(X) R(Y ) fr every X, Y P ω (f).a strict reversal is a fucti S : P ω (f) U such that S(X Y ) = S(X) S(Y ) fr every X, Y P ω (f). It is clear that every strict reversal is a reversal (which justifies the termilgy). Defiiti 2.5 (Almst everywhere). Let P(x) be a predicate i e variable defied D 0 (expressig sme prperty f the test fuctis). We say that P(ϕ) hlds almst everywhere i D 0 r, simply, P(ϕ) a.e. (where a.e. stads fr almst everywhere ), if {ϕ D 0 : P(ϕ)} U. Example 2.6 (Radius f supprt). Let R ϕ be the supprt f ϕ (cf. (1)) ad let N. The (R ϕ R + & R ϕ < 1/) a.e. because D {ϕ D 0 : R ϕ R + & R ϕ < 1/} implies {ϕ D 0 : R ϕ R + & R ϕ < 1/} U by #1 f Lemma 2.4. The justificati f the termilgy almst everywhere is based the bservati that the mappig M U : P(D 0 ) {0, 1}, defied by M U (A) = 1if A U ad M U (A) = 0if A / U is fiitely additive prbability measure D 0. 3 D 0 -Nets ad Schwartz distributis Defiiti 3.1 (Idex set ad ets). Let D 0, D 1, D 2,... be the directig sequece defied i (Defiiti 2.1), where D 0 = D(R d ).Let S be a set. The fuctis f the frm A : D 0 S are called D 0 -ets i S r, simply ets i S fr shrt [17,p.65].We dete by S D 0 the set f all D 0 -ets i S. The space f test fuctis D 0 is the idex set f the ets. If A S D 0 is a et i S, we shall fte write A ϕ ad (A ϕ ) istead f A(ϕ) ad A, respectively.

10 I this secti we preset several techical lemmas abut D 0 -ets which are clsely related t the thery f Schwartz distributis ad the directig sequece (D ) (Sect. 2). Our termilgy ad tati i distributi thery is clse t thse i Vladimirv [41]. We start with several examples f D 0 -ets. Example 3.2 (Nets ad distributis). 1. We dete by C D 0 the set f all ets f the frm A : D 0 C. We shall fte write (A ϕ ) istead f A fr the ets i C D 0. It is clear that C D 0 is a rig with zer divisrs uder the usual pitwise peratis. Ntice that the ets i C D 0 ca be viewed as cmplex valued fuctials (t ecessarily liear) the space f test fuctis D(R d ). 2. Let Q be a pe subset f R d ad E(Q) =: C (Q). We dete by E(Q) D 0 the set f all ets f the frm f : D 0 E(Q). We shall fte write ( f ϕ ) r ( f ϕ (x)) istead f f fr the ets i E(Q) D Let S be a set ad P(S) stad fr the pwer set f S. We dete by P(S) D 0 the set f all ets f the frm A : D 0 P(S). We shall fte write (A ϕ ) istead f A fr the ets i P(S) D Let T d dete the usual tplgy R d. Fr every pe set Q T d we let Q ϕ = x Q d(x, Q) > R ϕ, Q ϕ = x Q d(x, Q) > 2R ϕ & x < 1/R ϕ, where d(x, Q) stads fr the Euclidea distace betwee x ad the budary Q f Q ad R ϕ is defied by (1). Let χ Q,ϕ : R d R be the characteristic fucti f the set Q ϕ.the cut-ff et (C Q,ϕ ) E(R d ) D 0 assciated with Q is defied by the frmula C Q,ϕ =: χ Q,ϕ *ϕ, where * stads fr the usual cvluti, that is, C Q,ϕ (x) = ϕ(x t) dt, Q ϕ fr all x R d ad all ϕ D 0. Ntice that supp(c Q,ϕ ) Q ϕ [41, Chap. I, Sects. 4, 6.T]. 5. Let T D ' (Q) be a Schwartz distributi Q. The ϕ-regularizati f T is the et (T ϕ ) E(Q) D 0 defied by the frmula T ϕ =: T ϕ, where T ϕ is a shrt tati fr (C Q,ϕ T )*ϕ ad * stads (as befre) fr the usual cvluti. I ther wrds, we have T ϕ (x) = T (t) C Q,ϕ (t)ϕ(x t), fr all x Q ad all ϕ D 0.Here ( ) stads fr the pairig betwee D ' (Q) ad D(Q) [41]. 6. We dete by L Q : L lc (Q) D ' (Q) the Schwartz embeddig f L lc (Q) it D ' (Q) defied by L Q ( f ) = T f.here T f D ' (Q) stads fr the (regular) distributi with kerel f, that is, (T f τ) = f (x)τ(x) dx fr all τ D ' (Q) [41]. Q

11 Als, L lc (Q) detes the space f the lcally itegrable (Lebesgue) fuctis frm Q t C. Recall that L Q preserves the additi ad multiplicati by cmplex umbers. The restricti f L Q E(Q) preserves als the partial differetiati (but t the multiplicati). We shall write f ϕ ad f *ϕ istead f T f ϕ ad T f *ϕ, respectively. Thus fr every f L lc (Q),every ϕ D 0 ad every x Q we have ( f ϕ)(x) = f (t)c Q,ϕ (t)ϕ(x t) dt. (2) x t <R ϕ I what fllws we shall fte write K < Q t idicate that K is a cmpact subset f Q. Lemma 3.3 (Lcalizati). Let Q be (as befre) a pe set f R d ad T D ' (Q) be a Schwartz distributi. The fr every cmpact set K Q there exists N 0 such that fr every x K ad every ϕ D we have: (a) C Q,ϕ (x) = 1. (b) (T ϕ)(x) = (T *ϕ)(x). (c) Csequetly, ( K < Q)( α N d 0 )( N 0)( x K )( ϕ D ) we have α (T ϕ)(x) = ( α T ϕ)(x) = (T α ϕ)(x). Prf (a) Let d(k, Q) dete the Euclidea distace betwee K ad Q. It suffices t chse N such that 3/ < d(k, Q) ad > sup x K x + 1. It fllws that 3R ϕ < d(k, Q) fr all ϕ D because R ϕ 1/ hlds by the defiiti f D. Nw (a) fllws frm the prperty f the cvluti [41, Chap. I, Sects. 