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Math. J. Okayama Univ. 52 (2010), 159–177 EXPONENTIAL GENERALIZED DISTRIBUTIONS

M. GORDON and L. LOURA

Abstract. In this paper we generalize the from the space of tempered distributions to a bigger space called exponential generalized distributions. For that purpose we replace the Schwartz space S by a smaller space X0 of smooth functions such that, among other properties, decay at infinity faster than any exponential. The construction of X0 is such that this space of test functions is closed for , for Fourier transform and for translations. We equip X0 with an appropriate locally convex topology and we study it’sdual ! ! X0;wecallX0 the space of exponential generalized distributions. The ! space X0 contains all the Schwartz tempered distributions, is closedfor derivatives, and both, translations and Fourier transform,arevectorand ! topological in X0.Asnontrivialexamplesofelements ! in X0,weshowthatsomemultipoleseriesappearinginphysicsare convergent in this space.

1. Introduction Laurent Schwartz in [18] introduced an important subspace ofthespace ! of all distributions in RN :thespace ! of tempered distributions. This Dis the strong dual of the Schwartz spaceS .Forthereader’sconvenience we recall the definiton of and certain propertiesS needed later on. The Schwartz test spaceS is the vector space of smooth functions1 defined by S

(1.1) = ϕ C∞; α,β NN xα∂βϕ L∞ . S { ∈ ∀ ∈ ∈ } The topology of is determined by the following family (indexed on the N of nonnegativeS ) of :

α β (1.2) ϕ sup x ∂ ϕ L∞ , −→ |α|,|β|≤m % %

Mathematics Subject Classification. Primary 46F05; Secondary 32A25, 32A45. Key words and phrases. Distribution, Ultradistribution, Multipole series, Fourier transform. The first author was supported in part by CCM, University of Madeira through FEDER/POCTI-SFA-1-219. The second author was supported in part by FCT (Portugal) through the Research Units Pluriannual Funding Program. 1We always consider complex functions defined in RN .

159 160 M. GORDON AND L. LOURA where, for a bounded ψ, ψ L∞ is the usual sup- % % defined by ψ L∞ =supx∈RN ψ(x) .Thespace is a locally convex sep- arable (Hausdor% %ff) vector space;| it is| metrizable, Fr´echetS ,Montel,reflexive, bornological (Mackey) and is continuous and densely embedded in . The space is closed forD derivatives and the operators ∂Sα are S 2 linear continuous from into .Thetranslationoperator τa is a vector S S and topological in with inverse τ a. S − For w R 0 we define the Fourier operator w by ∈ \{ } F −iwx.ξ (1.3) wϕ(ξ)= e ϕ(x)dx , F RN ! 3 N where ϕ and x.ξ = x ξ + + xN ξN is the usual inner product in R . ∈S 1 1 ··· The operator w is a vector and topological automorphism in . The space ofF tempered distributions ! is defined as the strongS dual of .Itisalocallyconvexseparable(HausdorS ff)vectorspace,non-metrizable, Montel,S reflexive, bornological (Mackey) and is continuous and densely em- bedded in !.Moreover, is continuous and densely embedded in !.A distributionDT ! is temperedS iff T is of the form ∂αF where FS is a continuous function∈D of slow growth (that is, F is bounded by a polynomial). α ! The operators ∂ and τa are extended to by transposition, in the usual way: S (1.4) ∂αT,ϕ =( 1)|α| T,∂αϕ , & ' − & '

(1.5) τaT,ϕ = T,τ aϕ , & ' & − ' where T !, ϕ and .,. denotes the dual product between ! and .Thederivativesarelinearcontinuousoperatorsfrom∈S ∈S & ' ! into !S.The S ! S S translation operator τa is an automorphism in with inverse τ−a. S ! By transposition the operator w can also be extended to in the stan- dard way F S

(1.6) wT,ϕ = T, wϕ . &F ' & F ' ! The operator w is a vector and topological automorphisms in and we recall (for theF proofs see Schwartz [18]) some of its properties: S N −1 w (1.7) ( w) = | | w ; F 2π F− " #

β β (1.8) w∂ T =(iwξ) wT ; F F 2 N For a ∈ R and ψ continuous, the translation operator τa is defined by τaψ(x)= ψ(x − a). 3We shall use the same notation for the usual inner product in CN . EXPONENTIAL GENERALIZED DISTRIBUTIONS 161

β β (1.9) ∂ wT = w ( iwx) T ; F F − $ % −iwa.ξ (1.10) wτaT = e wT ; F F

iwa.x (1.11) w e T = τa ( wT ); F F & ' 2π N (1.12) w1= δ ; F w "| |# β β (1.13) w∂ δ =(iwξ) ; F

