Inversion and Representation Theorems for the Laplace Transformation Citation Rooney, Paul George (1952) Inversion and Representation Theorems for the Laplace Transformation. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/1WJR-ZQ41. https://resolver.caltech.edu/CaltechTHESIS:12122017-142045730 Abstract A study is made of the Laplace transformation on Banach-valued functions of a real variable, with particular reference to inversion and representation theories. First a new type of integral for Banach-valued functions of a real variable, the "Improper Bochner" integral is defined. The relations between the Bochner, Improper Bochner, Riemann-Graves, and Riemann-Stieltjes integrals are studied. Next, inversion theorems are proved for a new "real" inversion operator when the integral in the Laplace transformation is each of the above mentioned types. Lastly, representation of Banach-valued functions by Laplace integrals of functions in B p ([0,∞);¥), 1 ≤ p < ∞, is studied, and theorems are very like those proved, for numerically-valued functions, by D. V. Widder in his book "The Laplace Transform" (Princeton, 1941) page 312. The classes H p (α ; ¥), 1 ≤ p < ∞, are also studied in this section as is the representation of numerically-valued functions by Laplace-Stieltjes integrals. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Mathematics and Physics) Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Mathematics Minor Option: Physics Thesis Availability: Public (worldwide access) Research Advisor(s): Erdélyi, Arthur Thesis Committee: Unknown, Unknown Defense Date: 1 January 1952 Record Number: CaltechTHESIS:12122017-142045730 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:12122017-142045730 DOI: 10.7907/1WJR-ZQ41 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 10602 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 12 Dec 2017 23:22 Last Modified: 16 May 2023 22:14 Thesis Files Preview PDF - Final Version See Usage Policy. 18MB Repository Staff Only: item control page

INVERSION AND REPRESENTATION THEOREJIJS FOR THE LAPLACE TRANSFORlflATION Thesis by Paul George Rooney In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technolo gy Pasadena, California ACKNOWLEDGEMENT It gives me great pleasure to acknowledge my indebtedness to Professor J!.•• Erdelyi for his invaluable aid in this endeavour. His kindness and helpfulness to me will always command my gratitude. I must also extend r11y t h anks to Professors H. F. Bohnenblust and s. Karlin for considerable assistance. ABSTRACT A stuC.y is made of the Laplace transformation on Banach-valued functions of a real variable, 1T.i th particular reference to inversion and representation t heories . First a new type of integral for Banach- valued functions of a real variable, the 11 ImpropET Bochnertr integral is defined. The relations b etween the Bochner, Improp er Bochner, Riemann- Graves, and Riemann-Stieltj es intee;rals are studied. Next, inv ersion theorems are :;Jroved for a new 11 real 11 inversion operator vv-h en t he integral in th e Laplace t ransformat i on is each of t he abovementio ned t ype s. La s tly, re1)resentation of Banach-valued functions by Lap lace i ntegrals of functj_ons in B ( [ O, co ); ::f ) , 1 4 p ~ ro, is studied, p and t heo rems are prov ed giving nec essary and sui'fi cient conc.i.iti ons . The t l.1eore;ns are v er;y like t l1ose p roved, for nume r ically-valued fu nctions, by D. V. 7.'i dder in his b ook ''The Lap:W.ce t r ansfor m" ( Prine et on, 19L1.1 ) page 31 2. The cla ss es H ( ol. ; p x ), 1 ~ p <:. °"' are also stud:i.e d in this 3ection as is t h e re~) res entation of nmneric a.l lyvalued functions by L::.:;)la ce-Stieltj es int e grals. TABLE OF CONTENTS PAGE TITI.Z!: PAI~;T Intro clue ti on vi Chapter I 1 1. Introduction 1 2. Functions and Measure 3 34 Integration 5 4. 5. Improper Bochner Integral 8 Riemann- Graves Integral 15 6. Riemann-Stieltjes Integral 18 7. Relations Between Integrals 19 8. Vfoak Convergence of Operators 24 Chapter II 27 1. Introduction 27 2. Some Preliminary Lenunas 29 3. Inversion of Transformations of TY';ie I 34 4. 5. Inversion of Transformations of Types II and III 39 Inversion of Transf onnations of Type IV Chapter III 43 45 1• Introduction 45 2. Preliminary Lemmas 47 3. Fundamental Theorem 51 4. Representation Theorems for 1 K -C ' [r ( A ~ 54 v Page Title Part 5. Representation Theorems for f( ?-.. ) in H (ex.,;:?£:' ) 6. A Special Representation Theorem for a Class of p Numerically-valued Functions References 65 70 71 I NTRODUCTION The theory of the Laplace transformation on real or complex functions of a real variable is one which has been, for a number of years, of considerable interest to both pure and applied mathematicians. The interest of a number of these mathematicians has centered around inversion and representa.tion theories, the former of these theories being of interest from both the utilitarian and function-theoretic points of view, and the latter from a primarily function-theoretic viewpoint. Historically, the Laplace transformation arrived on the scene considerably before large developments in abstract spaces, and consequently was defined only on nUi-nerically-valued functions. However, the extension of the definition o.f the' transformation to the domain of Banach- valued functions of a real variable, that is functions on ( O,oo) to a Banach space, is one of the several e2..-tensions that were made, and it is with this extension that we shall be dealing here. In many respects the theory resulting from this extension is now approaching the degree of completeness enjoyed by the theory for nrunerically-valued functions. However, in the respects of j_nversion and especially representation· theories, the approach is not so close. In parti cular there are, as yet, no representation theorems for Banach-valued functions of what might be called the 11Widder 11 type. This terminology recpires some explanation. Let f( ?.. ) be a numerically-valued function of the real or complex vari2.ble 'A , and let LK,"C [f( 'A)] denote any fixed but arbitrary inversion op erator for the Laplace transformation. Then it has been vii shmm for many such inversion operators that if LK.,-c J°" l1K.)-C Lr( A )]I pd "C ~ M, p fixed, 0 M, K > K , 1 < p < °"' K > K. , or ess. sup \ LK)-C [t( I\. )]I exists, and either 0 ~ o~ -i:: < °" , f( 'A where M is independent of K ) is equal for 'A > 0 0 to the Lap1ace transform of a function in L (O ,oo) or L (O, oo) respectively. p 00 A theory of this type for a particular inversion operator is what we ~~ For examples see Widder ~ call a Tr\\fidder 11 type theory. [ 1 2, ch. 7, The task of developing 11Widder 11 -f:0rpe representation theorems for Banach-valu ed functions is t he one we have set ourselves. Since a 11 Viidder" type representation theory is stated in terms of a specific inversion operator, the opportunity was also presented both to enlarge the irwersion theor-y- for Banach-valued functions, and to study a new inversion operator for the Lap lac e transf ormation. We have grasped this opportunity and have developed the theory in terms of a new 11 real 11 inversion operator. A real inversion operator is one that utilizes the values of the generating function a~ising only from real values of the independent variable . Several of these are knovm; for examples see Widder [12 , ch. 7, § 6; ch. 8 § 25 J , or Hirschman [ 7] • The new operator in question is denned bv the formula 00 1 • 1 2K 2 I L Ki:- [f( A =:~ cos(2 K'll 2)f( K.( 11 +1 )/T: )d ?( • 3 ' I f f(?.) ~ 1l r .-'A-C~(l:: )d-C 0 = , then under certain conditions, 0 Hurribers in squa.re brackets refer to the biblio graphy at the end of t h e dissertation. viii lim I<-+ 1K ' 00 ~ [ f( 'A )] = ~ ( -c ). The fact that the representation theorems will be stated relative to this particuJar operator is no real restriction, for the method i s quite general , and will work equally well with any inversion operator for which the theorems are true in the numerically-valued case. This operator was originally given by A. Erdelyi [ 3] . H01'lfever the resulting inversion and repres errt.ation theories were not developed there. These theories were developed, by the author , originally for the numerically-valued case and have been acc epted for publication; see Rooney [ 11 ] • There is another operator related to I , which i s given by the formula 1 - K.e 2K. - Tt""'C' J°" sin(2 K'Y(2 )f( K( Y{ + 1 )/-c ) d 'Y( . 