4, 6.T]. (b) If K < Q, the there exists m N such that L =: {t Q : d(t, K ) 1/m} < Q. Hece, by part (a), there exists N (with m) such that C Q,ϕ (x)ϕ(x t) = ϕ(x t) fr all x K,all t Q ad all ϕ D. (c) fllws directly frm (b) bearig i mid that we have α (T *ϕ)(x) = ( α T * ϕ)(x) = (T * α ϕ)(x). Lemma 3.4 (Schwartz distributis). Let Q be a pe set f R d ad T D ' (Q) be a Schwartz distributi. The fr every cmpact set K Q ad every multi-idex α N d 0 there exist m, N 0 such that fr every ϕ D we have sup x K α (T ϕ)(x) (R ϕ ) m. Prf Let K ad α be chse arbitrarily. By Lemma 3.3, there exists q N such that α (T ϕ)(x) = ( α T *ϕ)(x) fr all x K ad all ϕ D q.let O be a pe relatively cmpact subset f Q ctaiig K ad let k N be greater tha 1/d(K, O). We bserve that ϕ x D(O) fr all x K ad all ϕ D k, where ϕ x (t) =: ϕ(x t t). O the ther had, there exist M R + ad b N 0 such that ( α T τ) M β b sup t O β τ(t) fr all τ D(O) by the ctiuity f α T. t Thus ( α T *ϕ)(x) = ( α T ϕ x (t)) M β b sup t R d β ϕ(t) fr all x K ad all ϕ D k. With this i mid we chse m = 2(b+d)+1 ad max{q, k, C, b},

12 t where C = M β b 1. Nw, fr every x K ad every ϕ D we have 2( β +d) m α 2(b+d) (T ϕ)(x) M (R ϕ ) C(R ϕ ) (R ϕ ), β b as required, where the last iequality hlds because R ϕ 1/ by the defiiti f D (Defiiti 2.1) ad 1/ 1/C by the chice f. Lemma 3.5 (C -Fuctis). Let Q be a pe set f R d ad f E(Q) be a C fucti. The fr every cmpact set K Q, every multi-idex α N d 0 ad every p N there exists N 0 such that fr every ϕ D we have sup α ( f ϕ)(x) α f (x) (R ϕ ) p. x K Prf Suppse that p N, K < Q ad α N d 0. By Lemma 3.3, there exists q N 0 such that α ( f ϕ)(x) = ( α f *ϕ)(x) fr all x K ad all ϕ D q. As befre, let O be a pe relatively cmpact subset s f Q ctaiig K ad let 2C k N be greater tha 1/d(K, O). Let max p, q, k, (p+1)!, where C =: t β = p+1 sup ξ O ( α+β f )(ξ). Let x K ad ϕ D. By ivlvig the defiiti f the sets D, we calculate: α ( f ϕ)(x) α f (x) = (Lemma 3.3 ad q) = ( α f *ϕ)(x) α f (x) = (sice 1) = α f (x y) α f (x) ϕ(y) dy = y R ϕ Taylr expasi fr sme t [0, 1] p ( 1) β α+β f (x) β ( 1) p+1 = y ϕ(y) dy + β! (p + 1)! β =1 y R ϕ _ =0 sice p β y ϕ(y) α+β f (x yt) dy β =p+1 y Rϕ

13 Rϕ p+1 Rϕ p+1 Rϕ p+1 = C ϕ(y) dy C (1 + 1/) < 2C R ( p + 1)! (p + 1)! (p + 1)! ϕ y R ϕ p, as required, where the last iequality fllws frm R ϕ 1/ (p + 1)!/2C. Lemma 3.6 (Pairig). Let Q be a pe set f R d,t D ' (Q) be a Schwartz distributi ad τ D(Q) be a test fucti. The fr every p N there exists N 0 such that fr every ϕ D we have (T ϕ τ) (T τ) (R ϕ ) p. (3) Prf Let p N ad let O be a pe relatively cmpact subset f Q ctait ig supp(τ). There exist M R + ad a N 0 such that (T ψ) M α a sup x O α ψ(x) fr all ψ D(O) by the ctiuity f T. Als, there exists q N 0 such that α (τ ϕ)(x) α τ(x) (R ϕ ) p+1 fr all x O, all α a ad all ϕ D q by Lemma 3.5. We bserve as well that there exists m N 0 such t that τ ϕ τ D(O) wheever ϕ D m.let ϕ D, where max{1, q, m, M α a 1}. Sice ϕ( x) t = ϕ(x) fr all x R d,wehave t (T ϕ τ) (T τ) = (T τ ϕ τ) M α a (R ϕ) p+1 = (R ϕ ) p (R ϕ )M( α a 1) (R ϕ) p as required. 4 Asympttic umbers ad asympttic fuctis We defie a field -C D 0 f asympttic umbers ad the differetial algebra f asympttic fuctis E Q (Q) D 0 ver the field -C D 0. N backgrud i -stadard aalysis is required f the reader: ur framewrk is still the usual stadard aalysis. Bth - C D 0 ad E Q (Q) D 0, hwever, d have alterative -stadard represetatis, but we shall pstpe the discussi f the cecti with -stadard aalysis t Sect. 7. The readers wh are ufamiliar with the -liear thery f geeralized fuctis [6 10] might treat this ad the ext sectis as a itrducti t a (mdified ad imprved versi) f Clmbeau thery. The readers wh are familiar with Clmbeau thery will bserve the strg similarity betwee the cstructi f - C D 0 ad the defiiti f the rig C f Clmbeau geeralized umbers [6, pp. 136]. The defiiti f E Q (Q) D 0 als resembles the defiiti f the special algebra G(Q) f Clmbeau geeralized fuctis [7]. We believe, hwever, that ur asympttic umbers ad asympttic fuctis ffer a imprtat imprvemet f Clmbeau thery because - C D 0 is a algebraically clsed field (Therem 4.2) i ctrast t C, which is a rig with zer divisrs. Defiiti 4.1 (Asympttic umbers). Let R ϕ be the radius f supprt f ϕ (cf.(1)).