2π N (1.14) w ( wT )= T ( ξ); F F w − "| |# w (1.15) wT = w T ξ . F F 0 w " 0 # In formulas (1.12) and (1.13) δ is the Dirac distribution at the point 0; in (1.15) w0 is a non null . We write for 1 and we call ϕ the Fourier transform of ϕ. F F F We need also the space c,wherec ]0, + [, defined by G ∈ ∞ ∞ − c |x|2 (1.16) c = ϕ C ; p ϕ(x)=p(x)e 2 , G { ∈ ∃ ∈P } N 2 2 where denotes the space of all polynomials in R and x = x1 + + xN . ··· We shallP write instead of .Thespace is important| | because it is a 1 ( dense subset of G .TheproofistrivialbecauseG G contains all the Hermite functions and theseS functions are dense in ;theproofofthislaststatementG is not an easy one and can be found in [22]S or, for the one dimensional case, in [6]. Incidentally we mention that the space has been studied in [13]. In that paper a locally convex topology is introducedG in and the dual ! is studied in detail. For our purposes we do not need to introducG eatopologyG in because we just use the property that the set is dense in ,andthis requiresG only the topology of . G S S The aim of this paper is to construct an extension of the space !,using the same duality method used by L. Schwartz in his distribution theory.S For that purpose, we construct a space of exponential generalized distributions ! 0 which is closed for derivatives, translations and Fourier transform. These operatorsX are linear and continuous of ! into itself. The translations and X0 162 M. GORDON AND L. LOURA

! the Fourier transform define a vector and topological automorphism in 0, generalizing the beautiful result of Schwartz for tempered distributions.X This paper is organized as follows: In Section 2 we construct the space ∞ N − of all C functions ϕ defined on R ,decayingexponentiallyatinfinity De as well as their derivatives: ∂αϕ(x) Ce−k|x|.Weshowthatthisspaceis closed for derivatives, for translations| |≤ and for the productbyfunctionsof slow exponential growth. We equip it with a locally convex structure and we study some of their topological properties. In Section 3 we study the space of exponential distributions,thatisthe ! ! of − .Thespace is closed for derivatives and trans- De− De De− lations. We also study the topological properties of ! and prove that the De− space ! is a continuous and dense subspace of ! . S De− Since e− is not closed for the Fourier operator, in Section 4 we construct the testD of all complex-valued functions ϕ defined on RN X0 such that both ϕ and ωϕ belong to − . This space is closed for derivatives, F De translations and Fourier transform. We also prove that any function of 0 may be extended to CN as an entire function. We introduce a locally convexX structure in 0,weshowthat 0 is continuous and densely embedded in and we proveX the continuityX of the derivative, translation and Fourier S operators in 0. X ! In Section 5 we study the strong dual 0 of 0,thatwecallthespace X X ! of exponential generalized distributions. The space 0 is closed for the above mentioned operators, and both translations andX Fourier transform ! ! are vector and topological automorphism in 0.Thespace is continuous ! X S and densely embedded in 0. In Theorem 5.3 we prove that all continuous X ! function with exponential growth are in 0.Astructuretheoremforthe ! X space 0 remains an open problem but we hope to solve it in a near future. To closeX the section we give some examples of convergent multipole series in ! appearing as solutions of ODE. X0

2. The test function space − De We say that a continuous function ϕ defined on RN decays exponentially at infinity iff, for each k N,thereisanupperboundofthetype ϕ(x) −k|x| ∈ | |≤ Ce ,whereC>0isarealnumberdependingonk.Wedenoteby e− the vector space of all C∞ functions defined on RN ,decayingexponentiallyD at infinity as well as their derivatives: (2.1) ∞ N N α −k|x| − = ϕ C ; α N k N C>0 x R ∂ ϕ(x) Ce . De { ∈ ∀ ∈ ∀ ∈ ∃ ∀ ∈ | |≤ }

Notice that the space c defined in (1.16) is included in − . G De EXPONENTIAL GENERALIZED DISTRIBUTIONS 163

We say that a continuous function f defined on RN has an exponential growth iffthere is an upper bound of the type f(x) Cek|x|,whereC>0 | |≤∞ and k N.Wedenoteby e+ the vector space of all C functions f defined on RN∈,suchthatf and itsD derivatives have an exponential growth: (2.2) ∞ N N α k|x| e+ = f C ; α N C>0 k N x R ∂ f(x) Ce . D { ∈ ∀ ∈ ∃ ∃ ∈ ∀ ∈ | |≤ } If f e+ we say that f has a slow exponential growth. It is∈ obviousD that the exponentials ez.x, with z CN , and the polynomials ∈ are functions of + and that − is a vector subspace of + . De De De The following theorem collects some properties of the space − . De Theorem 2.1. (i) The space e− is a dense subspace of the Schwartz space . D S N (ii) For all a R we have τa ( e− )= e− . ∈ D N D α (iii) For all ϕ in − and all α N we have ∂ ϕ in − . De ∈ De (iv) For all ϕ in − and all f in + we have fϕ in − . De De De (v) For all ϕ1 and ϕ2 in e− we have ϕ1ϕ2 in e− . D z.x D N z.x (vi) For all ϕ in − and all exponential e , with z C , we have e ϕ De ∈ in − . De (vii) For all ϕ in e− and all polynomial p we have pϕ in e− . D D N (viii) For any ϕ in e− , we have wϕ in and it can be extended to C as an entire function. D F S (ix) For all ϕ in − and all λ R 0 we have ϕ(λx) in − . De ∈ \{ } De Proof. Statement (i) follows from the fact that is included in e− and is dense in .Theproofsofstatements(ii)to(vii)and(ix)arestraightG D forward.G S To prove (viii) let ϕ in − . We have wϕ because ϕ and the De F ∈S ∈S operator w is a vector and topological automorphism in .Furthermore, F N S for each j =1,...,N and each z =(z1,...,zN )inC ,theintegrals

−iwx.z −iwx.z e ϕ(x)dx and ∂zj e ϕ (x) dx RN RN ! ! & ' N are absolutely convergent; this proves that wϕ can be extended to C as an entire function. F !