0 Both I and II are special cases of a third opera tor III LK -c [ f ( 'A ' ~ = (2-CK \I ( 2 K ))- K 1 co J 1 '>'l /' Jy (2K'>t2) f ( K(?t +1)/c )d1'/ , ...., 2 0 which can also be found. in Prof. Erdel~ri ts paper. The inversion and representation theories for these last operators have also been investigated, and were found to be sir.J.ilar in every respect to those for operator I. To avoid inessential difficulties we shal l restrict our attention to operator I. / Operator I has some points of resemblance to Phragmen' s operat or [l 2; ch. 7, § 2] in that both are "real' i nversion operators , involve only the values of f( A ) for large real values of A only elementary functions. , and involve IIowever, Phragmen's operator is not an inte gral operator. ?Te have, perhaps inevitably, been drawn into certain subjects which, wh:i.le of great interest and importance, are subsidiary to the main theme of this dissertation. Chief among these are certain diffi- culties concerning the relations between various integrals of Banachvalued functions. These difficulties are resolved in Chapter I, wherein are also the main theorems we shall need concerning these various integrals. Chapter II is devoted to invers ion theory. This theory is developed for several different kinds of integrals~ to the representation theory. This chapter may be considered to contain the principal results of the dissertation. Chapter III is given over -1- Chapter I 1. Introduction. In this chapter we collect and elaborate certain theorems concerning integrals of Banach-valued functions of real variables. The main tool in this regard is the Bochner integral, which is an analogue for this type of function, of the Lebesgue integral. sections 2 and J. Its theory is developed in Nearly all of the material of these two sections is abstracted, verbatim or paraphrased, from Hille [ 6] . for such theorems no proof is offered. Consequently, The reader who wishes to see proofs of these theorems should look in Hille [6] • In other places where known results are used, we shall give references to these results. Section 4 introduces a slight generalization of the Bochner integral which we call the "Improper Bochner Integral". It is the analogue of the improper Lebesgue integral. Less powerful, but nevertheless important, tools are the RiemannGraves and Riemann-Stieltjes integrals, which correspond, in the numerically-valued case, to the Riemann and Riemann-Stieltjes integrals respectively. Their theory is outlined in sections 5 and 6. In section 7 we develop theorems giving sufficient conditions for various of these integrals to be equivalent. Finally in section 8, we develop a weak sequential form of the Banach-Steinhaus theorem. The notation to be employed in this and subsequent chapters is, in the main, that of Hille f6) • That is, Banach spaces will be denoted -2- by Germ.an capital letters, and their elements by English lower case letters. The space of bounded linear functionals over a Banach space will be denoted by "starring" the symbol for that space, and its elements will be denoted by the 11 starred" elements of the Banach space. Heal or complex numbers will be denoted by Greek letters. - We shall denote the zero vector of a Banach space by e, and the void set of a collection of sets by 0. Several exceptions to this rule will be made, mostly in cases where long usage has prescribed symbols, which clash vvith the above notation, for certain quantities, e.g. LP' e, etc. One notable exception is that we shall often use English letters for subscripts. Other exceptions will be seen to occur. One other point is worthy of notice. 11 Vihenever we use the word limit 11 we mean the limit in the strong sense. will be pref aced by e:xplaining words. Other types of limits -3- 2. Let ~ be the k- dimensional Euclidean Functions and 11easure: space, ~ a measurable set in ~' and x( oe.. ) a function on the Banach space ~ Def ini ti on 1 • 2. 1 : to ..:f • • Let x( "'" ) and The sequence { xn ( °'- ) with m( 6c- )< t x( ol ) on 2) - L: e be functions on ~ xn ( o1., ) J converges to x( almost uniformly if to every t > (i) oL ) o, there is a set ~c such that { xn( c£.. ) } converges uniformly to • (ii) almost everywhere if there exists a null set that Hx( o<. ) - xn( cj._ ~( Theorem 1. 2.1: to E -+ 0 for L 0 C L; such in 2: - Z::0 • ol. The two types of convergence are related as follows: (i) implies (ii) , and if m(L') <. <»' (ii) implies (i). Definition 1.2.2: ( i) x( oL ) B is said to be finitely-valued in if it is constant on each of a finite number of disjoint measurable sets j B j with UI; . = ~ • (ii) J x(o<..) is said to be countably valued if it assumes at most a countable set of values in X each on a separate measurable set ~ .• J Definition 1. 2.J.: (i) ::0- if x.,..(x( ex:. )J for every x-Jf- E. ::X-,.. • x ( ol ) is said to be weakly measurable in measurable (Lebesgue) in L:; (ii) x ( o<.., ) is strongly measurable in B if there exists a sequence of countably-valued functions converging almost unif or-mly in x( °'- ) • is ~ to -4- Theorem 1. 2. 2: ( i) and If x( oc. ) and y( ol, ) are strongly measurable in ?f' 2 are constants, then E Y1 and er 1x( °" ) + 0 Zf( °"' ) is strongly measurable. (ii) If ce ( ot. ) is a finite numerically-valued function which is measurable (Lebesgue), then <e ( o(. ) • x( (){.,, ) is strongly measurable if x( cX. ) has this property. (iii) If x ( cX. ) is the limit almost everywhere of a sequence of strongly measurable functions, then x(<X. ) is strongly measurable. Definition 1. 2. 4: J: to the space A function x( ot.. ) on the closed interval (J , .3 ] 1 2 is of (i) bounded variation if supll .2; [ x( (3 . ) - x( o£. . )]II <. .,, for every i 1 1 choice of a finite number of non-overlapping intervals ( ol., A . ) ~ J. x(cx... 1 )1\ < 1 in (] 1' ] ] ; 2 (ii) strongly bounded variation if sup where all possible partitions of [ 'L;\' x( oL .) - . J. 1 J.- J 1 , J 2 ] are allowed. oo The two suprema are known as the total and strong total variations respectively. Definition 1. 2. 5: domain if _()_ A set _()_ of complex numbers will be called a is an open connected set. 'l'he closure of a domain will be called a closed domain. Definition 1. 2. 6: If 'S" is a complex variable, and x ( function on the open domain _i2_ 5 ) is a of the complex plane to then x ( -~ ) "Will be called holomorphic in _()_ holomorphic in Cauchy• s sense for every x *" x if x *" (x( ~ ) ) is 'Y..14- • in .x. , -5- 3. Integration. Definition 1. 3. 1 : A countably-valued fu..."lction x( ol ) on E is integrable (Bochner) if and only if (Lebesgue) . ( B) s E II x( ol )I\ to :X- is integrable By definition 00 x( tX.. ) d o£. : 6 j=1 The series converges since 00 00 II~ xJ. m( E J.)II~ L j =1 j=1 II xJ·ll m( Ej) = S.E \\ x( al- ) \\ d ol • Consequently " ( B) J ~ x( o£. ) d Definition 1. 3. 2: ~ l\ ~ Sl\x< «- ) \\ d o£.. • ~ A function x( 0(.. ) on ~ to ":;t is integr able (Bochner) if and only if there exists a sequence of countably- valued functions converging almost uniformly to x( D<J ) , and such that By definition ( B) 1 °" ) ~ = ~ x( d J lim (B) x ( o£. )d 0(, .. En n ~oo • We shall drop the "(B)" from the integral when there is no danger of confusion. Under t he postulated conditions , the integral exists uniquely. That is , for every sequence { xn} of countably-valued functions wit h - 6- the above postulated properties, J~ xn (oG )doc:. (B) \ lim n -i- co exists , and bas the same value. Theorem 1. J . 1: to X A necessary and sufficient condition that x( ol be integrable (Bochner) is that x( ) on 6 °'"' ) be str ongly measurable, and that 5~ Jlx< )II d 0(, Definition 1.3. 3: 0(, < .... B(E ; ~) , 1 ~ p< co , if x( OG p )t: xc "'- l\I p 11 to :X A function x( <X.. ) on E d oc. -<.. ) is strongly measurable on E x( Ol. ) belongs to B ( E co. belongs to 0) , and , ~ ) if x( .x. ) is strongly measurable and is bounded except in a null set. * If ~ Theor em 1. 3. 2: B (~ ; p is a real or complex Banach space, then x ) is a real or complex Banach space under the norm 1 . n x( • ) II p = { sE II x( U x( • )l\ ()() 0(.. ) ll p d 0(.} p J 1 ~p <. .., = ess.I; sup I\ x( ct. ) I\ • Theorem 1.3. 3: If x( <X.. ) is i n B( E ; x) , then \ 5x( oe. )d°'"' ~ E Theorem 1. J . 4: If x( 0(. ) is in B( ~ ; x) and x* is in ~*" , t hen x""'° (x( ov ) ) is in L('E' ) , and x*(~ ~ x( ot.. )dol-) = Theorem 1.3. 5: S. ~ x*(x(o(.. ))d°" • If x(ol.) is in B(E1 ; X) , then for almost all j E1' ~- We shall often use B(Z ; x) for B ( ~ ; ~ ) . 1 in -7- . (i) 1 im 1 - l~ 0 ~ ~j +~ Jj I\ x( QC.. ) - x(3 )fl dot. = O, and in particular 1 . 