14 1. We defie the sets f the mderate ad egligible ets i C D 0 by s m M(C D 0 ) = (A ϕ ) C D 0 : ( m N) A ϕ (R ϕ ) a.e., (4) s N (C D 0 ) = (A ϕ ) C D 0 : ( p N) A ϕ <(R ϕ ) p a.e., (5) respectively, where a.e stads fr almst everywhere (Defiiti 2.5). We defie the factr rig - C D 0 =: M(C D 0 )/N (C D 0 ) ad we dete by A -C D ϕ 0 the equivalece class f the et (A ϕ ) M(C D 0 ). 2. If S C D 0,welet S; =: A ϕ : (A ϕ ) S M(C D 0 ). We call the elemets f - C D 0 cmplex asympttic umbers ad the elemets f - R D 0 real asympttic umbers. We defie a rder relati -R D 0 as fllws: Let A - ϕ R D 0 ad AA ϕ = 0. The AA ϕ > 0if A ϕ > 0 a.e., that is {ϕ D 0 : A ϕ > 0} U. 3. We defie the embeddigs C -C D 0 ad R -R D 0 by the cstat ets, that is, by A A;. Therem 4.2 (Algebraic prperties). - C D 0 is a algebraically clsed field, - R D 0 is a real clsed field ad we have the usual cecti - C D 0 = -R D 0 (i). Prf It is clear that - C D 0 is a rig ad - C D 0 = -R D 0 (i). T shw that - C D 0 is a field, suppse that (A ϕ ) M(C D 0 ) \ N (C D 0 ). Thus there exist m, p N such that < =: {ϕ D 0 : (R ϕ ) p A ϕ (R ϕ ) m } U. We defie the et (B ϕ ) C D 0 by B ϕ = 1/A ϕ if ϕ < ad B ϕ = 1if ϕ D 0 \ <. It is clear that A ϕ B ϕ = 1 a.e. thus A A = 1 as required. T shw that - C D ϕ B ϕ 0 is a algebraically clsed field, let P(x) = x p + a 1 x p 1 + +a 0 be a plymial with cefficiets i - C D 0 ad degree p 1. Sice -C D 0 is a field, we have assumed withut lss f geerality that the leadig cefficiet is 1. We have a k = -A ϕ,k, fr sme mderate ets (A ϕ,k ). Dete P ϕ (x) =: x p + A ϕ,p 1 x p A ϕ,0 ad bserve that fr every ϕ D 0 there exists a cmplex umber X ϕ C such that P ϕ (X ϕ ) = 0 sice C is a algebraically clsed field. Thus there exists a et (X ϕ ) C D 0 such that P(X ϕ ) = 0 fr all ϕ D 0. Als the estimati X ϕ 1 + A ϕ,p A ϕ,0 implies that the et (X ϕ ) is a mderate et. The asympttic umber AX - ϕ C D 0 is the zer f the plymial p p 1 p P we are lkig fr because P(XA ϕ ) = XA ϕ + a p 1 XA ϕ + +a 0 = XA ϕ + Q A p 1 A = R D ϕ,p 1 X ϕ A ϕ,0 = PQ ϕ (X ϕ ) = ; 0 as required. The fact that - 0 is a real clsed field fllws directly frm the fact that -C D 0 is a algebraically clsed field ad the cecti - C D 0 = -R D 0 + i - R D 0 [39, Chap. 11]. Crllary 4.3 (Ttal rder). -R D 0 is a ttally rdered field ad we have the fllwig characterizati f the rder relati: if a -R D 0 the a 0iff a = b 2 fr sme b - R D C D 0-0. Csequetly, the mappig : - R D 0, defied by the frmula a + ib = a 2 + b 2,isa abslute value -C D 0 [31, pp. 3 6].