We introduce a locally convex structure in e− with the family (indexed in N)ofseminormsdefinedby D k|x| α (2.3) νk(ϕ)= sup e ∂ ϕ L∞ . |α|≤k % %

Notice that a sequence (ϕn)in − is bounded iff De N N k|x| α (2.4) α N k N C 0 n N x R e ∂ ϕn(x) C ∀ ∈ ∀ ∈ ∃ ≥ ∀ ∈ ∀ ∈ | |≤ 164 M. GORDON AND L. LOURA and converges to zero iff N k|x| α N (2.5) α N k N e ∂ ϕn 0uniformlyinR . ∀ ∈ ∀ ∈ | |→ The space is clearly included in − .Theinclusioniscontinuous D De because a filter (ϕj)convergesto0in iffthere exists a compact subset K N D N of R such that all the ϕj have their supports in K and, for each α N , α N k|x| α ∈ ∂ ϕj 0uniformlyinR .Thisimpliesthat,foreachk N, e ∂ ϕj 0 → N ∈ → uniformly in R ,thatis(ϕj)convergesto0in − . De The space is dense in e− .Toprovethis,wefixafunctionψ in such that 0 ψ D1, ψ(x)=1ifD x 1andψ(x)=0if x 2and,foreachD ≤ ≤ | |1≤ | |≥ natural n,wewriteψn(x)=ψ( x). Given a function ϕ in − ,itiseasily n De seen that the sequence (ψnϕ)isin and converges to ϕ in − . D De We saw in Theorem 2.1 that − is a dense subspace of .Fromthe De S seminorms (1.2) and (2.3) we conclude that the injection of e− into is continuous. Thus, we have4: D S

(2.6) , e− , . D →d D →d S In order to prove the main theorem of this section, Theorem 2.2below, we need the following useful lemma.

Lemma 1. Let (ϕn) be a bounded sequence in − and let ϕ be such De ∈S that ϕn ϕ in .Thenϕ − and ϕn ϕ in − . → S ∈De → De Proof. We first prove that ϕ is in − .Ontheonehand,theboundedness De of (ϕn)allowustouse(2.4).Ontheotherhand,asϕn ϕ in ,we N α →α NS know that, for each α N ,(∂ ϕn)convergesuniformlyto∂ ϕ in R .Fix α NN and k N,andlet∈ n + in (2.4): ∈ ∈ → ∞ ek|x| ∂αϕ(x) C. | |≤ As α and k are arbitrary, this means that ϕ is in − . De It remains to prove that ϕn ϕ in e− .Wewriteψn = ϕn ϕ and we → ND − prove that ψn 0in − .Fixα N and k N;fromtheboundedness → De ∈ ∈ of (ϕn)in e− ,weknowthat(ψn)isboundedin e− and, by (2.4), this implies theD existence of a real number C 0suchthat,forallD n N and all x RN , ≥ ∈ ∈ (k+1)|x| α e ∂ ψn(x) C. | |≤ We may rewrite this inequality in the form k|x| α −|x| (2.7) e ∂ ψn(x) Ce . | |≤ α N The sequence (∂ ψn)convergestozerouniformlyinR ,becauseitconverges k|x| α to zero in .Therefore,thesequencee ∂ ψn converges to zero uniformly S | | 4We use #→ for continuous injection and #→ for continuous and dense injection. d EXPONENTIAL GENERALIZED DISTRIBUTIONS 165 in the compact subsets of RN .ThefunctionCe−|x| tends to zero at infinity. k|x| α From these two facts we conclude, by (2.7), that e ∂ ψn converges to N | | zero uniformly in R .Thismeans,by(2.5),thatψn 0in − . ! → De Now we are ready to state the main theorem of this section:

Theorem 2.2. The space e− is: (i) Hausdorff; (ii) metrizable; (iii) Fr´echet; (iv) Montel; (v) reflexive;D (vi) Mackey (bornological).