1im ($~ 0 ~ (ii) = x( j Definition 1.3.4: ). V~e shall call the set of 3 where fornmla (i) of the preceding theorem holds true the Lebesgue set of x(ol ). If x( OC..) is in B(E ; :X), and if Theorem 1.3.6: 1 J is in the Lebesgue set of x(o(... ), then the Lebesgue set of x(oc:. ) is equal to ll x( o<.. ) - x( J )fl • the Lebesgue set of If x( o<. , ~ ) is a strongly measurable function of Theorem 1.3. 7: ( 0(. , (3 ) = ( ~ 1 , • • • , ~ m' (3 1 , • • • , (' n) , then x( 0(. , ~ ) is in B(E ; m+n x( o<., x ) i f there is a function y( oe.. , R ) such that y( ~ ~ ) for almost all ( ol , ~ ) and JE { n exists. In this case x( d 0(. d s °" ,~ ) ~ = ) E = ) { E m ~ y( d- , ~ )d~ 1 d 0(, EI\ y(oL m E n E m+n m+n J 0(.. , ,q I ) = ,~ ll\ d") d/3 E m • E n Theorem 1.3.8: If x ( O(; ) are in B( :E; n x ) for all n, and the sequence converges almost uniformly to a limit function x(ol ), and if there exists a numerically-valued function <e ( ()(., ) in L( B ) such that If xn ( cJ- )U ~ <e ( o1., ) for all o{, in L; and lim n~"" JE x n (o<.. )do(, = ~ x( oc. )doc:. • ~ , then x( oe.. ) is in B( 2i ; :l. ) , -8- 4. Improper Bochner Integral: '/. 5 ~ > A • If and all ; X ) for a fixed Let x( oC. ) be in B( [ 'A , w] Definition 1 • 4. 1 : w x( ot.,, ) do<.. converges (in the ~ strong sense) to a limit y, as (&) -+ °"' that is to say if for any €> 0 there is an (,,) 0 ( £ ) such that S: " y - x( c<. )doc,\I-" € for every W > w , then we say that 0 the improper Bochner integral of x(cx. ) over -+oo S"' put SW =y = lim x( ex. )d c:x. ,. A c.J-+ "" ['A ,"") exists and we x( ct... )d at.. • We shall prove tvro theorems concerning interchange of integrations when one of the integrals involved is an improper integral. For this we need the folJ.owing two lemmas. Lemma 1.4.1: Let x(OC.) be a strongly measurable function on the finite closed interval [ 3 ,11] to the Banach space ~ • 'I'hen x( 0(..) is the almost uniform limit of finitely-valued functions on this interval. Proof: By assumption there exists a sequence of countably-valued functions x (ex,), (n n there is a set = 1,2, ••• ) such that for every L: 1 <;;; [3, '>?) and an integer N( t m( .E, ) < e2' all in [3,?z] -E 1 • and there exist sets i "" f. j, and LJ i=1 l\ xn ( ct. ) - x( o(.. ) II~ E. E.,c'~o ) such that for n "'7' N( E, ) and Since x (C(.. ) is countably-valued, n 2: n,i. such that xn ( z. n,i. ) = xn,i. , !: n,i.n E n,J. = 0, m(E n,i) = '>'l - 3 • - 9- Thus for every E. 1 o, M( e , n) exists such that I > 00 i=M 00 Z:: 2 = 2::; 1 U ( U Let U 2:: . ). n=1 i ~ M Then n, i oo I ~ m(~1) + L::; ( E m(:62) n=1 Further, let x~ ( oC.. ) I M in E 2 t ~ + n2:=" 2~+1 = c • m(.8 n i))..:::: ' = xn ( °" ) in [ J , "l] =e Then the x i~ co ,, -~ 2 • are finitely-valued and on n - ~ 2 , llx~( o£-) - x( oL )I\<. e , n > N( £ ), since x~ = xn on [ J , 1(] - E 2 • (J , '>l] Lemma 1.4. 2: If x ').. ( oL ) are in B( [ 0 , oo) ; ~ ) for each 1• 2. lim '),. ~ in Lo, w] , each \\x ~ ( oL )\\ ~ <(! ( ol.- ) for x A. ( oL... ) = x( ()(... ) uniformly for oG 00 W> 01 J . · ~ ( OC) in L(O,oo) exists such that all A > 0, A> 0 and all o<. , 0 ~ o,(., ..:'.'.: ro; then ( i) x( oC. ) is in B( [ O, co) ; :X )J (ii) lim ( co A~ooJO x l. ( o<.. )doc. = ~ 00 x( ol )dol. • 0 Proof: (i) Since x( o(.) is the strong limit almost everywhere of strongly measurable functions , x( ()(, ) is strongly measurable 'UIJ theorem 1 • 2. 2 (iii) . Further, since \\ x 'A ( °'- )\\ ~ <e ( ol. ) , \\ x( OL.- )\\ ~ (( ( o<.. ), - 10- and t hus l\x(ol. )\\ is in L(o , ... ) . Thus by theor em 1 . 3 . 1, x(oc.) is in B( [ O, ro); X ). (ii) l\ x ?I ( Ol. )I\ ~ ce (0(. ) , and \I x( oc )l\ £: Since r S~ <e c l doc , anct co(. l 11 d o< "' 11 x , oL ~ WO r for every \lx(ol. )\\ 0 <e (o£...) is in 1(0, ro) , for every E. > o, Also , since exists such that J ce ( ol )d ol < ~ for ..:::.. 6 J® 11 xcoc )If w1 J.., llx~(oc.)\\ d ol 3' w1 > w(6 Choose an and W ( €.. ) c...> 1 > w ( E, ) , and thus ~ dac.. < for 3 w1 ). (.J 1 > w ( 6 ). A0 (,6 ) exists such that for By 3., o. c.v > WO W1 C(> (oc. ) , I\ > A 0 and Ol Ux ~ ( °" ) - x(o£. )\l ~ 3 7u1• Thus, for A>A 11 r 0 x ;I (o<. )doc. r / j"'1 0 -f xc "'" id ... 0 1 II (x A ( ()(. ) - x( c<- ) )do<. - 0 < e. . + ~~x )I ( o< ) d«- -r x(-')dool\ w1 0 "'" II = l( x A ( °" ) - x( ol. ) I\ do£, ct ot. + J.., II xc cu1 The differ ences between this l ast theor em and theorem 1 . 3 . 8 should be noted. cl, ) U d C(, -11- Let Ew . }. = ( '~ Theorem 1 • 4. 1 : If x ( o1., , (3 ) i s in B(E w 1• \-+ J x( CX> 2. \3 )d <>'- oL , ,r ; .?f ) for a fixed "s and all converges uniformly with respect to 0 (S" ~-+ CX> (i) ) (ii) J J 0 0 r~ (iii) x( ot.. , ~ ) d ~ ~ ~ x( o<- , ~ ) d o< 0 0 ~ ' , o~~~) then 01 lV > rs x( o(. ' exists, d °" 'S] ;?f ), is in B( lO , ti )d ~ d °" l \-+J· "' l = 0 0 x( ol. , ~ )dcX. d ~ • 0 Proof: (i) It is sufficient to show that for every e > o, UJ ( £ ) exists such that for By 2., for every f, > o, II ~ ~ x( "" , ~ )dol \\ L.. ~ 3( 8 6 if ) exists such that 4J 2 > UJ1> W 3 (6 ). W1 Let (.,.) ( 6 ) = lJ / ~ ). Then if W2> w 1 >w(E.) , - 12- = €. • The inter- change of integrations is justified by theorem 1. J . 7. (ii) By theorem 1. 2. 2 (iii) , ~°' J - x ( °" , ~ )d()(.. is a strongly measurable function of ~ , 0 0~(3 ~'s'" ~rm x( Thus by theorem 1. 3. 1 it is sufficient to show that • ol. ' ~ )do<..\l is in L ( 0 , l ). By lenuna 1 • 4. 1 , 0 \~°' .) x( °" , ~ )d oL. is the aJmost uniform limit of finitely- valued 0 functions , and consequently so is 5w~ 00 x( ct.- , fJ )do(. u~: x( °'- , (.3 )d each W.:>O , Further, by 1. and theorem 1. 3. 1, • «-\\ E, L(O , ') ) for each W ( 6 ) vJ > O. By 2., for exists such that for \\ S ~°' x( ~ , (3 )dot.\\< 6 • Since W1 is the w1 almost uniform limit of finitely- valued functions , \\~ ~°' x( ot. , ~ )do<.\\ is the almost uniform l imit of step functions w1 and, is thus measurable. Thus l\ J~ x( w1 c{, , ~ )doe.. \I , being a bounded measurable function, - 13- is in L( 0, S ) and thus so is \\ ) ~=x( "'- , p )d °"II • 0 (iii) II r= 8 > o, By (i) ' for each ~~ x("" . ~ )d ~ (,J doo II ( L_ ~. Thus O w1 ) ,~.., r = I\ J J xc (){. , (3 )d °" d ~ 0 f~°' (! 0 0 - J J xc()(, , ~)ct~ d()(,,I\ 0 (1 f~°' ~ l\ J J 0 e ) exists such that for w 1> w Ce ) x( o£ , ~ ) d Of.. d (0 0 + c/2 , by 1. and theorem 1.J. 7. e>O, (J ( E, ) exists such that for W J > w ( € ) But by 2., for each IJ ~ -+cc x( o'.- , (3 ) d °' JI < E. • 2 2 WJ • Letting C: -> Let E w = Theorem 1 • 4. 2: 1. Then +~2 = c • (3 ..c. co } o, the conclusion is reached. l( o£. , (3 ) \ 0 ~ °"' ~ W ; 0 ~ If x( ol. , ~ ) is in B(E w ; X ) all w > o, • -1 4- ~->., x( cL , p )dcL 2. 0 ?'f > for each converges uniformly in (3 O, all W I w 4! ( (l ) exists , in L(o, .. ) such that II 3. ' x( DL ' r )do<.11 "' <e( (3 ) (3 , O ~ {/> < "", O > 0 and all Then J -+<»x(ot.- , (3 )dOl (i) isinB([O,"" );:X ) , 0 ~ .... ~., x( o<- , ~ )d oL d (,I (ii) 0 (iii) converges, 0 J., 10 x ( Ol ' \3 )d 0 Let xw ( Proof : ~)= ) °" d ~ = w r ., r 0 x( Ol ' (3 )d (3 d o<. 0 x(o<- , (3 )do<. • Then, by 2. and 3., x u.> ( ~ ) satisfies all th~ hypotheses of lemma 1. ~-· 2. Thus lim W -+ <» r·~ x( OL , ~ )d ) "" 0 0 J:1:!"., w ( ~ ) ~ = x 0 equal. lim d r _. , ~ 0 ~ = lim W.+ oo J <lO x£J ( (3 )d(3 , and 0 x ( "' ' (3 ) do<. d (3 exist and are 0 But J" J w x( oc , (3 ) d Ol. d (3 r., r W .+ c:o = d oL. Q doC- 0 x ( 0(. ' p )d ~ d 0(, , the inter change of integrations 0 0 being permitted by (1) and theorem 1.3. 7. • - 15- 5. Riemann-Graves Integral. Let x( CL ) be a b ounded function on the finite closed interval [J , 12] to the Banach space [ J, Subdivide = '>l] :?f • into 3 1 < ••• < lo<. subintervals J?t.- ~ n = '1l • and let ol. . be an arbitrary interior point of 1 the subdivision of [ J , ?z] ~ i by points Let d.=j .- "2 1.- 1' 1 1 6.. 1. . We shall denote together with the points ~ . 1 TT , which we shall call a partition of [ 3,'>'l] • Let N(Tf ) by = max n i Let G(TI) = E x(o<.. . ) f .. By lim 1 i=1 limit, i f it exists , of max . 1 N(TT ~ o G( 1T) we mean the x(oe.i) 8. as n ~ co in such a way that L i=1 81. ~ o. Definition 1.5. 1: n 1 1. If for every sequence of partitions Yvi.th N( IT) ~ o, G(IT ) exists and equals the same vector y , t hen we say that lim N(TT~ 0 x( Ol ) is Riemann-Graves integrable over this comm.on limit y by (a{'l [ x( o<- ) d «. 'J , '>1] , and we denote . ~ We shall drop the n (R) 11 where there is no danger of confusion. It should be noted that the integral is defined for bounded functions only. If x( 0(., ) is Riemann- Graves integrable over [ j every 1(_ > J , and , '>1,] for -16- lim (RJ,"J x( o<, )do<. j denote this limit by "l -> exists in the strong sense, we shall 00 J 00 (R) 3 x(ol. )dot, ' l'i ,oo). and say that x( ot..) is Hiemann-Graves integrable over the range = °' 0 ( J , '>'i ) sup (I x( 1 ) - x(Ol- ) 2 x 3~.c., e(& ~ 1'\ the oscillation of x( ol ) on the closed interval ( j , 11] • If 0(.. Definition 1. 