15 Prf The algebraic peratis i ay real clsed field uiquely determie a ttal rder [39, Chap.11]. Thus the characterizati f the rder relati i - R D 0 fllws directly frm the fact that - R D 0 is a real clsed field. The existece f the rt x fr ay -egative x i - R D 0 als fllws frm the fact that - R D 0 is a real clsed field. Defiiti 4.4 (Ifiitesimals, fiite ad ifiitely large). A asympttic umber z - C D 0 is called ifiitesimal, i symbl z 0, if z < 1/ fr all N. Similarly, z is called fiite if z < fr sme N. Ad z is ifiitely large if < z fr all N. We dete by I(C - D 0 ), F(C - D 0 ) ad L(C - D 0 ) the sets f the ifiitesimal, fiite ad ifiitely large umbers i - C D 0, respectively. We defie the ifiitesimal - relati C D 0 by z z 1 if z z 1 is ifiitesimal. We defie the stadard part - mappig ;st : F(C D 0 ) C by the frmula ;st(z) z. The ext result shws that bth - R D 0 ad - C D 0 are -archimedea fields i the sese that they ctai -zer ifiitesimals. Lemma 4.5 (Caical Ifiitesimal i - R D 0). Let R ϕ be the radius f supprt f ϕ (cf.(1)). The the asympttic umber ρ; =: A R R D ϕ is a psitive ifiitesimal i - 0. We call ρ;the caical ifiitesimal i -R D 0 (the chice f the tati ρ;will be justified i Sect. 7). Prf We have 0 ρ< ; 1/ fr all N because R ϕ R + & R ϕ < 1/ a.e. (cf. Example 2.6). Als, ρ; = 0 because (R ϕ )/ N (C D 0 ). Defiiti 4.6 (Tplgy, Valuati, Ultra-rm, Ultra-metric). We supply -C D 0 with the rder tplgy, that is, the prduct tplgy iherited frm the rder tplgy -R D 0. We defie a valuati v : - C D 0 R { } -C D 0 by v(z) = sup{q Q z/ρ; q 0} if z = 0 ad v(0) =. We defie the ultra-rm - v : C D 0 R by the frmula z v = e v(z) (uder the cveti that e = 0). The frmula d(a, b) = a b - v defies a ultra-metric C D 0. Therem 4.7 (Ultra-prperties). Let a, b, c - C D 0. The (i) (a) v(a) = iff a = 0; (b) v(ab) = v(a) + v(b). (c) v(a + b) mi{v(a), v(b)}; (d) a < b implies v(a) v(b). (ii) (a) 0 v = 0, ±1 v = 1, ad a v > 0 wheever a = 0; (b) ab v = a v b v ; (c) a + b v max{ a v, b v } (ultra-rm iequality); (d) a < b implies a v b v. (iii) d(a, b) max{d(a, c), d(c, b)} (ultra-metric iequality). Csequetly, (C - D 0, d) ad (R - D 0, d) are ultra-metric spaces. Prf The prperties (i) (iii) fllw easily frm the defiiti f v ad we leave the verificati t the reader.

16 Remark 4.8 (Clmbeau thery). The cuterpart v f v i Clmbeau thery is ly a pseud-valuati, t a valuati, i the sese that v satisfies the prperty v( ab) v(a) + v(b), t v(ab) = v(a) + v(b). Csequetly, the cuterpart v f v i Clmbeau thery is pseud-ultra-metric, t a ultra-metric, i the sese that it satisfies the prperty ab v a v b v, t ab v = a v b v. Fr the ccept f classical valuati we refer the reader t Ribebim [31]. Defiiti 4.9 (Asympttic fuctis). Let Q be a pe set f R d ad R ϕ be the radius f supprt f ϕ (cf. (1)). 1. We defie the sets f the mderate ets M(E(Q) D 0 ) ad egligible ets N (E(Q) D 0 ) f E(Q) D 0 by: ( f ϕ ) M(E(Q) D 0 ) if (by defiiti) ( K < Q)( α N d )( m N 0 )(sup α f ϕ (x) (R ϕ ) x K ad, similarly, ( f ϕ ) N (E(Q) D 0 ) if (by defiiti) m a.e.), ( K < Q)( α N d )( p N)(sup α f ϕ (x) (R ϕ ) p a.e.), x K respectively. Here α f ϕ (x) stads fr the α-partial derivative f f ϕ (x) with respect t x ad a.e stads (as befre) fr almst everywhere (Defiiti 2.5). We defie the factr rig E Q (Q) D 0 =: M(E(Q) D 0 )/N E(Q) D 0 ad we dete by ;f ϕ E Q (Q) D 0 the equivalece class f the et ( f ϕ ) M(E(Q) D 0 ). We call the elemets f E Q (Q) D 0 asympttic fuctis Q. Mre geerally, if S E(Q) D 0, we let S;=: ;f ϕ : ( f ϕ ) S M(E(Q) D 0 ). 2. We supply E Q (Q) D 0 with the rig peratis ad partial differetiati f ay rder iherited frm E(Q). Als, fr every asympttic umber A -C D ϕ 0 ad every asympttic fucti ;f ϕ E Q (Q) D 0 we defie the prduct A ;f ϕ E Q (Q) D ϕ 0 by A ϕ ;f ϕ = Qf A ϕ ϕ. 3. We defie the pairig betwee E Q (Q) D 0 ad D(Q) by the frmula ( ;f ϕ τ) = (Qf ϕ τ), where ( f ϕ τ) =: Q f ϕ (x)τ(x) dx. 4. We say that a asympttic fucti ;f ϕ E Q (Q) D 0 is weakly equal t zer i E Q (Q) D 0, i symbl ;f ϕ = 0, if ( ;f ϕ τ) = 0 fr all τ D(Q). We say that ;f ϕ, g; ϕ E Q (Q) D 0 are weakly equal, i symbl ;f ϕ = g; ϕ,if(;f ϕ τ) = (g; ϕ τ) i - C D 0 fr all τ D(Q). 5. We say that a asympttic fucti ;f ϕ E Q (Q) D 0 is weakly ifiitesimal (r, assciated t zer), i symbl ;f ϕ 0, if ( ;f ϕ τ) 0 fr all τ D(Q), where the latter is the ifiitesimal relati -C D 0 (Defiiti 4.4). We say that ;f ϕ, g; ϕ E Q (Q) D 0 are weakly ifiitely clse (r, assciated), i symbl ;f ϕ g; ϕ, if ( ;f ϕ τ) (g; ϕ τ) fr all τ D(Q), where i the latter frmula stads fr the ifiitesimal relati i - C D 0.