Proof. (i) e− is obviously Hausdorffbecause, for all ϕ e− 0 ,there exists a RDN such that ϕ (a) =0,thus ∈D \{ } ∈ , ν (ϕ)= ϕ ∞ ϕ (a) > 0 . 0 % %L ≥| | (ii) − is metrizable because the familly of seminorms (2.3) is countable. De (iii) We just have to prove that − is complete. Let (ϕn)beaCauchy De sequence in − .By(2.6),(ϕn)isalsoaCauchysequencein and, as we De S know that is complete, there exists ϕ in such that ϕn converges to ϕ in S S .ByLemma1,weseethatϕ belongs to − and that ϕn converges to ϕ S De in − .Thisprovesthat − is complete. De De (iv) The space − is barrelled because it is a Fr´chet space. Thus, we just De have to prove that, if L is a bounded subset of − ,thenL is relatively com- De pact in − ,or,equivalently,thateverysequence(ϕn)inL has a convergent De subsequence (in the − topology). De The set L is bounded in − and, by (2.6), L is also bounded in .But De S is a , so there exists ϕ in and a subsequence (ϕm )ofthe S S n sequence (ϕn)suchthatϕm ϕ in .ByLemma1weseethatϕ is in n → S − and ϕm ϕ in − .Thisprovesthat − is a Montel space. De n → De De (v) The space − is reflexive because it is a Montel space. De (vi) The space e− is a Mackey (or bornological) space because all metrizable locally convexD spaces are bornological. ! Remark 1. Using the seminorms (2.3) it is obvious that the derivative op- α erators ∂ and the translation operators τa are linear continuous from − De into − .Moreover,asτa is bijective, with inverse τ a,weseethatthe De − translations are vector and topological automorphisms in − . De 3. Exponential distributions ! The dual of − is called the space of exponential distributions. We De− De consider in ! the strong . From (2.6) we have the inclusions De− ! , ! , ! . S →De− →D 166 M. GORDON AND L. LOURA

Obviously ! is dense in ! because is included in ! and it is well De− D D De− known that is dense in !. D D The space e− is continuously embedded in and, as we know that is D ! S S continuously embedded in ,weseethat e− is continuously embedded in ! S D! e− .Next,weprovethat e− is dense in e− .Itissufficienttoprovethat D D ! D every continuous f on e− that is null on e− is identically null. D ! ! D Taking such a form f,wehavef ( e− ) and, as e− is reflexive, f e− ; this implies that ∈ D D ∈D f,f = f(x) 2dx =0 & ' RN | | ! and this shows that f is the null form. Collecting all the previous results, we have the following chain of contin- uous and dense embeddings:

! ! ! (3.1) , e− , , , e− , . D →d D →d S →d S →d D →d D ! Theorem 3.1. The space e− is: (i) Hausdorff; (ii) complete; (iii) Montel; (iv) reflexive. D Proof. (i) The strong dual topology (as well as the weak dual topology) is always separated. ! (ii) The space e− is complete because it is the dual of a separated . D (iii) The dual of a Montel space is always a Montel space. (iv) All Montel spaces are reflexive. ! Similarly to what has been done in the tempered distributions, the op- α ! erators ∂ and τa can be extended to e− by transposition, using formulas ! D (1.4) and (1.5), where now T e− , ϕ e− and .,. denotes the dual ! ∈D ∈D & ' product between e− and e− .Theseoperatorsareobviouslylinearand D! D! continuous from e− into e− .Thetranslationoperatorisavectorand D D ! topological automorphism in − with inverse τ a.Alltheseoperatorsare De − the restriction to ! of the corresponding operators defined on ! and are De− D extensions of the operators on !. Like the case of tempered distributions,S there is a structureresultforthe space ! .AdistributionT in ! belongs to ! if and only if De− D De− (3.2) T = ∂αF, where α NN and F is a continuous function of exponential growth. The sufficient∈ condition is trivial; the proof of the necessary condition is based on the structure of the bounded distributions and may be done exactly in the same way as for the case of tempered distributions (see [18], page 240). EXPONENTIAL GENERALIZED DISTRIBUTIONS 167

4. The test function space X0 We call the space of all complex-valued functions ϕ defined on RN X0 such that both ϕ and ϕ belong to − : F De (4.1) = ϕ − ; ϕ − . X0 { ∈De F ∈De } It is clearly a vector subspace of − and we have the following results: De Theorem 4.1. (i) The space 0 is a dense subspace of the Schwartz space . X S N (ii) Every function ϕ in 0 can be extended to C as an entire function. N X (iii) For all a R we have τa( 0)= 0. (iv) For all ϕ ∈in and all α XNN weX have ∂αϕ in . X0 ∈ X0 (v) ( 0)= 0. (vi)F ForX all ϕX and ϕ in we have ϕ ϕ in . 1 2 X0 1 2 X0 (vii) For all ϕ in 0 and all polynomial p we have pϕ in 0. X N ib.x X (viii) For all ϕ in 0 and all b R we have e ϕ in 0. X ∈ N b.x X (ix) For all ψ in c and all b R we have e ψ in . G ∈ X0 (x) For all ϕ in and all w R 0 we have wϕ in . X0 ∈ \{ } F X0