5. 2: We shall call is an interior point of [ j , 'l1] "(. , we call 0 ( x the oscillatL on of x( oG ) at the point ot. • of the interval we use 0 ( 0(. , x Lemma 1.5.1: Proof: If 0 ( 0(.. ) x O(OC.-8 x , oG If °"' G) is an end point °" + S ), or 0x(oe.. - i , ot.., ) • ol, is an interior point of the interval. e > o, J' ( E, ) exists such that + J ' ) <:.t, • That is sup I\ x( oL ) - x( o'- ) 1 2 f:A-f ~"'· )0(.. ,,,.<><-4'd so that ll x( ol 1 ) - x(tX- 2) \I~ c; Set ol. 2 = ~ , and oG 1 = ~ ; then we have l\ x(ol. ) - x( ~ )\\ for \ o£, ~I< - cX- + =o, then x(cX. ) is continuous at ()('., • Consider first that By hypothesis, for each °" ) :=6->lim0 0x (o£. - f , II for>"·-&-=- oc. 1 , oc. 2 ~ r ( t). The extension t o the case of ~ OC, I\ +d • an end point is obvious, as is the converse of the lemma. Let r x be the set of discontinuities of x( oL ) in ( _j ,11_] Theorem 1 • 5. 1 : If x( oL. ) is a bounded function on [ J , ?( J • to _;f , and if m( r') = O, then xis Riemann-Graves integrable on (j' , ?/_] • Proof: See Graves ( 5 ] • It should be noted that the condition m( I:) =0 is a sufficient, but by no means necessary, condition that a bounded function be Riemann- -17- Graves integrable. There are functions which are everywhere dis- continuous and are yet Riemann-Graves integrable. Graves [ 5] • For example, see -1 8- 6. Riemann- Stieltjes Integr al. Definition 1 • 6. 1 : Let Cf_ ( ot..., ) be a bounded numerically- valued function on the finite closed interval TI Let H( to the Banach space ~ [J ,'>z] be a function on be a partition of n , and let a( °" ) , 7i_] [ j as in § 5. [ J , '>?.] TT ) = 2: (/J ( ol . ) [a( M • ) - a( oc, • )) . 1 'C l J. l - 1 • • Let Then if for every 1= sequence of partitions vii.th l'J( IT)~ o, equals the sa111e vector y , we say that J , ?{_] integrable over [ lim H( TI) exists and N(TT) ~ O ~ ( cL ) is Riemann-Stieltj es with respect to a( cl. ) , and we denote this coF~mon limit y by i"l <e ( 3 If 11 r J3 ~ (~ JJ <£ ( lim ~~ cc )da(oi, ) . 0(. '>] )da( ol- ) exists for every '1. >] , and ot.., )da( ex:.. ) exists (in the strong sense) , then we shall denote this limit by <e ( ()(, )da( "'). Theorem 1 . 6. 1 : If <£ ( ~ ) is a continuous nlimerically- valued function on the finite interval of bounded variation on s'J? <e ( ()(_ )da( Proof: ol. ) [ J , '1] exists . See Hille [ 6 ] ; page 52. ( j , 'l(_] , and if a( ol ) is a function to the Banach space X , then - 19- 7. Helations Between Integrals (A) Bochner and Improper Bochner Integrals. The relation between these two types of integrals is provided by the following theorem. Theorem 1.7.1: If x( oc:, ) is in B( [ 1. 2. o, w] ; ~) all w > o, is in L(O,oo) , then (i ) x( ol.. ) is in B( [ O, co) ; % ) , (ii) J-+<» x( ol. )d ol.. 0 ClO (iii) I x( oL )d converges, °" =5...)-oo x( d.. )doc:- • 0 0 Proof: By theorem 1.3.1, it is sufficient to show that x( ol ) is strongl y measurable over [ 0 , co) • By 1 ., x( ciJ ) is strongly measurable over Let lo, w] for each "->>0. 1 w ( cl.. ) = 1 =O Obviously 1 w cL > t.U is Lebesgue measurable for each value of w Then x w ( ci. ) = 1 w ( ol )x( Q., ) is strongly measurable over by theorem 1. 2. 3 (ii) ' and since x w ( is strongly measurable over Obviously x( ol ) = lim W-+ co °" ) = e for a(, >w • [ o, lcJ] ' x w ( oL- ) [ O,oo ) . x w ( o0 ) , where the limit is in the strong -20- sense, and thus, by theorem 1.2.3 (iii) over (ii) x(ol-) is strongly measurable [O,oo). r Since II x( Ol ) \\ 2 II (.,.)1 x( I){, )def. \\ l n ~~ x( (iii) o(, is in 1(0,oo) we have ,f II 2 <J1 x(o< r lll d" ~ 0 as x(o<- )do<.11 = )do<- 0 0 111w _..,x(~ )docll ~ (B) 1 Bochner, Riemann-Graves, and Improper Bochner Integrals. Sufficient conditions for a function to be integrable in both the Bochner and Riemann-Graves sense, and for these integrals to be equal, are given by the following theorem. Theorem 1.7.2: [J ' 'rt] If x(ol) is a bounded function on the finite interval to :f ' and the set r of discontinuities of x( oi, ) has Lebesgue measure zero, then x( o<. ) is both Bochner and RiemannGraves integrable over (B) Proof: J , ?J. J , and further, 11 (l J3 [ x( oG )d ex.. = (R) S x( cX. )do£. • j By theorem 1. 5. 1 , the Riemann-Graves integral of x( ot., ) exists. Let {TI n 1 be a sequence of partitions of ( j , '>l) N( TT n ) tending to zero. Let G( \f ) = L d . be the ¥1 ¥1 In these sums we may assume that the ~ . ¥1 n associated Riemann sums. with i x( Ol. . ) -21- are chosen so that they do not fall in the set ~ • x (()I.; ) = 2:- x( O£. • ¥1 ) <p . where Jfl function of the interval f:),,. . • n . J. :yi Then if ~ not in ~ . Let is the characteristic ~ is a point of [j , '>1) r ' nlimcoII x( ~ ) - xn( ~ )II = o. For, if b,,. ¥1. be the -+ I\ x( (3 ) - xn ( ~ ) \\ = \\ x( ~ ) - x(oc.. :i,n)\\ , and since x( ()(, ) is continuous at oG = (3 and l °'- . - ~ I -+ 0 as :i,n interval in which ~ lies, Jl x( ~ ) - xn( ~ )\( -+ o. m.-+ co, Then, by theorem 1 • 2. 1, lim x ( °" ) n = x( tX. ) almost uniformly, so that, since x ( of.- ) are finitely-valued functions, x(""' ) is strongly n But, by hypothesis, x(°" ) is bounded, so that, by theorem measurable. 1• 3 • 1 , x( oc ) is in B( [ ~ , '>'}] ; :£ ) • We have then, (B) s: = lim n -+ co x( o<- )doc. G( Tf n =ni:m~ S f""' S~ 3 is bounded on the intervals set ¥1) ) = (R) } '>1 x( <:J.. )doc,, ., If x( oc ) is a function on Corollary: =n:.im~ ~ x(O£. xn( oL )da<. [J , '>7] (j, co) to X , if x( oG ) for every 1'/ >] of discontinuities of x(ol) in , and if the ( j ,co) has Lebesgue measure zero, then (R) ~; x( r-+co cL )do<. = JJ x( el )dOL if either of the integrals exist. By the preceding theorem, Proof: (R) iJ'l x( oL )do<.- = (B) t'l x( "'- )do<.- , for every '1') > 3 , and -22- taking .limits on both sides we have if either integral exists. (C) Stieltjes Integral. The relation of the Stieltjes integral to other integrals depends on the fallowing lemma. Lermna 1. 7.1: [j , 11,] If a( <::L ) is a function of strongly bounded variation on to .):'. , then the set r of discontinuities of a(('){., ) is at most countable. Proof: Let M be the strong total variation of a(°'- ) over \n and co ={()(.,) 1~ r= n=1u r n , and r o<: n ~~ ; O(cx.)> ~J [ j , 11] , r is Then • a contains fewer than 2n elements. Thus at most countable. We can now prove a theorem relating Stieltjes, Riemann-Graves, and Bochner integrals. Theorem 1. 7.3: on [J , '>?] If a( o<... ) is a function of strongly bounded variation to ~ , and <e ( ol- ) is a numerically-valued function with continuous first deri va tive on [ J , 17] , then s~(o<.)da(cx.) = <e('l1)a(?j) - <e< 3 )a(~) - )1~' ( )a( o<.)do< • o<. The integral on the right hand side may be taken in either the HiemannGraves, or Bochner sense. Proof: The proof is the same as that for numerically-valued functions once it is noted that since a(ol ) is of stronely bounded variation it is bounded, and by the preceding lemma its set of discontinuities has -23- measure zero, so that, by theorem 1.5.1, the integral exists in the Riemann-Graves sense, and by theorem 1. 7.1, it exists in the Bochner sense. Corollary: If a(d.-) is a function on strongly bounded variation on [ j , ?z] X (3 ,co) to for each which is of 11 >j ' if re (°'- ) is a numerically-valued function on ( J ,co) which has a continuous first derivative on [.j ,co) , and if lim <{'( '1 )a( '7 equals y, then sj~ <e ( o(. co )da( a<. ) = y - =y- <e ( j )a( j <e ( 3 )a(] if any one of the three integrals exist. Proof: ) exists and The proof is obvious. ) - (R) ) - SJ r·~ j te' (o<. )a(o<. )do<. <e .(o<. )a( o<. )d "'- ' -248. Weak Convergence of Operators. The following theorem, which is, in a sense, a weak sequential analogue of the Banach-Steinhaus theorem, will be used in Chapter III in connection with certain representation theorems. Theorem 1 • 8. 1 : { T rr If J ,O .c:: co, is a set of linear er '- transformations on a separable Banach space Banach space "()'for all <r .> ~ to a reflexive M independent of , and i f rr ' o, then there exists an increasing unbounded sequence {tri} , and a linear transformation T on to 'J/j with \\ T II ~ M, such that lim y*(Tcr (x)) =y-Jt-(T(x)), i ~ co i .X and every y * in ri.r;--'lf- . Let D =[ xnl be a countable set dense in ?f for every x in Proof: . Since '2rJ is reflexive, it has, by Gantmakher and Smulian [ 4] , a weakly compact unit sphere, so that there exists an increasing unbounded f i, 1} and an element y1 of 1!J in 11j* , sequence y*" CJ . lira y * 1 ~ co (T <r. . (~ ) ) = y 1 ,1 such that for every * (y 1 ) • Further, there exists an increasing unbounded sequence and an element y 2 in ?!J' . lim y ].