Kučera, Jan

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Institute of Mathematics AS CR, v. v. i.

Czechoslovak Mathematical JournalJan KučeraLaplace L2-transform of distributionsCzechoslovak Mathematical Journal, Vol. 19 (1969), No. 1, 181–189Persistent URL: http://dml.cz/dmlcz/100885Terms of use:© Institute of Mathematics AS CR, 1969Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documentsstrictly for personal use. Each copy of any part of this document must contain these Terms of use.This document has been digitized, optimized for electronic delivery andstamped with digital signature within the project DML-CZ: The Czech DigitalMathematics Library http://dml.czCzechoslovak Mathematical Journal, 19 (94) 1969, Praha LAPLACE L2-TRANSFORM OF DISTRIBUTIONS JAN KUCERA, Praha (Received november 13, 1967) In our paper [11] Hubert spaces L\, where к ranged in the set of all integers, were defined. It was shown that ^ = f]L\ с . . . с L^ с L^ с L^^ с .. . с [JL^ = ^ \ where У is the space of all rapidly decreasing functions together with their derivatives and the dual 5^' of .9^ is the space of tempered distributions (see [1]). Then Fourier transform #", based on the classical definition with the kernel exp ( —27cfÇ, x), is a unitary automorphism on every L\, The purpose of this paper is to transfer these results on Laplace transform. We make use of the following notation. ^ is the linear space, over the field С of complex numbers, of all functions f : R^ -> С with compact support s u p p / which possess continuous partial derivatives of all orders. The space ^ ' of distributions is the dual of ^ , where ^ is provided by usual topology, cf. [1]. Finally, we denote ^ \ - { / 6 ^ ' ; s u p p / c = < 0 , (X))"}. By a we denote a multiindex (a^, a2 , . . . , a„) e R", where all â  are non-negative integers. We write |a| = ^a^, x"" = П ^ } ^ for x e i^", D'' = д^^^Кдху. If there are for example variables (x^, . . . , x„, y^,..., j ^ ) and D^ acts only on variables (x^, . . . , Xj^^ then we indicate this by D^. If for a given multiindex a and a function f : R" -^ С there is a function g : R"" -^ С such that j^n gcp dx = ( — 1)''"' 5кп/^°'(р dx for all (p e ^, then we call g the generahzed derivative of/ of order a and denote it by D*/. Inequalities x ^ j ; and x < j , where x, y e R", mean that Xj ^ j j and, respectively, Xj < ур J = 1, . . . , n. As the function exp (a, x), with a fixed, will frequently appear throughout our paper, we denote it briefly by e^. Let us remind of definition 1 from [11]: For к ^ 0, integer, let L^ = {/ : i^" ~-> C; there exists D ^ for every ß, \ß\ й К and X U- x^^JD^P dx < +00}. There is defined an inner product in L \ by (/, g\ = JRn [ D J , D^g] dx, where D . = (1 + Z i^^i^J + ^ / ^ ^ # = ^ ^Mnixf D^ and 181 The space L2^, /с ^ 0, integer, is the dual of L2. If elements/, g e L^^' are, according to Fréchet-Riesz theorem, represented by (р,ф E LI^, then (/, g)-k = {Ф, (p)k defines an inner product in L]^. All L~2^, к integer, are Hubert spaces. Definition 1, For each integer к and each 7 e JR" we define the Hubert space Щ^у = = { / e ^ V ; e „ y / e L ^ } , with an inner product {f, g)k,y = {^-~yf,Q-yg)b ^ = = 0 , 1 , - 1 , 2 , - 2 , . . . From definition 