Proof. (i) is a subspace of the Schwartz space because − and X0 S X0 ⊂De − .Toprovethedensity,letc>0andϕ in c.Thenϕ − because De ⊂S G ∈De c is included in e− .Itiseasytoverifythat ϕ 1 , thus ϕ e− ; G D F ∈G2c F ∈D this means that c is included in 0.Astheset is dense in and 0 this implies thatG is dense in X. G S G⊂X X0 S (ii) Let us fix ϕ in 0.Itisclearthat ϕ belongs to e− and using Theorem 2.1 we may concludeX that both F( ϕ)and 1 D ( ϕ)=ϕ F−1 F (2π)N F−1 F can be extended to CN as entire functions. N (iii) Let a R and ϕ . As the space − is closed for translations ∈ ∈X0 De (see Theorem 2.1), then τaϕ is in e− . It remains to show that (τaϕ) D −ia.ξ F ∈ − . From (1.10) we have (τaϕ)=e ϕ.Weknowthat ϕ is in − , De F F F De because ϕ is in 0,and,by(vi)inTheorem2.1,weseethat (τaϕ) e− . X α F ∈D (iv) Since the derivative operators ∂ are linear continuous from e− into N α D e− ,thenforallϕ in 0 and all α N we have ∂ ϕ e− .From(1.8) D α Xα ∈α ∈D we have (∂ ϕ)=(iξ) ϕ. As (iξ) is a polynomial and ϕ is in e− , F F α F D then (vii) in Theorem 2.1 shows that (∂ ϕ)isin e− and consequently ∂αϕ . F D ∈X0 (v) If ϕ 0, then by definition of 0 (4.1), ϕ belongs to e− .From ∈X N X F D (1.14) we have ( ϕ)=(2π) ϕ ( ξ) . Since ϕ ( ξ) − ,then ( ϕ) F F − − ∈De F F also belongs to − .Thus (ϕ) ,thatis ( )= . De F ∈X0 F X0 X0 (vi) Since the space − is closed for the product (cf. Theorem 2.1 (v)), De then for all ϕ and ϕ in ,weobtainϕ ϕ − .Nowweneedtoprove 1 2 X0 1 2 ∈De 168 M. GORDON AND L. LOURA that the Fourier transform of ϕ1ϕ2 is in e− .FromTheorem2.1(viii), N D ∞ (ϕ1ϕ2)canbeextendedtoC as an entire function, so (ϕ1ϕ2) C . FixF α NN and k N.UsingtheclassicpropertiesoftheconvolutionweF ∈ have ∈ ∈ 1 N ek|ξ|∂α [ (ϕ ϕ )] = ek|ξ| ϕ ek|ξ|∂α ϕ . F 1 2 2π F 1 ∗ F 2 " # k|ξ| k|ξ| α On the other hand, e ϕ1 and e ∂ ϕ2 are in e− (cf. Theorem 2.1). k|ξ| k|ξ| Fα 2 F D Therefore e ϕ1,e ∂ ϕ2 L because e− is continuously embedded in and we knowF that Fis continuously∈ embeddedD in Lq, for all q 1 (seeS [18]). Thus, we can deduce,S from the general theory of theconvolution≥ k|ξ| k|ξ| α N product, that e ϕ1 e ∂ ϕ2 exists for a.a. ξ in C , is a continuous function and F ∗ F N k|ξ| α 1 k|ξ| k|ξ| α lim e ∂ [ (ϕ1ϕ2)] = lim e ϕ1 e ∂ ϕ2 =0 |ξ|→+∞ F 2π |ξ|→+∞ F ∗ F " # ) * which is equivalent to say that (ϕ ϕ )belongsto − . F 1 2 De (vii) From Theorem 2.1 we can say that pϕ belongs to − ,forallϕ in De 0 and all polynomial p. It is easy to show that (pϕ) e− ,becausefor X j F ∈D p = j∈NN , |j|≤n cjx we have

|j| j + (pϕ)= cj i ∂ ( ϕ) . F F NN j∈ ,, |j|≤n N (viii) Let ϕ 0 and b R .From(iii)and(v)abovewehaveτb ( ϕ) ∈X ib.x∈ F ∈ 0.Butτb ( ϕ)= e ϕ (cf. (1.11)). Using again (v) we can say that X ib.x F F ib.x e ϕ belongs to 0 and by (1.14) finally we obtain e ϕ 0. F F & X ' c 2 ∈X RN − 2 |x| &(ix)& Let ψ'' c and b .Thenψ = pe ,wherep .Weknow − c |x|2 ∈G ∈ − c |x|2 ∈P that e 2 0,thusτ b e 2 0, by (iii). Since 0 is closed for the c ∈X ∈X X 2 ) * − c x− b product of polynomials (see (vii) before), we have pe 2 | c | 0.Asthe set is a vector space, then ∈X X0 2 2 b − c x− b b.x e 2c pe 2 c = e ψ . | | ∈X0 (x) It results from (1.15) and (ix) of Theorem 2.1. ! Now we introduce in the following family of seminorms: X0 µk(ϕ)=νk(ϕ)+νk( ϕ) F (4.2) k|x| α k|ξ| α =sup e ∂ ϕ L∞ +sup e ∂ ϕ L∞ . |α|≤k % % |α|≤k % F %

With the family of seminorms (µk) is a locally convex space. X0 EXPONENTIAL GENERALIZED DISTRIBUTIONS 169

From the seminorms (2.3) and (4.2) we get a continuous injection of X0 into e− .Havinginmind(2.6)wehavethefollowingchainofcontinuous injections:D