17 6. Let ;f ϕ E Q (Q) D 0 ad let O be a pe subset f Q. We defie the restricti ;f ϕ I O E Q (O) D 0 f ;f ϕ t O by ;f ϕ I O = fq ϕ I O, where f ϕ I O is the usual restricti f f ϕ t O.The supprt supp( ;f ϕ ) f ;f ϕ is the cmplemet t Q f the largest pe subset G f Q such that ;f ϕ I G = 0i E Q (G) D Let Q, Q ' T d ad ψ Diff(Q ',Q) be a diffemrphism. Fr every ;f ϕ E Q (Q) D 0 we defie the cmpsiti (r, chage f variables) ;f ϕ ψ E Q (Q ' ) D 0 by the frmula ;f ϕ ψ = fq ϕ ψ, where f ϕ ψ stads fr the usual cmpsiti betwee f ϕ ad ψ. It is clear that M(E(Q) D 0 ) is a differetial rig ad N (E(Q) D 0 ) is a differetial ideal i M(E(Q) D 0 ). Thus E Q (Q) D 0 is a differetial rig. We leave t the reader t verify that the prduct A ; f ϕ is crrectly defied. Thus we have the fllwig result: A ϕ Therem 4.10 (Differetial algebra). E Q (Q) D 0 is a differetial algebra ver the field - C D 0. 5 A sluti t the prblem f multiplicati f Schwartz distributis I this secti we cstruct a caical embeddig E Q f the space D ' (Q) f Schwartz distributis it the algebra f asympttic fuctis E Q (Q) D 0. Thus E Q (Q) D 0 becmes a full algebra f geeralized fuctis f Clmbeau type (see 1). The algebra f asympttic fuctis E Q (Q) D 0 supplied with the embeddig E Q ffers a sluti t the prblem f the multiplicati f Schwartz distributis similar t but differet frm Clmbeau s sluti [6]. Defiiti 5.1 (Embeddigs). Let Q be a pe set f R d. 1. The stadard embeddig σ Q : E(Q) E Q (Q) D 0 is defied by the cstat ets, that is, by the frmula σ Q ( f ) = ;f. 2. The distributial embeddig E Q : D ' (Q) E Q (Q) D 0 is defied by the frmula E Q (T ) = T Q ϕ, where T ϕ is the ϕ-regularizati f T D ' (Q) (# 5 i Examples 3.2). 3. The classical fucti embeddig E Q L Q : L lc (Q) E Q (Q) D 0 is defied by the frmula (E Q L Q )( f ) = Qf ϕ, where f ϕ is the ϕ-regularizati f f L lc (Q) (# 6 i Examples 3.2). Lemma 5.2 (Crrectess). The cstat ets are mderate i the sese that f E(Q) implies ( f ) M(E(Q) D 0 ) (Sect. 4). Similarly the ϕ-regularizati f a Schwartz distributi (# 5 i Examples 3.2) is als a mderate et, that is, T D ' (Q) implies (T ϕ) M(E(Q) D 0 ). Prf It is clear that the cstat ets are mderate. T shw the mderateess f (T ϕ), suppse that K < Q ad α N 0. By Lemma 3.4 there exist m, N 0 such that D {ϕ D 0 : ( x K ) α (T ϕ)(x) (R ϕ ) m } implyig {ϕ D 0 : sup x K α (T ϕ)(x) (R ϕ ) m } U, as required.

18 Ntice that the embeddig E Q is caical i the s sese that it is uiquely defied i terms already used i the defiiti f the family E Q (Q) D 0 (Defiiti 4.9). Q T d Therem 5.3 (Prperties f embeddig). Let Q be a pe set f R d. The: (i) We have (E Q L Q )( f ) = σ Q ( f ) fr all f E(Q). This ca be summarized i the fllwig cmmutative diagram: E(Q) L Q D ' (Q) σ Q E Q E Q (Q) D 0 Csequetly, E(Q) ad (E Q L Q )[E(Q)] are ismrphic differetial algebras ver C.Als, E Q L Q = σ Q preserves the pairig betwee E(Q) ad D(Q) i the sese that Q f (x)τ(x) dx = (σ Q ( f ) τ) = ((E Q L Q )( f ) τ), fr all f E(Q) ad all τ D(Q). Csequetly, E Q L Q = σ Q is ijective. (ii) E Q is C-liear ad it preserves the partial differetiati f ay rder i D ' (Q). Als, E Q preserves the pairig betwee D ' (Q) ad D(Q) i the sese that (T τ) = (E Q (T ) τ) fr all T D ' (Q) ad all τ D(Q). Csequetly, E Q is ijective. (iii) E Q L Q is C-liear. Als, E Q L Q preserves the pairig betwee L lc (Q) ad D(Q) i the sese that Q f (x)τ(x) dx = ((E Q L Q )( f ) τ), fr all f L lc (Q) ad all τ D(Q). Csequetly, E Q L Q is ijective. (iv) Each f the abve embeddigs: σ Q, E Q ad E Q L Q,is sheaf preservig i the sese that it preserves the restricti t a pe subset. We summarize all f the abve i E(Q) L lc (Q) D ' (Q) E Q (Q) D 0, where: (a) E(Q) is a differetial subalgebra f E Q (Q) D 0 ver C;(b) L lc (Q) is a vectr subspace f E Q (Q) D 0 ver C ad (c) D ' (Q) is a differetial vectr subspace f E Q (Q) D 0 ver C. We shall fte write simply T istead f the mre precise E Q (T ) fr a Schwartz distributi i the framewrk f E Q (Q) D 0. Prf (i) Suppse that K < Q, α N d ad p N (are chse arbitrarily). By 0 Lemma 3.5 there exist N 0 such that D ϕ D 0 : sup α ( f ϕ)(x) α f (x) (R ϕ ) p. x K