1 and from [ U ] it follows immediately that . . . cz L^^y cz Ll^y a cz L^y cz Lj j , cz L j J c: ... , and that the identity-operator , / : L^^^ -» L[^y, к ^ I, is continuous. If cr ^ y, a,yE JR", then £2^^ cz L2,^, fe = 0, 1, — 1, 2, —2, . . . , and again the identity-operator J^ : Ll^y -^ L\^„ is continuous. Let us show that also the operator djox^ : L\^y -^ L\~y^ is continuous. First we prove the inclusion {djdxi) (Ь^^у) cz L27ŷ - Actually, for / G L2 у we have iSldx,)f={dldx,)(OyQ_yf) = {{dldx,yy)o^yf+^^^^ Evidently y^febly and (^/öxi) (e_y/) GL2 \ Moreover, supp (5/c)xi) (e_y/) cz cz <0, 00)". This implies that supp ey(5/3xi) (e .^ / ) c: <0, 00)" which proves Qy{djôxi) (o^yf) e L\~y^ and hence {djdx^jfeL^'y^. The continuity of djdx^ -^2,7 "^ -^ L\~y^ follows then from the continuity of д\дх^ : L\ -> L2~^ In [11], Фр^^, p ^ q "^ 0, signified the normed space of all such functions (p : R" -^ -> С that the mapping ij/ -> (pij/ maps continuously L^~^ into L|~^' for к = 0,1, ..., q. The norm was given by \\(p\\p,q = max sup ||<Р^|д-А-k = 0,l,...,q ll^!|p_fc=l Then it was shown that L "̂̂ " cz Ф^^^ cz L|~^~', where r = 1 + [^n], and that the identity-operators J^ : Ц"̂ *" -^ ф^^^, J : Фр^^ -> Ll"^'' are continuous. Every poly­nomial of degree к is an element of Ф^+и,а^ ^ è 0. For cp e (P̂ ^̂ , / e L2 ,̂ jp ^ (? ^ 0, we have defined the product cpfeLI^ by {(pf) ф = /{(рф), ф e L^. Then evidently \\(f>f\\--p й \\(ß\\p,q \\f\\-q- Hence the mapping {(p,f) -> (pf of Фр^^ x L~^ into L"^ is hypocontinuous (i.e. continuous in each variable locally uniformly with respect to the other one). This definition is in accordance with the classical Schwartz's definition of multiplication. Now, we are able to define the product cpf, where (p e ^p ,^ , /e L^jl, as an element of LJ^^ by (pf = Qy{(pQ^yf). The mapping {(p,f) -> (pf is then hypocon­tinuous. In particular, the m a p p i n g / - » {-xYf of L^̂ ^ into Ьз^у"'""' is continuous. For each real number с > 0 we define the homogeneity operator j f ^ : L2 -^ L2 as follows: Let / e L ^ . If fe ^ 0, then {ffj){x) = fixjc), xeRWt кй 0, then (Jfj) {(p) - c"/(^i/,(p), where (p e ЦК Proposition. Given с > 0, a multiindex a, and integers k, p, q, p ^ q ^ 0. Then 1) Же is a homeomorphic automorphism of L\ for which Ц^сЦ ^ (<̂  + l/<^)c^", ^ С — ' ^ lie-2) Ж^^ = c^J^Jf 1/̂ , where ^ is Fourier operator, 4) For cp e (Pp.,,/e L2" we have Ж^с/) = (Ж ,̂<р) ( Ж J ) . 182 Definition 2, Given y e JR" and an integer к. Then we define Laplace transform jSf as a mapping of L2̂ y into the space of all mappings of the set {a e R"; a "^ y] into L2 by the formula (1) {^f\ = ^^2nH^-J). feLl^, G^y, Lemma 1. Given y,XeR^, an integer k, a multiindex a and f e L\.^, Then the following formulae are valid: (2) ^-ixYfX = Щ^1\, where a^y, (3) i?(e, T>JX ={a-X+ hf ^{tJX, where a^y + X. In particular, for A = 0 we get (3a) i f(Dy), = ((7 + / т ) ^ ( ^ Д . Proof. 1) ^ ( ( - i x ) V ) . = ^^2^^(e-.(-ïx)V) = (27i)"^^,/4(-~rx)^e„J) = 2) ^{tldfldx,)X - {a, ~ X,) ^{^J\ = ^^,^{t,^ldf\dx,)- {G, - X,)o,.J) = = J^2nH^ldx,) (e,_ J ) = ^2.