(4.3) , − , . X0 →De →S Moreover, since is a dense subspace of ,cf.Theorem4.1,then X0 S (4.4) 0 , . X →d S Before we state the main theorem (Theorem 4.2 below) for the test func- tion space we need the following lemma: X0 Lemma 2. Let (ϕn) be a bounded sequence in and let ϕ − be such X0 ∈De that ϕn ϕ in − .Thenϕ and ϕn ϕ in . → De ∈X0 → X0 Proof. First we prove that ϕ is in .Wehavetoshowthat ϕ belongs X0 F to e− . The hypothesis of boundedness of the sequence (ϕn)implies,in particular,D that (4.5) N N k|ξ| α α N k N C 0 n N ξ R e ∂ ϕn(ξ) C. ∀ ∈ ∀ ∈ ∃ ≥ ∀ ∈ ∀ ∈ | F |≤ Using the dominated convergence theorem and the hypothesis ϕn ϕ in α → N e− ,wemaydeducethatthesequence(∂ ϕn)convergesuniformlyinR toD ∂α ϕ. Fix α NN and k N and let nF + in (4.5); then F ∈ ∈ → ∞ ek|ξ| ∂α ϕ(ξ) C. | F |≤ As α and k are arbitrary, this implies that ϕ is in − . F De It remains to prove that ϕn ϕ in 0.Wewriteψn = ϕn ϕ and prove →N X − that ψn 0in .Fixα N and k N;fromtheboundednessof(ϕn) → X0 ∈ ∈ in 0,weknowthat(ψn)isboundedin 0 and, by (4.5), this implies the existenceX of a real number C 0suchthat,forallX n N and all ξ RN , ≥ ∈ ∈ k|ξ| α −|ξ| (4.6) e ∂ ψn (ξ) Ce . | F |≤ α α N As we know that ∂ ϕn ∂ ϕ uniformly in R ,weseethat F → F α N ∂ ψn 0uniformlyinR . F → k|ξ| α Therefore, the sequence e ∂ ψn converges to zero uniformly in the | F | compact subsets of RN .ThefunctionCe−|ξ| tends to zero at infinity. From k|ξ| α these two facts we conclude, by (4.6), that e ∂ ψn converges to zero N | F | uniformly in R .Ontheotherhand,weknowthatϕn ϕ in − , that is → De ψn 0in − ;thisimplies → De k|x| α N e ∂ ψn 0uniformlyinR . | |→ This prove that ψn 0in . ! → X0 170 M. GORDON AND L. LOURA

Theorem 4.2. The space 0 is: (i) Hausdorff; (ii) metrizable; (iii) Fr´echet; (iv) Montel; (v) reflexive;X (vi) bornological (Mackey). Proof. (i) Let ϕ in such that ϕ =0.FromTheorem2.2(i)weseethat X0 , ν0(ϕ) =0.Thenµ0 (ϕ)=ν0 (ϕ)+ν0 ( ϕ) =0andthisimpliesthat 0 is aHausdor, ffspace. F , X (ii) is metrizable because the familly of seminorms (4.2) is countable. X0 (iii) We just have to prove that is complete. Let (ϕn)beaCauchy X0 sequence in .By(4.3),(ϕn)isalsoaCauchysequencein − and, X0 De applying Theorem 2.2 (iii), we may conclude that there exists ϕ in − such De that ϕn converges to ϕ in − .Since(ϕn)isaboundedsequencein , De X0 then ϕ belongs to and ϕn converges to ϕ in (cf. Lemma 2). X0 X0 (iv) The space 0 is barrelled because it is a Fr´echet space. Therefore, we just have to proveX that, if L is a bounded subset of ,thenL is rela- X0 tively compact in 0,orequivalently,thateverysequence(ϕn)inL has a convergent subsequenceX (in the topology). X0 The set L is bounded in and, by (4.3), L is also bounded in − .From X0 De Theorem 2.2, − is a metrizable and a Montel space, so there exists ϕ in De − and a subsequence (ϕm )ofthesequence(ϕn)suchthatϕm ϕ in De n n → − .ByLemma2weseethatϕ is in and ϕm ϕ in .Thisproves De X0 n → X0 that 0 is a Montel space. (v)X is a reflexive space because it is a Montel space. X0 (vi) The space 0 is a Mackey (or bornological) space because all metriz- able locally convexX spaces are bornological. ! From Theorem 4.1 we can conclude that the derivative operator ∂α, the translation operator τa and the operator ϕ pϕ with p are linear from → ∈P 0 into 0.Weknowthattheinjectionof 0 into e− is continuous (see (4.3)),X thenX using the seminorms (4.2) and theX fact thatD theseoperatorsare continuous from − into − ,itiseasytoprovethecontinuityofthese De De operators in .Moreover,asτa is bijective, with inverse τ a,weseethatthe X0 − translations are vector and topological automorphisms in 0. These results are summarized in the following corollary: X N Corollary 4.3. (i) For each a R ,thetranslationoperatorτa is a vector ∈ and topological automorphism in 0 with inverse τ−a. (ii) For each α NN the derivativeX operator ∂α is linear continuous from ∈ 0 into 0. (iii)X ForX each polynomial p,theoperatorϕ pϕ is linear continuous from into . → X0 X0 Theorem 4.4. The Fourier operator w is a vector and topological auto- N F |w| morphism in with inverse w . X0 2π F− ) * EXPONENTIAL GENERALIZED DISTRIBUTIONS 171

Proof. Consider first w =1.Astheoperator is the restriction to 0 of the operator defined in (see (1.3)), then TheoremF 4.1 allows easily toX conclude S the linearity of this operator from 0 into 0.Toprovethecontinuityitis enough to use the seminorms (4.2).X It is simpleX to show the bijectivity and 1 N we see that the inverse operator is 2π −1.As 0 is a Fr´echet space (cf. Theorem 4.2 (iii)), then it follows from theF open applicatioX ntheoremthat is an open application, therefore& −'1 is also continuous. F The case w =1isadirectconsequenceoftheformula(1.15)and(ix)inF Theorem 2.1. , !