19 Thus {ϕ D 0 : sup x K α ( f ϕ)(x) α f (x) (R ϕ ) p } U. The latter meas that the et ( f ϕ f ) is egligible (Defiiti 4.9) thus (E Q L Q )( f ) = Qf ϕ = ;f = σ Q ( f ) as required. Csequetly, we have (E Q L Q )[E(Q)] = σ Q [E(Q)]. Thus E(Q) ad (E Q L Q )[E(Q)] are ismrphic differetial algebras because E(Q) ad σ Q [E(Q)] are (bviusly) ismrphic differetial algebras. Als, E Q L Q preserves the pairig because σ Q preserves (bviusly) the pairig. (ii) I Q is C-liear because the mappig T T ϕ is C-liear. T shw the preservati f partial differetiati we have t shw that fr every multi-idex β N d 0 the et β T ϕ β (T ϕ) is egligible (Defiiti 4.9). This fllws easily frm Lemma 3.3 similarly t (i) abve. T shw that E Q preserves the pairig, we have t shw that fr ay test fucti τ the et A ϕ =: (T ϕ τ) (T τ) is egligible (Defiiti 4.1). The latter fllws easily frm Lemma 3.6. (iii) (E Q L Q ) is C-liear because the mappig f f ϕ is C-liear. The preservig f pairig fllws frm (ii) i the particular case T = T f. (iv) The preservig f the restricti a pe subset fllws easily frm the defiiti ad we leave the details t the reader. We shuld meti that if f E(Q) ad T D ' (Q), the E Q ( f )E Q (T ) = E Q ( ft ) is false i geeral. That meas that the multiplicati i the algebra i E Q (Q) D 0 des t reprduce the Schwartz multiplicati i D ' (Q) (multiplicati by duality). Similarly, let C(Q) dete the class f ctiuus fuctis frm Q t C. If g, h C(Q), the E Q (g)e Q (h) = E Q (gh) is als false i geeral. That meas that the multiplicati i the algebra i E Q (Q) D 0 des t reprduce the usual multiplicati i C(Q). Of curse, all these are ievitable i view f the Schwartz impssibility results [35]. Fr a discussi we refer t [10, p. 8]. Istead, we have a smewhat weaker result. Therem 5.4 (Weak preservati). Let T D ' (Q), f E(Q) ad g, h C(Q). The: (i) E Q ( f )E Q (T ) = E Q ( ft )(Defiiti 4.9, #4), that is, (E Q ( f )E Q (T ) τ) = (E Q ( ft ) τ) fr all τ D(Q). (ii) E Q (g)e Q (h) E Q (gh) (Defiiti 4.9, #5), that is, (E Q (g)e Q (h) τ) (E Q (gh) τ) fr all τ D(Q), where i the latter frmula stads fr the ifiitesimal relati i the field - C D 0. Prf (i) We dete f ϕ,τ := ( f (T ϕ) τ) = f (T ϕ f τ) ad calculate f (E Q ( f )E Q (T ) τ) = ;f T Q ϕ τ = f (QT ϕ) τ = -f ϕ,τ = T Q ϕ f τ = (T f τ) = ( ft τ) = (E Q ( ft ) τ) as required. (ii) This fllws frm the fact that fr each N ad K < Q we have sup x K (g ϕ g)(x)h(x) < 1/ ad sup x K (g ϕ)(x)(h ϕ h)(x) < 1/ a.e. i D 0 (Defiiti 2.5) which ca be see by elemetary bservati. Let Q, Q ' T d ad ψ Diff(Q ',Q). The E Q (T ) ψ = E Q '(T ψ) des t geerally hld i E Q (Q) D 0. That meas that the family f algebras {E Q (Q) D 0 } Q T d is t diffemrphism ivariat (see Sect. 1). Here T ψ stads fr the cmpsiti i the sese f the distributi thery [41]. Istead, we have the fllwig weaker result.

20 Therem 5.5 (Diffemrphisms). E Q weakly preserves the cmpsiti with diffemrphisms i the sese that fr every Q, Q ' T d, every T D ' (Q) ad every ψ Diff(Q ',Q) we have E Q (T ) ψ = E Q ' (T ψ), that is, (E Q (T ) ψ τ) = (E Q ' (T ψ) τ) fr all test fuctis τ D(Q ' ). Prf The prf is aalgus t the prf f part (i) f Therem 5.4 ad we leave the details t the reader. Example Let δ D ' (R d ) be the Dirac delta fucti (delta distributi) R d. Fr its ϕ-regularizati (#5 i Examples 3.2)wehave δ ϕ = δ ϕ = δ*ϕ = ϕ. Thus E R d (δ) = ; ϕ. Similarly, E α R d ( α δ) = -ϕ. 2. We have E R d (δ) = (; ϕ) = ϕa, = 1, 2,.... We express this result simply as δ = Aϕ. Recall that the pwers δ are meaigless withi D ' (R d ) fr Let H(x) be the Heaviside step fucti R. Fr its ϕ-regularizati (#6 i Examples 3.2)wehave H ϕ = (H ϕ).let K < R. We bserve that fr every x x K we have H ϕ (x) = (H *ϕ)(x) = ϕ(t) dt a.e. i D 0 (Defiiti 2.5). Thus x Q x Q E R (H) = ϕ(t) dt. We express this result simply as H(x) = ϕ(t) dt. Sice the embeddig E ' R preserves the differetiati, f we have H = δ. x 4. We have E Q R (H)E R (δ) = ; ϕ ϕ(t) dt = ϕh - ϕ. We express this result simply as Hδ = -ϕ H ϕ. Recall that the prduct Hδ is t meaigful withi D ' (R). f x 5. We have (E Q R (H)) = ϕ(t) dt = ( QH ϕ ) which we write simply as H = (QH ϕ ). Sice E Q (R) D 0 is a differetial algebra, we ca apply the chai rule: (H ) ' = H 1 δ which als is meaigless i D ' (R) fr Ntice that H = H, = 2, 3,... i E Q (R) D 0. Actually H = H, = 2, 3,..., fail i ay differetial algebra. Ideed, H 2 = H implies 2Hδ = δ while H 3 = H implies 3Hδ = δ thus 2 = 3, a ctradicti. Fr a discussi we refer t [14, Example (1.1.1)]. 6 Distributial -stadard mdel The distributial -stadard mdel preseted i this secti is especially desiged fr the purpse f the -liear thery f geeralized fuctis (Clmbeau thery). It is a fully c + -saturated ultrapwer -stadard mdel (Therem 6.3) with the set f idividuals R based the D 0 -ets (Defiiti 3.1). Here c = card(r) ad c + stads fr the successr f c. The cecti f the thery f asympttic umbers ad fuctis (Sect. 4) with -stadard aalysis will be discussed i the ext secti. We shuld meti that a similar ultrapwer -stadard mdel (with the same idex set ad differet ultrafilter) was used i Berger s thesis [1] fr studyig delta-like slutis f Hpf s equati. Fr readers wh are familiar with -stadard aalysis this secti is a shrt review f the ultra-pwer apprach t -stadard aalysis itrduced by Luxemburg [22] almst 40 years ag (see als [36]). Fr the reader withut backgrud i -stadard aalysis, this secti ffers a shrt itrducti t the subject. Fr additial readig, we refer t Davis [12], Lidstrøm [21] ad Chap. 2 i Capiński ad Cutlad [4].