{2niT,) ^{^,-J) = iTi^,,#-(e^_ J ) = iz, ^{tj%. The mathematical induction completes the proof. Theorem 1. Given y G R", an integer к and f e L\^y. Then for every G > у Laplace image {^f)a is a function. Moreover, if we denote the variable of (i^/)^ by т, then {^f)a (T) IS a holomorphic function of variable G + IT on the set [G + it e C"; G > y}. Therefore we will further write (=âf/) [G + ix) instead of {^f)^ (т). 1/ к ^ 0, then for each pair of multiindices a, ß, |a| + \ß\ g —/с, there exists a polynomial P^ß of degree S\ß\ ^^d gp e L^^y such that (4) {^f){u)= E P^ß{u)D:{^gß){u), R e u > 7 . Remark. We shall see in the proof that if we differentiate with respect to (f Im и) instead of w on the right-hand side of (4), then (4) is valid on the set [u e C"; Re г/ ^ y). Having already known that ^f, feL^y, is holomorphic for Re м > у we might replace the operator D^ by D^^ in (2). In this way we would get (2a) ^-xff) (и) = ( D ; ; ^ / ) {и). Re t/ > у . Proof of Theorem L We first prove the second part of Theorem 1, then the first one. Put m = ~k. According to Theorem 2 of [11] for each multiindex jS, \ß\ S ш, there are hß e L2 and a polynomial Qß of degree ^m — \ß\ such that e„y/ = = Z Qßi~i^) D^̂ /î- As supp e_y/ c: <0, 00)", we may suppose that supp hß с: 183 c= <0, oo)" for every ß, \ß\ й т. Hence, using (2), (3), и = a + it, we get for a ^y, (5) ( ^ / ) . = ^(e,e_,/)„ = ^ ( e , I Q,{-ix) T>%\ = \<m = Z ßДD.)((^-r + '̂̂ )'̂ (eЛ).)• If we carry out the indicated differentiation we can write \oi\ + \ß\um where P^ß is a polynomial of degree ^ \ß\ - (m - \ß\ ~ \oc\) й \ß\- It remains to put Qß = Qyhß which is evidently an element of L2,y for every ß, \ß\ g m. To prove the first part of Theorem 1 it suffices to show that {<^gp)a is holomorphic on {u; Reu>y}, In fact, {^gßX = ^^2л^{^-<у^р) =" JK« ^^P ( - (0" + fi), x) gß{x) dx = == Jijn exp (7 — u, x) hß{x) dx. Let us take Ä > 0, then evidently H^{u) = |<о,л>« ̂ xp (y - w, x) hß{x) dx is an entire function of u. Put for brevity M = <0, 00)" - <0, A}"". Then for each г e e (0, 00)", Re w ^ у + г, we have Г |exp (y - w, x) hß{x)\ dx\ й S Г |exp(y - и,х)\Ых Г |й^|Мх ^ \\h\\lfl ; i - e x p ( - 2Ле,) . JM JM 5=1 25̂  Thus ИтЯ^(м) = О uniformly on [и; Re м ^ у + г}. According to well-known Weierstrass theorem the Hmit-function {^gß)<r is holomorphic on {ti; Re м > y}. The proof is complete. Corollary. It follows from the proof of Theorem 1 that there exists a function (p{u), holomorphic on {u; Re w > y} and bounded on {w; Re w ^ у + e} for every e e e (0, oo)^ such that (6) {^f){u) = <p{u)U{^j-yjT, R e u > y . Hence if we write ^f = F, then the integral g(x) = ^  ^ exp ((7 + IT, x) dx , j = i is absolutely convergent for every a > y, does not depend on a %ßd is equal to zero for X Ф <0, сю)", see [10], Lemma 1. 184 Evidently ^̂  is a distribution from П ^2,^ and can be differentiated. Let us write (T>y n / ß xm+2 п /ß \m+2 n - П — - Ti %^-Щ(т + 2nix) П (aj + IniTj - yj)---') = j=i\dxj J i=i = o,^-\F{a + 2niT)) = e , ^ ~ 4 r i / 2 . % + n) = = e ,# ' : -^# ' (e_J) = e , e . J = / . We have got an inversion formula Theorem 2. Given у eR\ a non-negative integer m and f eL2'y> Let F = S£f. Then for every a > y we have (7) / == (2710- П — " ^̂ 1 ^(^) П K- - 1^"-' ^̂ P («̂  A dti , where the indicated differentiation is in the sense of distribution theory. Definition 3. Given yeR"" and an integer fc ^ 0. Then we put H^2,y = {^l ^ is holomorphic for Re м > у and sup ^ ^^n \{o + ixfY • \^^F{G + 1т)р di < < +oo}. "^^ И + 1/̂ Î fc Lemma 2, Le^ у G i^" anJ к "^ 0 be an integer. Then 1) Laplace transform S£ is an isomorphism of L\.