5. Exponential generalized distributions ! We call the dual 0 of 0 the space of exponential generalized distribu- X ! X tions. We consider in 0 the strong dual topology. From (4.4) we haveX the inclusion ! , ! . S →X0 The space 0 is continuously embedded in and, as we know that is X ! S S ! continuously embedded in ,then 0 is also continuously embedded in 0. S X ! X Next, we prove that 0 is dense in 0.Itissufficienttoprovethatevery X ! X continuous linear form f on 0 that is null on 0 is identically null. Taking X ! ! X such a form f,wehavef ( 0) and, as 0 is reflexive, f 0;thisimplies that ∈ X X ∈X f,f = f(x) 2dx =0 & ' RN | | ! and this shows that f is the null form. Collecting all the previous results, we have the following chain of contin- uous and dense embeddings:

! ! (5.1) 0 , , , 0 . X →d S →d S →d X ! Theorem 5.1. The space 0 is: (i) Hausdorff; (ii) complete; (iii) Montel; (iv) reflexive. X Proof. (i) The strong dual topology (as well as the weak dual topology) is always separated. ! (ii) The space 0 is complete because it is the dual of a separated Mackey space. X (iii) The dual of a Montel space is always a Montel space. (iv) All Montel spaces are reflexive. ! ! Similarly to what has been done in e− (see (1.4) and (1.5)) we get, by Dα transposition, the derivative operator ∂ and the translation operator τa in 172 M. GORDON AND L. LOURA

! 0,whichextendtheseoperatorsdefinedin 0. They are linear and contin- X ! ! X uous from 0 into 0.Wecanstillseethatτa is a vector and topological X X! automorphism in , with inverse operator τ a. X0 − α ! Theorem 5.2. The kernel of the operator ∂ in 0 is the same as its kernel in !. X D Proof. It is sufficient to prove that, in dimension 1 (N =1),if∂T =0,then T is constant. First, it is easy to show the following equality: +∞ (5.2) ∂ ( )= ψ ; ψ (x) dx =0 . X0 ∈X0 - !−∞ . This shows that ∂ ( 0)isahyperplaneof 0. Write X X 2 1 − x ϕ0 = e 2 . √2π The function ϕ belongs to and 0 X0 +∞ (5.3) ϕ0 (x) dx =1 , !−∞ thus is the direct sum of ∂ ( )andspan ϕ : X0 X0 { 0} (5.4) = ∂ ( ) span ϕ . X0 X0 ⊕ { 0} If ∂T =0then,forallϕ in , ∂T,ϕ = T,∂ϕ =0.Thismeansthat X0 & ' −& ' ψ ∂ ( ) T,ψ =0 . ∀ ∈ X0 & ' Using (5.4) we have T,ϕ = T,ψ + λ T,ϕ0 = λ T,ϕ0 , where λ C, and & ' & ' & ' & ' ∈ +∞ +∞ +∞ ϕ (x) dx = ψ (x) dx + λ ϕ0 (x) dx = λ !−∞ !−∞ !−∞ In the second equality we have used (5.2) and (5.3). Therefore, T,ϕ = c, ϕ ,withc = T,ϕ C,whichimpliesthatT is constant. & ' ! & ' & 0'∈ As the Fourier transform w is a vector and topological automorphism F ! in 0, it can be extended this to 0,bytransposition,inthesamewayas X ! X for (see (1.6)). Moreover, the operator w is a vector and topological S ! F ! automorphism in 0 and the properties of w in 0 are still valid in 0,i.e., the properties (1.7)X to (1.15). F X X We are now able to identify some elements of the space !: X0 Theorem 5.3. If f is a continuous function defined on RN with exponential growth, then f ! . ∈X0 EXPONENTIAL GENERALIZED DISTRIBUTIONS 173

N Proof. Call e+ the set of all continuous functions defined on R with ex- ponential growthC and define ! : + T Ce −→ X 0 f Tf = (f) 1−→ T by

ϕ 0 Tf ,ϕ = f (x) ϕ (x) dx . ∀ ∈X & ' RN ! The existence of RN f (x) ϕ (x) dx and the linearity of Tf are straightfor- ! ward. To prove that Tf is in we only have to show the continuity of / X0 Tf . Let (ϕn)beasequencein ,suchthatϕn 0in . Using the domi- X0 → X0 nated convergence theorem and the hypothesis ϕn 0in 0,weseethat C → X ! Tf ,ϕn converges to 0 in ;thisshowsthatTf belongs to 0. & To prove' that is injective, we have to show that Ker =X 0 .Suppose T N T { } now that Tf =0;then,forallα N , ∈ −|x|2 (5.5) mα fe =0 , ) * −|x|2 −|x|2 where mα fe are the moments of order α of the function fe . It is easy) to see that* α −|x|2 |α| −|x|2 ∂ fe (0) = ( i) mα fe . F − Therefore, by (5.5),$ we) have *% ) *