21 Defiiti 6.1 (Distributial -stadard mdel). 1. Let S be a ifiite set. The superstructure S is defied by V (S) =: =0 V (S), where V 0 (S) = S ad V +1 (S) = V (S) P (V (S)). The level λ(a) f A V (S) is defied by the frmula λ(a) =: mi{ N 0 : A V (S)}. The superstructure V (S) is trasitive i the sese that V (S) \ S P(V (S)). Thus V (S) \ S is a Blea algebra. The members s f S are called idividuals f the superstructure V (S). 2. Let S = R. We bserve that V (R) ctais all bjects i stadard aalysis: all rdered pairs f real umbers thus the set f cmplex umbers C, Cartesia prducts f subsets f R ad f C thus all relatis R ad C, all biary algebraic peratis R ad C, all real ad cmplex fuctis, all sets f fuctis, etc. 3. Let R D 0 be the set f all D 0 -ets i R (Defiiti 3.1). The set R f -stadard real umbers is defied as fllws: (a) We defie the equivalece relati U R D 0 by (A ϕ ) U (B ϕ ) if A ϕ = B ϕ a.e. r, equivaletly, if {ϕ D 0 : A ϕ = B ϕ } U (Defiiti 2.5). (b) The equivalece classes i R d = A R D 0 / U are called -stadard real umbers. We dete by A ϕ R the equivalece class f the et (A ϕ ) R D 0. The rig peratis i R are iherited frm the rig R D 0. The rder i R is defied by d A A ϕ > 0if Aϕ > 0 a.e., that is, if {ϕ D 0 : A ϕ > 0} U. (c) We defie the d caical A embeddig R R by the cstat ets, that is, by A A ϕ, where A ϕ = A fr all ϕ D 0. We shall write simply R R istead f R dr.alsif A (A ϕ ) is a cstat et, we shall write simply (A) istead f A ϕ. 4. Let S = R. The superstructure V ( R) ctais all bjects i -stadard aalysis: rdered pairs f -stadard real umbers thus the set f -stadard cmplex umbers C, all Cartesia prducts f subsets f R ad f C thus all relatis R ad C, all biary algebraic peratis R ad C,all -stadard fuctis, all sets f -stadard fuctis, etc. 5. Let V (R) D 0 stad fr the set f all D 0 -ets i V (R) (Defiiti 3.1). A et (A ϕ ) i V (R) D 0 is called tame if ( N 0 )( ϕ D 0 )(A ϕ V (R)).If(A ϕ ) is a tame et i V (R) D 0 its level λ((a ϕ )) is defied (uiquely) as the umber N 0 such that {ϕ D 0 : λ(a ϕ ) = } U, where λ(a ϕ ) is the level f A ϕ i V (R) (see #1 abve). 6. Fr every tame et (A ϕ ) i V (R) D 0 we defie d A d A A ϕ V ( R) iductively the level f the ets: If λ((a ϕ )) = 0, the A ϕ is defied i #3 abve. Suppse d A A ϕ is already defied fr all tame ets (A ϕ ) i V (R) D 0 with λ((a ϕ )) <. If d A (B ϕ ) V (R) D 0 is a tame et with λ((b ϕ )) =,welet B ϕ =: (A ϕ ) V (R) D 0 : λ((a ϕ )) < & A ϕ B ϕ a.e., where, as befre, A ϕ B ϕ a.e. meas ϕ D 0 : A ϕ B ϕ U (Defiiti 2.5). Let (A ϕ ) be a cstat et i V (R) D 0, that is, A ϕ = A fr all ϕ D 0 ad sme d A A V (R). I the case f cstat ets we shall write simply (A) istead f A ϕ.