^ onto H^y. 2) For each F e H\y and each multiindex ß, \ß\ ^ к, there exists hm D'̂ î ((7 + ii) a-*y + in the topology of L2 {with respect to the variable т) and this limit is the generalized derivative of order ß of Hm F{a + IT). We denote Hm F{a + /r) = F(y + IT). 3) For each oc, ß, |a| + \ß\ й К the function Ĵ ,̂ |(Ö- + ixf DfF((j + ix)\^ d t zs non-increasing in all variables on the set {a e R!^; a ^ y]. In particular. sup a>y \{a + ixf DfF((7 + ZT)|2 dr = \{a + ixf DfF(y + 1т)|2 dt 4) Я2,у is Hilbert space with an inner product (8) ^^'^^""^-^i^jl [ад^ + '̂ )'ß*G(7 + iT)J dt, 185 where \ j=l ÔTjJ W\ + \ß\uk and [D,F{y + it), D,G(7 + ii)] = 5) J^ is a unitary mapping of L^^ onto H2^y. Proof. Points 1, 2, 3 follow immediately from [10], Lemma 6, point 4 is evident. We prove point 5 with the help of [11], Theorem 1. Compute at first = ^ , / 2 . ( l + l(iTj + 2n -^)\ = yf./^ß • Now for f, g e L^^, we have J R» J JR« (2я)-« [D.if/, D,i?0] d. = (if/, ifg)H.,„ . Definition 4. Given у e R" and an integer fe > 0. Then we denote Hj,^ the hnear hull of all functions u" D > , |a| + [jS] S К where Ф e H _̂̂ . For F = u"" D > , G = !<" D »̂//, Ф,ф€ Hj,^, [ос] + 1JS| й к, \x\ + \À\ й К we define the inner product (9) (F, С)-.я,.^ = (D^d-x)" if-^Ф) , D'^a-x)^ ^-'Пь-'^... • Theorems. Laplace transform ^:L\y-^H\y is a unitary isomorphism for each integer k. Proof. The case к ^ 0 has been already proved in Lemma 2. Further let к < 0. As it follows from Theorem 1 Laplace transform if is a one-to-pne mapping of L2^y onto H\ y. The hnearity of ^ is evident. The inner product (9) was defined so that «Sf 186 turns out to be unitary. Indeed, let F, G e Я2 y, F •== ^ â ŵ"" DfФ«^, G = = E b.X 0„>.я, Ф«/>, •/'.я е Я«, . Then И +1/»! ^ -* |и| + | Я | а - * = Е «.д Е и^"Ч«^оЧ,),А'-Ч«''о''А.я)и,= Corollary. Я2,у is the image of a unitary mapping of Eilbert space L\^y. Hence it is also Hilbert space. Definition 5. Given у e R" and integers p, 4^ P ^ q ^ 0. Then we define a nor med space Фр^^^у = { / G ^ + ; there exists ge(9p,q such that Q-yf = ^g}. For/e^P*^^^^ we put | | / j |L ,y= l l^~ 'e_ , / | | ^ , , . Remark . It was shown in [11] that 6?*,g,y ^ Щ,~у^~'' and that the identity-operator J : {P*̂ ĝ ,̂ ~> L i ; / " " is continuous. Lemma 3. Given integers p, q, p ^ q ^ 0, f E ^ + n =^(^р,^) cind g e ^ + n L~2^. Then f ^ g eS)'+. [The convolution (.*.) was defined in [11] as a mapping from #-((P^J X L̂ ^̂  ШГО L^^ byf^g == ^-\^f.^g)). Proof. We show at first that supp #"̂ 0^ с (—oo,0>". Take cp e Ь\ for which supp ^ n (— 00, 0>" = 0. Then supp ^^cp n <0, oo)" = supp (p{ — x) n <0, oo)" = 0. Hence [З^^д) (p = g{^^(p) = 0. As ^ is dense in Lf, it suffices to prove that ( / * ^) (̂  = 0 for every cpeS^ fulfilhng supp cp n <0, GO)" = 0. We have {f^g)(p = ^^'g{3^~^cp * / ) . Hence it suffices to show that supp(.^~^ç9 * / ) n ( - o o , 0>" = 0. Actually, {^~^(p ^f)e ЬГ^"" . Take ф e L^^'-^ for which supp ф a ( - oo, 0>". Then {^~^(p^f)ф = (c^V) {^~^(p^ ^^Ф) = 0. We have received the last equality from the following equality supp {^~^(p * ^~^ф)п{~00, 0>" = supp (J^n (p{y - x) ф{-у) Ay) n {-co, 0 / = 0. Remark . It follows from Lemma 3 t h a t / G (P*̂ ^̂ ,̂ g e L^̂ ^ implies (e_y/ * e.