~co such that for every y + * (T CT. i,2 ( x 2) ) =y in J./j* , * (y 2) • Inductively, there exists an increasing unbounded sequence and an element yn of 1IJ- such that for -25- * in 11) * every y . lim y ii' ( T O". 1-+ "" 1,n (x )) n = y "*" (yn ) • Thus, using the diagonal sequence, we have, for every y -Jt- lim y i-+"" *" (T O" i,j_ = lim ( x. ) ) J =y* ( y . ) • J y* (T rr. . (xj)) ~ ll y*ll M \\ xjl\ 1,1 so that, by Hille [ Vfe define tr i on D, \\ T \ \ ~ M. . J-"""" = ().1,1. , and T(xn ) = yn• D such that. lim xn. J-+co n. \\ Yn. - J and Then there is a sequence Further, if y J n. = T(xn. ) , J J J Il_t. M( = x. ?f- • Obviously we have for T exists; for, Fur~her if if y J J J lim y ' 6; thm. 2.12.3], Uy.\\~ M Ux.n. Let x be an arbitrary element of { xn ,) c; in ! xnR.1 ~ = T(x D is any other sequence whose limit is x, then, ), 11.f ;ntl\ =\I T(xn.J - ~n,t II x - xn. \\ + J )\\ ~ M Hxn. - xn! II J \\x - xnt l\ ) -+ O as j, 1 have the same limit. -+ ""' . We define T(x) so that =• lim y11. • J _,,. "" It is evident that T is bounded and linear on J :I- , and, in fact, -26- Also vre have lim y*(T er (x)) i i-)o oo = y "'* (T(x)) for every y*" in 11)*'. For , I ylt (T er (x)) - y* (T(x))I i = y * (T <r. (x - xn _)) + (y* (T 0-. (xn.)) - y T(xn.))) + y * (T(x-xn. ))f J J J J ~ 21\ y•l\ Mn x- xn.ll + r y*(T . . (xn . )) -y* (T(xn. ))I . .)- 0 jt- ( l l tr J as i,j ...)- °"• Thus the theorem is proved. i,J. J J -27- Chapter II 1. Introduction In this chapter we shall consider the inversion of the Laplace transformation of Banach-valued functions of a real variable. In particular we shall consider four kinds of Laplace integrals, namely f( A ) = (B) I Jco e-?.-c 'x (l:" ) d-r:- 0 =J~oo e-?.r x(-r)d-c II f( A) III r( A) = IV f( A ) = ~R) f .-Ar x(-c)d-c re~'ll-c 00 cta(-c). 0 We shall consider the inversion theory with respect to a certain 1treal11 inversion operator 2K 11-<,-c [r( /\ )] \co 1 1 = ~ e-i: (B) J '(_- 2 cos(2 1/. 2 )r( "' ( ryz + 1 )/c)d "Y( • 0 Vfo shall also use the "Improper Bochner" generalization of this operator, namely -+ L K )"C K e2K [re'A J = rr-c \-+"' - ~ ~ J 11. cos(2 i/. )f(~( ?'/ + 1)/r)d?(. 0 Whether we use 11.-<)-C [f( A )] -+ , or L Kft: [r( A )] does not depend, as one might suppose, on whether we are inverting transformation I or II. Onihe contrary, the use of 11-<)"C [f( A il -+ or LK ,"C [r( A U -28- depends upon the behaviour of f( A ) in the neighbourhood of 'A that is on the behaviour of x(r) in the neighbourhood of r = m, = o, while whether x(l:') has a transformation of type I or II depends on the behaviour of x('C) in the neighbourhood of r =~. In section 2 we prove several preliminary lenuna.s concerned lNith the evaluation of certain singular integrals. Section 3 contains the inversion theory for transformations of type I, and section 4 for transformations of types II and III. Finally, in section 5, we prove an inversion theorem for transformations of type IV. - 29- 2. Some Preliminary Lemmas: For the various inversion and representation theories we shall need the following lemmas. Lemma 2. 2. 1 : If x( ol.) is in B( [ o, w] ; 1. ~ J 2. x ) for each w > o, CX) e -~Ol X ( OL A = ?.. 0 • converges for )dOl 0 Then for each t: > 0 and for all (i) K .::> ?.. '"C" 0 converges, 0 (ii) for each -C> 0 in the Lebesgue set of x(Ol- ), lim I ~ = x(-C). K~oo Proof: M w 2 > c.u 1 , and let Let K > ~~, let (i) = sup a< w< co Thus, \\ \we- :;i. oOl J0 ~ GJ ...L-K(OC.-c: +- ::£) Ol - z:- X ( J ()£, 1 H . By 2., M I (&)l... \\ x( Ol. )doL Ot'..) &. «, \\ =- £. 00• -30- +M (w' { 4_ (JZ.-K(~ + f)+/.. 0C-oct} d~ 0 j<.J;i = zM( J.<X- ...Q..- K ( ~ +- ~ ) + A 0 lJ I cu,- i-) as (ii) It should be noted that 00 ..Q2.K (;-erk ~ ...a..- Kl~+ ~) oc -4: J.~ = 0 <J 1 , w 2 -+ oo . -31- (where ~2= ~ By Peirce [ 8; page 63, f ornmla 495_) • ;;:: \ · Let "'C be in the Lebesque set of x( oL ) , and let • > '). 0-c. K Let w> r Then, \\ IK - X<r)ll " II ..e...2 !; K (;,:) ± I":- ~ K ( + ;f)ol -i X too) d.. "" - :<. t-cJI\ H~~K (~\Y~J~ :-Kl~+-;, )cl_± ( 'Z(ol) - X(-c))<l rL \\ f' 0 .; _J!,l. ... <:-c)i + II ...a..->K(:,Jl: +II x <-c>I\ r:- £K( ~ ~)oG-~ II x K \ i- ~ + ~ ) .,d: _,{"'l:-i L£K(~ (0(.) - x (-c)ll .I. 0(. X(o<.) J.. o<. \l 00 + ,;} (){,- i.LoL - J, + L. + J3. Let K 0 be a positive number, and ~ 0 < I'{ • Then ) - 32- J; = I\ ;r, lrll\ ..a.'" (;\)i = \\ (_.c" l":: ~) °'-l: + <l"'- X(r)\\ fK (;:-c)t r~_t,-Ko(.~ +~) ol-L..e..-(K-Ke)(~+ ~)cl°" (U :6 \\ -X,(i:)ll ...e, .2.K (TC~)~ .Jl- (K-Ko) ( ~ ~ ~) t~ £Ko(~+~)()(,- ~J, ~ , cw + w-c)....Q...-K ( w + ~ 2.>r)W X,( r)\I ( .::-c) _q, 00 == \\ -+ "i 0 as K o(.., -+ -:C: ~) (c(. £Ko\ "'E+;;;:. ol-i do£. (ij - + ..:E. attains its minimlUll value of 2 at co, since !!:.. 17 o/,; = -c, and UJ > 'C = 11 ~1 K (;\)i ,;;, ..z M Ko 1;" ..Q. 1>< .. r {L <£.i ":; + ~)+ "·"'.::- l) C'r\.)!: L""l- L(£ :-c)± £ ::: 2 M ( K ( H."'"'. . -•. ~ +;iyA,"' K ( ~ .\- ~ - 2) +-A 0 w ot.- l JJ."" -k 0)- -+ 0 as K By Widder [ 1 2,; theorem 2b , corollary 2b. 1 , page 278] J if r is in the Lebesgue set of ti x( Of., ) f' Xlf3l .1.1311 - x(t; )tl • -+ oo . 1 -+ -+ .,, However , by theorem 1. 3. 6, this is exactly the Lebesgue set of x( cL ) . lem_rna is proved. 0 as K Thus the -33- (i) x( ~ ) is in B( [ O,"°); X ) , then If e- 'A oct. Corollary: . . . "" for each T: > 0 and all K e- t< ( ~ + .§.) .... - T"" -> .. 0 I\/ IN - 1 ~ x( ol ) is ' in B( ( 0, ro); X ) , (ii) lim I K = x(-c) for each -c in the Lebesgue set of x( <£ ) , where 1 co J e 2 x( ex.. ) d ()(.; Q(, - - K ( ~ + L.:) -r OI., • 0 Proof: (i) This follows from the fact that for e-K ( ~ +£ )£illx(ol )l\~ K > 1. -c 0 K e-'Aocx. ltx(oL )l\ • (ii) (ii) now follows from theorem 1.7.1 and the preceding lemma. Lemma 2.2.2: 1• 2. If x( Of... ) is in B( [ 0 , 5-.- ?.o1. W) ; ~ ) for each x( o< )d oL W> O, ~ = '). 0 ' converges for 0 then j (i) for en.ch -C: > 0 and all ...+co e- K (~ + :E.) 't: QI. 3 Cl-- ~ x( o<... )doc.. converges 0 K >'/. l;, 0 (ii) for each "l:" > 0 in the Lebesgue set of x( ol ) , lim IK. =x(-c:). K-+ oo Proof: lemma. The proof is almost identical with that of the preceding -34- 3. Inversion of Tr ansformations of Type I . Conditions for the inversion of f( A ) = (B) J~ e- A~ x (r)dr, 0 are provided by the following theor~~ and its corollary. -+ gives conditions for the inversion by L ,,. [f( A The theorem B , and the "'- ' "'C coroliary for L K , "C [f ( I\ Theorem 2. 3 . 1: )] • If e- '.Aot. x( o1- ) is in B( [ O,<» ) ; -+ then Jor each -C > 0 and all lim L K i; [f( ';;\ )] ~-+oo K > x ) for all or, L K , 1: [f ( ~ )] 'A> o > O, exists , and =x(-c ) ' at every point , "C > 0 of the Lebesgue set of x( 'C ). -+ Proof: Vfo shall show that 00 = { rrK-ci'e 2 K ) e- K ( LK [f( ;t: 1 A )] ~ + ~ ) ol- -;ix ( ol. )do<- , and the conclusion 0 will then follow from the corollary to lerrnna 2. 2. 1. Let (wher e e::: K , r be fixed , positive, and ~ > )f • r{¥1L-if¥ ) . Then, -35- Thus , - - 1(1) "' ...., + J w. (2) -36- -+ 0 as w -+ ""• Further, since - ~ "-'"Ct' rig+ 'Vy liiKw I - e.. J e ] = o .ltrn, [ 1--I ..fl w~oo 1rr -i.~ -,P;w for cl, in almost everywhere lo, E] , we have, by theorem 1. 2.1, that the limit equals zero almost uniformly in ol for ot., in [ o, 6] • Also, since and where M is independent -37- where N is independent of cf., integrable function of • ol, and cu , and this bound is an Thus , by theorem 1 • 3. 8, = o, and the theorem is proved. Corollary: If e- 1 'A <X.. - '15 '- x(ci) is in B([ O,ao); :I: ol ) for all ?. > 't' > o, then (i) f( A ) exists for (ij_) 1 lt<JL" [f( A )] 'A. :> ¥ , "L > 0 and all K > ¥ "C exists for each , (iii) for each -c > 0 in the Lebesgue set of x( ex. ) = x(-c). lim L..:>-i: [f( A )] ao 'I\+¥ - ?. 0(. e- ( 2 )~ e- ( 2 ) Since e K _,. Proof: = and sufficiently large <L for A> o , e- ( )\ - ir) 2 'A - ,. 1 Ol.. £.. , and for oL - ~ , f( I\. ) exists • Further, exists for each -C > 0 and all K > o-C , since -36- Finally (iii) follows f ram theorems 1 • 7. 1 and 2. 3. 1 , since theorem 1. 7. 