22 7. A elemet A f V ( R) is called iteral if A = d A ϕ A fr sme tame et (A ϕ ) V (R) D 0. We dete by V (R) the set f the iteral elemets f V ( R) (icludig the -stadard reals i R). The elemets f V (R) \ R are called iteral sets. The iteral sets f the frm (A), where A V (R) (i.e. geerated by cstat ets), are called iteral stadard (r simply, stadard). The elemets f V ( R) \ V (R) are called exteral sets. 8. We defie the extesi mappig : V (R) V ( R) by A = (A). Ntice that the rage ra( ) f the extesi mappig csists exactly f the iteral stadard elemets f V ( R). The termilgy extesi mappig fr is due t the fllwig result: Let S V (R) \ R. The S S ad the equality ccurs iff S is a fiite set. 9. It ca be shw that A is iteral iff A A fr sme A V (R). It ca be shw as well that a elemet A V (R) is iteral iff A R r A is a fiite set (tice that V (R) V ( R) sice R R). The ifiite sets i V (R) \ R are called exteral stadard sets. Fr example, the familiar N, N 0, Z, Q, R, C are all exteral stadard sets. d 10. A pit ζ C is called ifiitesimal if ζ < 1/ fr all N. Als, ζ C d is called fiite if ζ < fr sme N. Similarly, ζ C d is called ifiitely large if < ζ fr all N. We dete by I( C d ), F( C d ) ad d L( C d ) the sets f the ifiitesimal, fiite ad ifiitely large pits i C, respectively. We fte write ζ 0 istead f ζ I( C d ) ad ζ 1 ζ 2 istead f ζ 1 ζ 2 I( C d ). Mre geerally, if S C d, the I(S), F(S) ad L(S) dete the sets f ifiitesimal, fiite ad ifiitely large pits i S, respectively. d 11. We defie the stadard part mappig st : F( C d ) C by the frmula d st(ζ ) ζ. We bserve that st is a vectr hmmrphism frm F( C d ) t C. I particular, st : F( C) C is a rder preservig rig hmmrphism frm F( C) t C (relative t the partial rder i C). 12. We call ρ R, defied by ρ = d R ϕ A (cf. (1)), the caical ifiitesimal i R.Itis caical because is defied uiquely i terms f the idex set f the distributial -stadard mdel. It is a psitive ifiitesimal because 0 < ρ< 1/ fr all N (Example 2.6). 13. Let x R d ad X R d.the mads f x ad X are defied by s μ(x) = x + dx : dx R d & dx 0, s μ(x) = x + dx : x X & dx R d & dx 0, respectively. Als, μ 0 (x) =: μ(x) \{x} is the deleted mad f x. Therem 6.2 (Extesi priciple). R is a prper extesi f R, that is, R R. Csequetly, V (R) V ( R). Prf We bserve that ρ R \ R (#12 i Defiiti 6.1). I what fllws we assume a particular case f the ctiuum hypthesis i the frm c + = 2 c.

23 Therem 6.3 (Saturati priciple). Our -stadard mdel V ( R) is c+ -saturated i the sese that every family (A γ ) γ f f iteral sets i V ( R) with the fiite iter secti prperty ad card(f) c has the -empty itersecti γ f A γ =. Als V ( R) is fully saturated i the sese that V ( R) is card( R)-saturated (cf. [5, Chap. 5]. Prf We refer the reader t the rigial prf i Chag ad Keisler [5] (fr a presetati see als Lidstrøm [21]). We shuld meti that the prperty f the ultrafilter U t be c + -gd (# 6 i Lemma 2.4) is ivlved i the prf f this therem. T shw that V ( R) is fully saturated, we have t shw that card( R) = c +. Ideed, card( R) card(r D 0 ) = 2 c fllws frm the defiiti f R i the distributial mdel ad card( R) 2 c fllws frm the fact that V ( R) is c + -saturated. The ext result demstrates the remarkable feature f -stadard aalysis t reduce (ad smetimes eve t elimiate cmpletely) the umber f quatifiers cmpared with stadard aalysis. Therem 6.4 (Usual tplgy R d ad mads). Let X R d ad x R d. The: (a) xis a iterir pit f X iff μ(x) X. Csequetly, X is pe iff μ(x) X. (b) X is clsed iff st( X) = X, where st : F( R d ) R d stads fr the stadard part mappig. (c) x is a adheret pit f X (i.e. x X) iff X μ(x) =. (d) X is a cluster pit f X iff X μ 0 (x) =. (e) X isa buded set iff X csists f fiite pits ly. (f) X is cmpact iff X μ(x). Prf We refer the reader t the rigial prfs i Rbis [32] (r, t a presetati i Salbay ad Tdrv [34]). T cmplete ur survey -stadard aalysis, we wuld have t discuss several mre imprtat priciples: the rder cmpleteess priciple i R fr iteral sets, differet spillig priciples (uderflw ad verflw), trasfer priciple ad iteral defiiti priciple. The trasfer priciple is csidered by may as the heart ad sul f -stadard aalysis. O these tpics we refer the reader t Davis [12], Lidstrøm [21] ad Chap. 2 i Capiński ad Cutlad [4]. Fr reader with experiece i mathematical lgic we recmmed Rbis [32]. 7 Clmbeau s thery f geeralized fuctis ad -stadard aalysis We shw that the field f asympttic umbers - C D 0 (Defiiti 4.1) isismrphict a particular Rbis field ρ C [33] f ρ-asympttic umbers. We als prve that the algebra f asympttic fuctis E Q (Q) D 0 (Defiiti 4.9) is ismrphic t a particular algebra f ρ-asympttic fuctis ρ E(Q) itrduced i [28]. Bth ρ C ad ρ E(Q) are defied i the framewrk f -stadard aalysis (see Defiitis 7.1 ad 7.7 belw). - As faraswetreat C D 0 ad E Q (Q) D 0 as mdified ad, we believe, imprved versis f Clmbeau s C ad G(Q), respectively, these results establish a cecti betwee Clmbeau thery ad -stadard aalysis. Recall the defiiti f Rbis s field ρ R [33] ad its cmplex cuterpart ρ C.