2 ∂α fe−|x| (0) = 0 . F $ )2 *% 2 All the derivatives of fe−|x| are null at the point 0. Since fe−|x| F F can be extended to CN) as an entire* function (see Theorem 2.1)) and the* Fourier transform is injective, we can conclude that f 0. This proves the injectivity of . ≡ ! T Corollary 5.4. Let f be an entire function with exponential growth in CN ! and let (pn) be the corresponding sequence of Mac-Laurin polynomials in 0. ! X Then pn f in . → X0 N |α| Corollary 5.5. Let α N such that α =0. If the sequence α! bα ∈ | |, | | α ! is bounded, then the multipole series NN bα∂ δ is convergent) in . * α∈ ( X0 −|x|2 + ! Notice that although e may be identify with an element of 0 (see Theorem 5.3), the corresponding sequence of Mac-Laurin polynomialsX do not 174 M. GORDON AND L. LOURA

−|x|2 ! converges to e in 0.ThisfactisnotincontradictionwithCorollary 2 X 5.4 because e−|x| do not have an exponential growth in CN .

Next we give some examples of exponential generalized distributions which can be represented by convergent multipole series. Applications to ODE in ! are given. X0 x C x ! Example 5.6. It is clear that e has exponential growth in .Thene 0 (cf. Theorem 5.3) and, from Corollary 5.4, ∈X n xj ex in ! . j! −→ X0 j ,=0 ! As the Fourier transform is a vector and topological automorphism in 0, we have F X ∞ 2πij (ex)= δ(j) . F j! j=0 , ! The previous multipole series is convergent in 0 and its sum is the Fourier transform of the exponential function. X R iax ! Example 5.7. Let a .Thefunctione is in 0 because it is bounded. Therefore the corresponding∈ sequence of Mac-LaurinX polynomials converges ! in 0: X n (ai)j xj eiax. j! −→ j ,=0 Applying Fourier we obtain ∞ ( a)j (5.6) eiax =2π − δ(j) . F j! j=0 & ' , On the other hand, and having in mind the properties of the Fourier trans- form in !,wehave X0 iax e = τa ( 1) = 2πδa . F F Then, from (5.6), we obtain& ' ∞ ( a)j (5.7) δ = − δ(j) . a j! j ,=0 We have just proved that the distribution δa has the representation (5.7) as a convergent multipole series in !.Moregenerally,forn in N,wehave X0 ∞ ∞ ( a)j ( a)j−n (5.8) δ(n) = − δ(j+n) = − δ(j) . a j! (j n)! j j n ,=0 ,= − EXPONENTIAL GENERALIZED DISTRIBUTIONS 175

Example 5.8. Let us consider the following ODE: (5.9) x2T ! + T =0 . ∞ (j) Let us suppose that T has the form T = j=0 bjδ .From(5.9)wehave ∞ + (j) (bj+1(j +2)(j +1)+bj)δ =0 j ,=0 and we conclude that, for all j in N, ( 1)j bj = b − , 0 (j +1)!j! with b C.Therefore,thesolutionof(5.9)is 0 ∈ ∞ ( 1)j (5.10) T = b − δ(j) . 0 (j +1)!j! j ,=0 ! Now we show that the multipole series (5.10) is convergent in 0. Note that the sequence X j j ( 1) j! b0 − 1 (j +1)!j! 0 2 2 3j∈N1 2 2 is bounded, because it is convergent.2 From2 Corollary 5.5 we can finally 2 2 deduce the desired result. ! In the previous examples we showed that some elements of 0 have a representation as a multipole series. Next we show that not all theX elements ! of 0 have such representation. X x ! For example e belongs to 0 but does not have a representation as a multipole series. In fact, ex is theX solution of the following differential equa- tion: (5.11) T ! T =0 . − ∞ (j) If T = j=0 bjδ is a solution of (5.11), then ∞ + (j) (bj bj)δ b δ =0 . −1 − − 0 j ,=1 Thus, for all j in N,wehavebj =0andconsequentlyT =0.

Acknowledgement We wish to thank professor Jos´e Lu´ıs da Silva from Universidade da Madeira and professors Lu´ısa Ribeiro and Francisco Viegas from Instituto Superior T´ecnico, Lisboa, for their valuable suggestions. 176 M. GORDON AND L. LOURA

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M. GORDON Departamento de Matematica´ e Engenharias Universidade da Madeira Campus Universitario´ da Penteada 9000-390 Funchal, Portugal e-mail address:[email protected]

L. LOURA Departamento de Engenharia Electrotecnica´ e Automac¸ao˜ Secc¸ao˜ de Matematica´ Instituto Superior de Engenharia de Lisboa Rua Conselheiro Em´ıdio Navarro, 1 1950-062 Lisboa, Portugal e-mail address:[email protected]

(Received April 8, 2008 )