1 tells us that when both L KJ -c -+ L,, ""> r [f( A )] exist, they are equal. [f( I\ )] and -39- 4. Inversion of Transformations of Types II and III. If x( Ol ) is in B( [ o, w] ; :X Theorem 2.4.1: and if r( A ) = J -e- ~~ x("' 0 )d"" A converges uniformly in ) for each w > o, for A> o > 0 , then for each r: > 0 -+ o r, 1 K/i:- [ f ( A TI K > and all -+ il = x (t) lim LK,r [ f( I\ K -+ exists, and co at every point -C> 0 of the Lebe~gue set of x( ~ ) . -+ We shall show that LV< -c [ f( A )] Proof : ) ii<(;-c?J-+co e- K ( :C°" + Ol ) '?:' l of., - = 1 "2 x(ol )doc:., , and the conclusion will 0 follow from lemma 2.2.1. Let K. , -c be fixed, J..o , o<'A 0 < r ooe-'Aoot. x ( 'r</r. positive and K > 0-C Let M = Choose Since Dl. )d oL r··e_ '.).o"' x( OL )d ()(. converges, 0 bounded. • 0 0:~< I\ r·•e- ). o°" x ( ol ) d o<. \\ 00 • 0 ...p - K...e.2Kr w I w-rrr J fr2.(,0S(2.»<1t£)/(K(?(-t-J)jr)J?'( ..!. 0 , Then M -< oo• Consider is -40- Since )doc converges uniformly for A > 'lf ' 0 and K /r:: > ~ , we may, by theorem 1 • 4. 1 , interchange the order of the integrations. Thus Then -+-oo Jtt.1= \I (:;.r;)±.-£~K-1._Q_-K.(~-r~{x_-:1: :z:,rO(,Jdac, - Iwll = J, (1) w + J-w. (4) -41- Now J~ ) is the same as the 1 every t > o, /~) -+ 0 as w converges uniformly in w /2) of theorem 2.3.1, so that for -+ "'• for W Also > O. For, for every -42- c ' ~ for any large enough that /~) E. Thus we may choose If x( ot, ) is Riemann-Graves integrable over [ o, w] ) , 0 6 for each o, <.A.I> The set I 2. e. ' > o, and thus lim Jw = o, and the theorem is proved. Theorem 2.4.2: 1. = \\ J S3) I\ . :; of discontinuities of x( O£.. ol L.. m, has Lebesgue measure zero, f( A ) = (R) 3. A>~ > -+ 1 K,t" [ f( A o, then il Jm e- ').()(. x ( oe- )do£. 0 exists, and for -c > -+ lim K-+ converges uniformly for LK, -c [ f( /\ )] 0 = x (t: ) almost everywhere. a> Proof: By 1., 2., and the corollary to theorem 1. 7. 2, x( al- ) is in r B( [ o, W] ; :X ) for each W > o, and ( R) e- :\ "'- x ( o< ) do< = 1-+m e- A°'- x( °" )do£.. • Consequently 0 0 we can make use of the previous theorem, which yields the above conclusions. -43- 5. Inversion of Transfonnations of Type I V. J 00 Let f( A ) : : : e- AOC. da(Ol ) . Conditions for the inversion 0 of this type of transformation are given by the following theorem. To avoid complications, we restrict ourselves to the operator L KJ-c[f( A ] . Theorem 2.5.1: 1. If a( ol ) is of str ongly bounded variation on [ o, w] for each o, and a(O) =e W> J" 2. e- 'AO(, da( ct., ) converges for "/I. >'A 0 and uniformly for , 0 ?..> A. 1' <(? ( A ) exists such that J. Jwe- 'AO(,, da(cx...) ~ (€ ( ?. 2) is in L(O ,oo) , and <e ( /l ), for all GJ.>O , and a ll 0 then lim K-+ co 1 K) -c [ f( A )] d-c ::::: a( t:) almost everywhere. 0 By t heorem 1 . 7.J, corollary, and 2., Proof: re 'A ) ) 1:' ::::: re-'A"' da( r:i., ) = 0 A rM e- Aot. a(oc. )dcx'.. • 0 L K, "C" [ f ( ). )] Thus , Ke A >J. 1 , 2K =~ 1 S 'l- 2 0 00 1 cos(2K.1'{ 2 )( K( 11+ 1 )/ c )d 'Yl I-+"" 0 e- K('l( +1 ) °"/ca (()(. )dOl. . - 44- Choose K >J... 1 r exists and • satisfies the hypotheses of theoreJU 1 . 4 . 2. We have then Thus by 2. and theorem 1. 4.1, [ f( A il -+co J 0 almost everywhere, as K -+ "°' by lemma 2. 2. 2. e - t( 3 (~ + .f) ex, - 2a( ol. ) doi: -+a( oG ) -45Chapter III 1. Introduction. We consider here the representation of Banach-valued functions f( A ) of a real or complex variable A Banach- valued functions of a real variable. by Laplace integrals of We shall consider representations in the forms oo I . J f( A ) = (B) - ?t-c e x(-c )d -c 0 ' oo II . f(A) = J e_?it: da(L: ). 0 In the latter case however, we shall restrict ourselves to numerically-valued functions. The reason for this is the lack, at the present time , of theorems for Banach-valued functions corresponding to those theorems for real functions which derive from 1-Ielly' s selection principle. Our tool in this task will be the 11 real 11 inversion operator which we used in the last chapter. Section 2 of this chapter contains certain prelinlinary lemmas r which yield conditions ensuring the existence of L K ' In section 3 we derive the 11 Fundamental Theorem". [re A )] . This Theorem shows that under certain conditions the Laplace transform (i.e. the function found by an integral of type I above) of has f( A ) for its limit as K tends to infini ty. -46- Section 4 contains the conditions that f ( 'A ) be r epresented as the Laplace integral of a function of BP ( [ 0 ' "°); 1 <. p ~ "°• l- ), p fixed, The cases 1 .: : : p < "° and p = co are treated separately there, as they are of a very differ ent nature. Section 5 contains a r epresentation theory for the case that f( A ) belongs to the class H ( C>l p class being defined there. ; :X ) , p fi xed 1 ~ The cases p = 1 , and 1 <. p ~ oo, this p <.. "° are treated separately, again because of their very different nature. Lastly, section 6 contains the conditions that f( 'A ) be represented as a Laplace- Stieltjes transform, (an i ntegr al of type II above) , but as mentioned before , we restrict ourselves to numer ically- valued functions . -47- 2. Preliminary Lemmas . '!Ve first prove the two following lemmas which are preliniinary to the "Fundamental Theorem" which is proved in the next section. Lemma 3 . 2.1: If 'A - 1 <('(A) is in 1( 8 ,oo ) for all J > 1. O, J /l.f ( 3 ) = 1( - 1 \re ('7( )! d11 = o( 3 -m) with m > o, as 3 J -+ m, and ')/J ( ] ) = 0( e 0 /3 ) with ff ~ 0 , as j -+ 0 +, 2. CX> m+ n > 3. o, then (i) 1 /\ n-1 <e (A - ) is in L(O, c.J) for all (ii) 6J ( r ) =fc?(n-1 \<e('i(-1)1 d?'{ = O( -r;m+n) as -c -+ o+, (.A)::. o, 0 (iii) S ( c and n ) = O(rne~'C m, if' either ) as -r: -+ 'tJ > '( = 0 O or o, ~ = 0( 1 ) as "[, -+ m if o = 0 and n . .:::. 0, J ~o(.ol re ( 00 (iv) n-1 e- 1 o1..- )d oG exists for 'A> "I ' and is 0 0( A -m-n ) as 'A -+ ""• Proof: (i) Clearly w > d> 0 ~ Thus , . "~ n-1 Ir\ c1 -1 ) I 0 1 1 Jc( j n- I <e ( 3 - ) I d j is in L( d , VJ ) for all · · ~ = = -c n '1p ( c -1 ) _ e n 1l' ( £ -1 ) _ n J3 13 c r: ~ nd 1jJ ( J -1 ) n-11/! ( j -1 ) d j , -48- and by 2. , the right hand side tends to a finite limit as c since, by 3., m + n > e( z:: ) = z:; n o. O, 'I'hus J-rj n-1 'W ( j -1 ) d j ~ ( r; -1 ) _ n e( r: ) = r; n"f' ( <:: -1) _on (ii) -+ fJ • n-1 '1fJ ( J - 1 )dj = o(l': m-m) 0 by the last equation and 2. e( 't; ) ( iv ) rt: 3 = t: nO(e o'C ) - n o = O, n If o, or n ~ o. Let either {( > (iii) 4- Since m + n > o, as r n-1 o(e ¥3 )d 3 -+ ""J = 0( t: ne ~-c ). 0 O, e( 'C ) is clearly bounded. I 't:/n ( ol -1 ) I is in L(o" , w ) for all Clearly e - A ol oG n-1 w >S > o. d oc, = e _'Aw 8( <U )- e - ~G r ) e( o + Convergence as 8 as (JJ -+ oo 0 follows from (ii) since m + n > O. -+ ~ > follows from (iii) i f Convergence o . Moreover, from Wi Q.der [I Z. , page 181 , theorem ] , the integral is 0( A -m-n) as ?. (and is O((~ - ~ )-n) as 'A -+ '( + if n ::> -+ oo, 0) . A ) is in B(( o ,w] ;X ) for all W > 8 > and if satisfies all the requirements of ~ ( /\. ) of lemma 3.1 . 1 vdth m > 1 then for each 2' J;emma 3 . 2. 2: L K >""C exists. [ f ( I\ If f( ~) - K. e2 K. lJ TI -c; 00 J 0 K. > 0 , and almost all 'C > 1 'Y[ - 2 cos( 2 K l'1 "2)f( K. ( '>'t +1 )/ t: )d 7f.. 1 o, o, -49- In particular, L K) r [r( A 3 ' L > exists when K 0 and K /-c is in the Lebesgue set of f( A )• r ~ Proof: It is sufficient to show that the integral ?{ - eos( 2K1/ 1) f( K ( 'f{. +1)/ <: ) d 11 converges at the 0 origin and at infinity. If K /-c is in the Lebesgue set of f( A ) , we have ~ JII W ( lf ) = f( K ( "1 +1 )/-.: ) - f( IC/ 't: )I\ d '11 = o( If ) , f4 ? 1 Thus J ?( - "2 If cos(2K '1l 'jr( K. ( 17 +1)/ -C )I\ d 'fl 8 ;5 ~ 1 s c J'1'1- = 0(1) + £ 0 = o(1) + ~L 'J'l J?( - 2 & f(~) I f'fl-2. d?/ + 1 3 £ 1 2 d,., ("'! ) = o(1) + 11 f(K(7(+1)/-C) -f(K/c )l\ct-q 3 J!'E l'[- 2 w ( ?/ )d'J? -22 o( "l )d 1l = 0(1) as t , S -+ O, and the integral converges at the origin. from lenm1a J. 2.1 we have f 3 S0 3 - 2 \\ f( 3 -1 ))I d 3 ~ "'• Here we put v'.? -1 -- "' ''- ( "l'l." +1 )/..,.v have1 <:)2 j .., "l: - Ke , an ct choose c; <::: c.... "L I tc. 1 (1 + ?'i )- "2 \\ f( K('t(_+ 1 )/ "C )\\ d 1 and the integral converges at infinity. 11 <. ..,, • We then -soLennna 3. 2.3: 1. If ~ - 1 ~ ( ?.. ) is i n 1( I , r.o ) for all 2. 'Ip (3 1 -+ oo ) = and Sjm A- 1 \<e( 'Al\ d:;I. = 0( j E> -m) wi t h m > "lf ~ 0 as '¥' (3 ) = 0( e 0 / J ) with then, for each 8 > o, J -+ o, as 0 + , 0, Jj~ :x -1 e_ f;l. I'(' ('A ) I d 'A = o( 3 - n) for every n > o, as J -+ 00 = 0( e ({ /~ ) , as 3 -+ 0 + • Proof: SJm ). -\ - EA lie(?. JI d ',\ L e- d Jjm:i, -1l<e( ',\ )\ tlA = e- c j rw ( 3 ) = 0 ( 3 - n) as J -+ 0 / j ) as J -+ 0 + • = O( e 00 -51- 3. Fundrunental Theorem. The followin g theorem is fundamen tal in the representation theory. Theorem 3.3.1: If A ~1 r (A.) is in B( (J' ,co); l: ) for all 1. 