^^^) e e ^ + . This enables us to state Definition 6. Let y e JR", integers p, q, p ^ q ^ 0,fe &p,q,y and g e L^^y be given. Then we define the convolution f ^ g as an element o / L J ^ by {Щ f'^g = ey(e_y/*e_,öf). 187 Remark . For /еФ*^^^^, ^ e L , , « we have | | / * ОГЦ ,̂-̂ ^ ^ = | |e_X/* â')||-p = = | |e_, /*e_,0 | l_p_^ | | ^ ( e_ , / ) l | , , , | | e_ ,0 | l_ ,= | | / | |* , , , i |öIU-„„. Thus the mapping {Â 9) -^ f "^ 9 of ^*.g,7 ^ ^2,7 iî^to L2 ^ is hypocontinuous. Lemma 4. Let y e î ", integers p, q, p ^ q '^ 0, /еФ^^у and g e LJ^^ be given. Then for every cr ^ 7 the equality (11) e^(e_ J * e.^é') = e / e _ / * e.^^f) holds. Proof. Put F = Q.yfe^{0p^g) n 9+, G = e^yg e L";̂  n ^V- We have to show that (Cy^^F) * (ey_^G) = e^„^F * G) holds for every a ^ y. According to the hypocontinuity of convolution and density of ^ in L2^ we may assume that G e eS^ n 9+. Take (pe^. Then (e,_,F * e ,_G) (p = (e,_,F) {^'(Oy-^G) * ф) = F(e,_,(#'^(e,..,G) * cp)), e ,„ , (F*G)(p = F(J^^G*e,„ , (p) . However, e,_,(#-^(e,_G) * Ç)) (x) = е,_Дх) еу„Ду - x) G(j; - x) (p{y) dy = - f G(3; - x) e,_ Ду) <p(};) dy = (.^^G * е,_,ф) (x ) . As ^ is dense in L^, the proof is complete. Theorem 4. Given fe (Pp^q^y and g e LJ^^, P ^ ^ ^ 0, integers, y e JR". Тйеп (12) ^ ( / * g) (u) = ^ / ( w ) . ^g{u) , Re w > 7 . Proof. Take a > y. Then according to (U) we may write ^{f * ^) (^ + 2niT) = J^(e_,( / * g)) (т) = J^(e_ J * e_,^) (т) -= J ^ ( e - J ) (T) # '(e_,^) (T) = ^/( (7 + Inir) ^g{c7 + Inix) . References [1] X. Schwartz: Théorie des distributions, vols I, II, Hermann, Paris 1950, 1951. [2] L. Schwartz: Transformation de Laplace des distributions, Medd. Lunds Univ. Mat. Semi-narium, tome supplémentaire, Lund 1952, pp. 196—206. [3] L. Schwartz: Méthodes mathématiques pour les sciences physiques, Hermann, Paris 1961. Russian trans. Математические методы для физических наук, Москва 1965. 188 [4] A. H. Zemanîan: The distributional Laplace and Melîin transformations, J. SIAM AppL Math., Vol. 14, No 1, January 1966, pp. 41 -59 . [5] A. H. Zemanian: Inversion formulas for the distributional Laplace transformation, J. SIAM Appl. Math., Vol. 14, No 1, January 1966, pp. 159-166. [6] Ditkin, Kuznecov: Справочник no операционному исчислению, Москва, Ленинград, 1954. Czech trans. Prîrucka operâtorového poctu, Praha 1954. [7] Ditkin, Prudnikov: Операционное исчисление по двум переменным и его приложения, Москва 1958. [8] S. Bochner, К. Chandrasekharan: Fourier Transforms, Princeton 1949. [9] / . Necas: Une note sur la propriété caractéristique de la transformée de Laplace d'une fonction et sur certaines espaces de Hubert dont la somme est l'ensemble des transformés de Laplace de distributions, Cas. pro pëst. mat., 83 (1958), pp. 160—170. [10] / . Kucera: Multiple Laplace integral, Czech. Ma h. J. /8 (93), (1968), 666-674. [11] / . Kucera: Fourier L2-transform of distributions, Czech. Math. J. 19 (94), (1969), 143—153. Author's address: Praha 1, Zitnâ 25, CSSR (Matematicky ùstav CSAV). 189 

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