8 > o, 1¥1 ( j ) = J.i ~1t- 1 11 r < ?1 Jll d 1l = 0( 1 -m) , with m > 21 ' oo, and 1J ( J ) = O( e 0 /j ) , with 0 >,. o, as 3 ~ o +, as j 2. ~ is in B([ O, co) ; ~) for e- )-C LK,i: [f( A)} 3. all K > K 0 S> 't 1 , and , then = f( ~ ) at every point '.r > ({ 1 lim K ~ co of the Lebesgue set of f( Proof: 1 K;c transform when [r( A)] ~ A ). exists by lemma 3.1.2, and has a Laplace > If 1 by 3. To prove the assertion we shall use theorem 1. 3. 6 and lemma 2.2. 1, corollary. Operating formally we have 00 t_.-:S--c LK,-t [ i'.OJ] d.-c = -52- ao oo = 1.~..£.~ttjo ..a,-~ 1 't" 'C- £l --c Jcos(2 K 3) f (K (f+l)/"C) Glj 0 00 00 Jor == .2. K...e.. te. f COS(,2,te3 )d.j ....Q.-S-c -c- 1 ~ ( K ( jl.+I )/c) cl;c 2 le = Jo -C!.2"<:r }~-r:-K<r~ +<~~r·>\'d f-< 13-• >d (3 ...)' f ( .} ) as K ...)' °"• These formal calculations will be justified if the two interchanges of integrations are justified and the conditions of lenrna 2. 1 • 1 are met. For the first interchange of integrations it is sufficient to show that -53- But by 2. and lemma 3. 2. 2, if 1 -2m) as v~ 0 ( .:J ~ K'S > '6 , the inner integral is . °", and m > 21 • 'Thus b y t h eorem 1 • 3. 9 t h e im,er1- change is justified. For the second interchange it is sufficient to show that J e-tc'5~@ CX) 3 - 2\\f(@ - )\\ d ~ 0 1 (°" 2 1 J e- ol \cos(2(K/'S~ )20G)l d ol <.. 0 But this is true since the inner integral is less than ~ ,ff( 00 since J e- K. )" ~ (.3 - ~ \\ f( f3. - 1 ) \\ d ~ 0 and lemma 3. 2. 1 (iv) . , and converges by 1., 2., co . -54- l_f• Representation Theorems for L K >1:' [r( A )] in B ( [ 0, °') ; X ) . p In this section we find conditions that a function f ( A ) on [Ol ,co) to a Banach space X be represented as the Laplace integral of a function in BP([ o , ..,);~ ), where 1 L. p ~ °'• In order to obtain such conditions for these general classes of functions, we find i t necessary, in the cases 1 . : : : p <. "" ' to postulate some sort of compactness condition on B ( [ O,co ); J: ). p We have chosen the weakest condition at present lmown, namely weak compactness of the unit sphere in BP( [ and Pettis [ 9] compactness is that o,..,);~ ). By Bochner and Taylor [ 1] , a necessary and sufficient condition for this X be reflexive, and this is the manner in which we have set the condition. It is well lmown that B ([ O,oo);X') 1 has never a weakly compact unit sphere. Thus, to obtain a represen- tation theorem for B it would be necessary to postulate some con1 vergence condition on L K, "'C [f( A )] , and we would obtain a theorem ver'J like that of Widder [ 12; page 318 J . We have not chosen to do this, since the results are quite obvious . The first theorem of this section gives sufficient, and in the cases p > 2 . necessary, conditions that f( A ) be represented as a Laplace integral of a function in B ([ O,ro); 1: ), 1 ~ p p ' °'• The fact that these conditions are not necessary for p ~ 2 is a consequence of the fact that, as the follo-v·Jin g example shows, f( -A ) may be the Laplace transform of a function in B ([ O,co); X ), p .6. 2, and yet p -55[f( A )] 1 K> -c may not exist. ( r(;))-1 oL - 213e- <X- • f( 'A = For example, l et x ( ol ) l'hen ) = I ~ .-;\CG x ( ()(, )d °" = ( 'A + 1 )- 1/3 , 0 [f(A )] But LK>r ; :: 2K ro does not exist sinc e - 1 J}7l 1 -2 2 c OS ( 2 K ?(_ ) f ( K ( 'Yl + 1 ) I c ) l ?'{ d 0 2KJ = ~~ 00 .'.!_ !cos(2Kl( 2 )1(_ _.'.!_2 _ .'.!_, (~(?'(+1)+1) 3 d'f/. =w. 0 I n order to cope with this phenomenon, we resort to what is essentially Cauchy 1 s method of sumJnation. This yields, in t heorem 3. 4. 2, necessary and sufficient conditions tha t f( ~ ) be r epresented as t he Laplace integral of a functi on in BP ( [ 0 , co); J. The case of B ( [ (lO this section. ), 1 L p ..::.. co. O, ro);"°J:.) is treated in the final theoren of It will be noted that the methods used are very different from those of the t wo previous theorems. Theorem 3. 4.1: If :f i s a reflexive Banach space , t hen the f ollow- ing conditions are sufficient for f( ~ ) to be equal a lmost everywhere for A > 1. 'A - 1f ( '-) is in B( [ ~ ,oo); X) for all 8 > 2. Jco 1l-1 Hf( 1l )\\ d rrt 3. 0 to the Laplace integral of a function in B ( [ O, oo) ; p O, = 0( 3 -m) with m > ~' as J = O(e ~/j) with M , p fixe d, 1 L p '= ), l'l'L- 3 -+ oo o> O, as J -+ O+, p "°' K > ~0 • L... 00 • -56- Conditions 1. and 3. are necessary for every p, 1 < p < co, and 2. is necessary if p > 2. Proof: Suppose f( A ) = Necessity: J~ e- kt: x( t: )d 1: a.e • , and x( t:: ) 0 is in BP( [ O,oo ) ; '".f. ). Then using Holders inequality we have , almost everywhere x' II :fDJll ~-r:-H 11.:1 t:lll d.1:,,; X' {L:-y,.A "J.-c}t L I =Al\_,_\- r 00 Thus J J u; 'h1\; lt( t:Jll so that 1. is necessary. ~ 00 A- 1 l\f(A )\Id?.~ A J '.3 1 ;\ - ( 1 + -) d.:l q so that 2. is necessary if ~ > ~ , i.e. is if p > = B .;;\. 2. From theorem 2.3 . 1, we have 1 LK.>T ff(?.)] = (rr~ )2 e2K so that, Joo 0 1 e-IC(~ +£,) ol-2 x (oL )do<:- ' 1 q ' - 57- t II ()() Thus , ~ (~ )-i = fu LK,"t[ iC?.lJll pd.-c )0 cl{: I, ()() __Q,l, l(. _Q- K ( ~ + ~ ) ( -c ex.)-± \\ x. (.oc.)U" <i.ol X(Ol.)11 p d._ct, ti 1 K> . [f ( A )] ll p Hence From 1., Sufficiency: ) > o, f() ) = lirn all Cl) K..+ex> r ~ \\x( •) \l p , so that 3. is necessary. 2., 3., and theorem 3.3.1, we have for almost .- ~-.: L 11:,-c [r( 1 )) d "C • 0 By Bochner and Taylor [ 1], and P ettis[~] , if X is a reflexive Banach space, BP( [ o,CX>); X) is a reflexive space for < °"' and by Gantrnakher and Smulian [ 4] , a reflexive Banach 1 < p Thus BP( [ O,co ); X) has a space has a weakly compact unit sphere. weakly compact unit sphere, so that there exists an element x(•) of BP( [ O,co ); X) and an increasing unb ounded sequence that for ever-J functional y*" y* in B*" ([ O,co); X p Let x i-+ex> x ~( J e-!ol g(ol )d Ol.) = 0 on B ([O ,co ) ; X) , (i.e. for every p i *" be an arbitrary element of ~ p such )) , lim y*(L K ,. (f( 'i\ )] ) = y*(x(•)). element of B ( [ 0, oo ) ; l:- ) , 00 {Ki} * • .i.'hen if g( • ) is an -58- defines an element Yr of BP ( l O,oo); I: ) for each j > o. For, y *"is obviously linear, and using Holder's inequality we have 1 \yf( g( • ))\ =\ ~ro e-)o(. x*(g(o<..))dot\ ~ ( r 0 ~ - 1-1 II x"II { l > O. Thus we have, for each x Y .*' * in .r. = x*~lim 1 l e-)~ 00 i~oo O and for almost all j > J® e-~c 1 <. -c fr( A j] d-.:: ) l~oo O x*(LK i' T i' [f( ~ D )d"t; =. lim y;* (1 K. • ~( A ~ ) = Yf (x( •) ) l~oo = f" i ' J e- 'p: x * (x( c: ) ) d 'C =x ~( 0 and thus, for almost all r('\' )= ) > o, r.- ~-cx(t:)dt: . 0 To obtai.. n necessary and sui'ficient conditions we define 1 ~ )C 0 (q1)<i 0 is bounded for each = lim dol)q{fnx*(g(oc)11p1 0 II g( ol )Ii p do<} i = II x*I\ II g(.) II p' so that y" (q))q x*(f() )) r' e-q~o1. ~( A ~ = L K, -c: [e- £A f ( ll )J • The following t.heorem yields the mentioned conditions. o, i -59If ~ Theorem J.4. 2: is a reflexive Banach space, then t he follow- ing conditions are necessary and sufficient for f( A ) to be equal '). > almost everywhere for in BP( [ O,co ) ; :! O to the Laplace integral of a function ). 1. A - 1f ( ~ ) is in B([J ,co ) ; l ), t.f >o,, 2. Jj A l\r( /I. lll A 00 -l = 0( .3 -m) vvi th m > 0, as d = O(et /j ll 1 ~ J. l ., [ JI\ L f(A) o> o, as ~ ~ O+, Mp' wher e MP is independent of K p and E , p fixed, 1 <- p...:: ) with ] ~ co oo, K > K0 • Proof: 1. was proved necessary in the previous theorem as was 2. Necessity: The proof of the necessity of 3. is almost exactly the srune as in the previous theorem. Sufficiency: e- € l f ( By 1., 2., 3., lemraa 3.2.3, and theorem 3.3.1, S ) = lim K~ co r e- r-c L :.,_ >r [r( A )} d c • 0 As in the previous theorem , BP( [ O,co ) ; :X ) has a weakly compact t unit sphere, so that for each > o, there exists an element x t:. ( • ) of BP( [ O,co ) ; X ) and an increasing unbounded sequence such that for every y * . lim y*(Le: 1.~co cK ., "'1. Further, since * ([ O,co); X) in B p • [ f ( A ~)=y~(xE ( • )) . {€.Kd -60- y*(LEeKi' we have ~ [f( 'A)] ) ~ \ly*\\ ·\\ L~Ki'. [re A )]llP~\\y'*H MP' ll x ~ ( • )\\f ~ Mp• Let x • be an arbitrary element of B~( [ O,oo); X ) as in the previous theorem. p and almost all ~ > , and define Y!* of Then for each £ > o, ~* ;x 0 (e-n :rO )) = x*" (lim r~e- lt: L \ If(;\. )] di:: ) i-+ JO E i'"'C' = lim JCX>e- r-c x *( LE K ~[f( I\ )] )d 'C x* oo i-+ CX> = 0 c i' J~e- ~ 10 x"(xi;(-C ))d "C = x*( J~ e- 3 " x E ( -c )dr ). 0 0 in E ., l. oo - E· "t" e ' x e .< -c )d "'C S 0 l. • -61- Let L = U . 1 Let x * 2:.1 . 'L: Then has measure zero. be an arbitrary element of B*([ O,oo); ?f) as previously. Then for every p x * (f() ) ) = lim x* ( e J - £.1 ') X* , and define y S of L not in f( S ) i -+ co 00 =.lim = y *' e-fc x*(x t .<r ))d>= O l-+0:> co (x( • ) ) = x "' ( 1 J Jr y! (x E .<·)) =.lim l-+oo e- l "C x( L ) d t:: 1 ) , so that for almost all 0 s> o, 00 f( ~ ) = e- x( "C ) d "C t:: • 0 The follo-vdng theorem deals with B ( [ O,co); X ). CV Theorem J.4.J: If J:_: is a unif orm.ly convex Banach space, then the following conditions are necessary and sufficient that f( A ) be A> equal almost everywhere for function in Bco ([ O,m); X 0 to a Laplace integral of a ). 'A-1f( A ) is in B( [ 8 ,co); l: ) for all 0 > 1. O, Jjoo r 1 \\ f ( '1 lll d 11 = O( § -m) with m > ; , as 3 -+ ,,._, = O(e "(/J ) with O > o, as ,j -+ O+, 2. 3. M, co K>K. 0 Proof: Necessity: Suppose f( 'A ) in B ([ O,cv); :f ). 00 = J cv - -;\ '{;" e x( -C )d "'C" , where x( r 0 ' ) is -62- J CX> • e -'r"C d "l: 6 'A- 2 \\ x( • ) \I , so that 1. is necessary. CX> 0 then 00 53 A- ll 1 f C A )\\ d A ~ .3 - 1 \\ x( • ) \l , 00 Further, so that 2. is necessary. Finally, from theorem 2.3.1, we have LK,~ \,.. [f( A )] 1 = (-K-)'2 e2K lT-C 5CX> 0 1 e- K (~+£\ii_- 2x(ol )doc. ' so that By 1., 2., J., and theorem 3.3.1, we have, for almost Sufficiency: all o, ) > re r ) = lim K 4- r.-~1: LK,T (!c Al] dt:" • 0 (IC By Pettis [ 10], a uniformly convex Banach space is reflexive, so that X Let <e is reflexive. be in L (0,w). 1 J 'l' ( Define 00 1' K ( <e ) = 0 -c ) L K, -c [ f CA l] d " • -63- II T K ( ~ ) II .; Then, r 0 <; II L K '. [r< A l] II m I <e ( -c ) I II L k 'T [r<A J1 II d" rI~ ( -c ) I dL ,;, Mm II re (.)II, 0 Thus { Tlo() is a set of linear transformations on a separable , and 11 TK: \l ~ space, L ( 0 ,eo ) , to a reflexive Banach space X 1 Thus, by theorem 1.8.1, there is an increasing unbounded sequence { and a linear transformation T on ~ to X , such that for everi.; * in ~ * , and every lim x *" (T K ( ~ ) ) = x * (T ( 'f ) ) . ~ functional x i-+ eo M00 • Kd , in ~, i But by Dunford [2] , every bounded linear transformation on X L 1 (O,eo ) to a uniformly convex Banach space T(<(l ) = Jm <e( r )x( r )d-c is of the form , where x(-C) is in Bm( [O,m); l: ), 0 Thus x( -C ) in B00 ( [ O,eo); X ) exists so that T( <(J ) has the above form, -Y*° and then we must have , for every x *" in J:. , lim x i-!)- co * ( J <e ( -C ) L t< > -c ~ ( A ~ d -c = lim x * (T ( <e )) ro 0 i, J ~( i-+ eo Ki 00 =x *" (T( (€ )) =x*( L )x(-C )d-C ). 0 Let '" "\:. ( "C ) = e- )-c , 't'J > O. Then, for aJmost all ~ ':::> o, -64- )) =x*(.lim J<X> e-r-c LK.,-C x*(f( 'S' J. .-)- =limx*(f~e-)"T: J. LK.,-C[f(A)] 0 i-+co 0 00 d"t:) J. . =. lim x* (T K. \ e-3"-c )) = x* (T(e- t-c )) 1-+co = x*( J. f"' 'f e- z:- x( "C )d "C ) , 0 so that for almost all f( )" ) = ) lfc A)] dt" > x( 'L )d -C- o, • ) -65- Representation Theorems for f( A ) in H ( oL ; p 5. The class H ( Ol ; p Definition J.5.1: x) . X ) is defined as follows. f( A ) will be said to belong to the class H ( Dl ; X ), p fixed, 1 ~ p p ..:.. oo, if (i) f( .A ) is a fure ti on on the complex numbers to the Banach space X sup (ii) '\ ~c which is holomorphic" for Re J p>iX. \ J 11 c f + ?'/ )ll ro f i -co A > oG • 1 111 P = l\ f Hp p ct = '1. ) lim f( f + i '1( ) f( cX.. + i exists for almost all E' ~ o(, values of /f(_ , and f(oL + i1() is in Bp (( _,,,oo); -:J: ). (iii) For a discussion of the dependence of (iii) on (i) and (ii) see Hille [ 6]. 'rhe following two theorems give the conditions under which a function in H ( OL ; J::: p Theorem 3. 5.1: ) can be represented as a Laplace integral. If f( ~ ) is in H ( OL 1 ; "X) where lim 1 K.. C [f ( ~ ~ . exists and equals I(.~ oo ' J~+ ).oo 1 g( r ) = - e-cfA' f(f- )df2K i . ti. - ).oo f('A) = f~ .-·h: g( -.:: )d T. of.. > o, then , and • 0 Proof: By Hille [ 6] , page 213, theorem 1O.h.1, we have for Re A>~ ~(- See defi ni ti on 1 • 2. 5. -66- 1 f( 'A ) 1 dr- =--=2 I\. 00 (f CA ~ Thus L K -c ' = :l. I( ..Q. :!. K.. n;-z, 2 K. = : e ._/) ""C" J Db j J- J(X) f(c1.. +i'>l) -co d?1 /I -( rx.. + i 11 ) 1 -, . 1 "2 cos( 2 K ) 2 )f( K-( J + 1 )/-c )d) 0 oeos(..u<j) -r(K.(f1·+1)/c )J3 The interchange of' integrations is valid for K. > r o<.. by Theorem 1.J. 6 since Obvi ous ly, for each lim oo and ?'} , 1 'C • 1 2 2 1 1 't.(°"- +i'VJ ) • e K. . e - K ( - K:( ~ + '>1 ) )2 • (1 - :!:: ( Cl + i"'1'1))_2_ -1 - e IC. Further, for each - -C < "">'] 4- '""• "C this limit exists uniformly in ?( for For , a lengthy, but straightforward, calculation - 67- shows that the maximum value of 1 1 \ e2 K(1 -(1 - ~( ol + i 1() )2) . (1 occurs at 1( = o. e ol "'(; - j:( t o, > K 0 2 - e C:(aG + i'Y'l )\ . ??)) - oc, + i -, e 2 K ( 1- ( 1 - Thus since lim I{.-:,. , for each t: ( ~)~) • (1 00 ( E- ) exists such that f or K >K"(t' ) Then for K > K ( c ) , 0 Now choose )( > K 0 ( 2Tl€ ) • Then BLK>--C [~(A}] - 2'n: llf ll 1 r:.:(cx+i.'1) ~(Ol.~17)d~IJ -.c QO = (\2.'rr J(...Q.:z...c C, _ <' - ~ <o<. +- L 'Yl >) ~)- ~ (()(. +- \. '1(>f ± -:..t. -c (111..+<.'>z)) tcO(.t-t.T)) J.11 \\ -00 ~ 1-rrJ t..e.·u<:.{l-{I- ~(0£.+'-"l>J~l,-~ <oc+.:~>r~ -._Q.-c<ot.+'-'l'J>\n ~lo£+~~1ll il.'Y} -~ ~ 00 ~ .i" 1..2.2.c (I- (1- ¥>'i) (I - " : r ~ -_£,'""I S_ 1' f.coo +i. 'l'J) II d. 'YJ I ~ Tf€: I ~ ll 1 =C., · < .1:n: · II ':f-\1, · I "\Thus lim I(_,,. co = -12ll i ~ «+ ol. - ioo e-Cf. loo f(,... )dr-- = g( /A- ). -68- Hence vve have = <:tc1). r r The interchange of integrations is valid for Re ') > oG II • <"' + i Tl )r<.,.. + i '1 lll d"l =e"'" "C -oo To deal with the cases p since llr(<>'+i"l)\\ d'Y(. > 1, we must take cognizance of the fact that ct,+ ico 1 2Tli J ~ - may not exist. is in H ( cX.. ; .:f 1 p e~<- f(f- )d}A- "f 1 1 ioo t DU ' l p + P' = 1 ' ~p 1 and 7 1, J . 4.2: If 1• r( A. ) is in H ( ~ ; ":£ ) p 2. 1 1 -+- p 3. P' = 1, @ P' > 1 ' then f( '). ) = f(A) ) so that we may apply the previous theorem to it, with the following result. s. Theorem /\-f" p 'A <5 r.-'). 0 i: :;r g (3 1, cl > ( -c: )d c o, where L -69- g ~("C) =lim LK -c K.-+ co ' ~-(> f( A) is in H ( ol; X Proof: ), for by applying Holder's 1 inequality we have r II (ol +i rl.) - ~ f( ol + i 71 )I\ ct "l~{r 11 r(oL + i "llll Pct~ ~. -ex;, -co {f" lol +i 'Y{\ - ~P 1 I d ?'(} P' <._ co. -co Thus applying the previous theorem, we have lim 1 K -C Krl co f ' and f ( -:X ) [~- (3 f ( A ~ = - 1 -. I «. + ioo e-c = A~J 0 2TI l co J d., - . lco e-~-c: g \Oc~ )d~. />- /'-- p f (/" ) d /'" ; g fl (-.: ) , -70- 6. A Special Hepresentation Theorem for a Class of Numerically- valued Functions. The fo l lowing theorem gives sufficient co nditions for a numericallyvalued function to be represented as a Laplace-Stieltjes integra l . If f( 'A Theorem 3. 6. 2: ) is a numerically-valued function which satis- fies conditions 1. and 2. of theorem 3.3.1, and r1 3. L K, -c [re 'A ij Idto ,,; M, 0 then there e..'::ists an ol. ( "'C ) , of bolm.ded variation in [ o, cu) all W > O, such that I\ ) f ( "' = Jf (X) e - ?."t: d oG ( -C: \ almost everywhere. 1, 0 Proof: By ~'~idder [ 1 2] , page 31, theorem 16.4, there exists an increasing and unbounded sequence of numbers CX- ( L: ) , of bounded variation in . 1 im i-+oo [ 0, w] J e- 'J""C L K., --C [f(A)] d -C: C0 O l lKi) , and a function all UJ > o, such that = But because of 1., 2., and J., f( ~ ) satisfies all the postulates of theorem 3.3.1, so that for aLuost all =f( ') ). d cX.. ( c ) almost everywhere. Thus we have -71- REFERENCES s. Bochner and A. E. Taylor, Linear functionals on certain 1 spaces of abstractly monotone functions, Annals of Math., v .39, 1938. 2 • N. Dunford, Integration and linear operations, Trans. Arn. Ifath. Soc., v. 40, 1936. 3 • A. ErcYelyi, The inversion of the Laplace transforr-..D.8.tion, Hathematics Magazine, v. 29, 1 9.50-.51 • v 4 • V. Gantmakher and V. Smulian, Sur les espaces dout la sphere unitaire est faiblement compacte, c. R. (Doklady) de l'Acaci.. des Sc. d l'U.R.s.s., v. 17, 1937. 5. • L. M. Graves, Riemann integration and Teylor 's theorem in general analysis, Trans. Am. Math. Soc., v. 29, 1927. 6 • E. Hille, Functional Analysis and Semi-Groups, Amer. Math • Soc. Colloq. Publications, v. xxxi, New York, 1948. 7 • I. I. Hirschman, Jr., A new representation a.~d inversion theory· for the LapJace inte gral, Duke Math. J., 1948. 8 • B. o. Peirce, A short Table of Integrals, Ginn and Co., Boston, 1929. 9 • B. J. Pettis, Differentiation in Banach spaces, Duke Math. v • .5, 1939. 10 • B. J. Pettis, A proof that every uniformly convex space is reflexive. Duke Math. J., 1939. J., -72- 11 • P. G. uooney, A new inversion and representation theory for the Laplace transformation, Canadian Journal of i,iathematics, to be published shortly. 12 • D. V. Widder, The Laplace Transform, Princeton, 1941.