Linear Functional Equations on a Composite Range Citation Kennison, Lawrence Sanford (1931) Linear Functional Equations on a Composite Range. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/e918-z167. https://resolver.caltech.edu/CaltechTHESIS:09062024-212755151 Abstract No abstract. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Mathematics) Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Mathematics Thesis Availability: Public (worldwide access) Research Advisor(s): Michal, Aristotle D. Thesis Committee: Unknown, Unknown Defense Date: 1930 Record Number: CaltechTHESIS:09062024-212755151 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:09062024-212755151 DOI: 10.7907/e918-z167 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 16701 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 09 Sep 2024 19:05 Last Modified: 09 Sep 2024 20:31 Thesis Files PDF - Final Version See Usage Policy. 17MB Repository Staff Only: item control page

LINEAR FUNCTIONAL EQ,UATIONS ON A COM:POSITE RANGE. Thesis by Lawrence Sanford Kennison In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy. California Institute of Technology, Pasadena, California. 1930 Introduction and Summary. In Part I of this paper we develop systematically a general theory of linear functional equations, where the unknowns are a function and n independent variables. 'rhe Fredh olm theory of integral equations , and the classical theory of linear algebraic equations are included if we specialize the equations. We may regard the set ·o f equations as a linear trans:formation on the composite range consisting of t h e function and then variab l es. I t is proved that the transformations for which the bordered Fredholm determinant, which is defined in § 2, do not vanish form a group. A general- ization of the transformation is brought in at the end of Part I. In Part II we apply the generalized transformation to the invariant theory of quadratic forms with continuous coefficients on the composite range. {•! The algebraic theory, and the theories worked out by A. D. Michal and T. s. Peterson (references on page 40 ) , are special cases of this. The theory of quadratic forms with con- tinuity of order one is reduced to the case studied earlier in Part II. In Part III the theory worked out in Pa.rt I is applied to projective transformations in fun ct ion space. In particular all the results given by L. L. Dines ( reference on page ? ) are obta,i ned somewhat simpler. U> Part II includes some unpublished work of Professor A. D. Michal. It is proved for the first time that the one-parameter family of finite transformations generated by an infinitesimal projective transformation in function space forms a one-para.meter continuous group. I wish to put on record my indebtedness to Professor A. D. Michal, who suggested the problem and has kindly supervised and assisted me in carrying it through. Table of Contents Part I: Linear Functional EQuations in a Function and n Independent Variables 5 1. Intro duct ion 2. Def in it ions 3. Convergence of the Series Defined 8 4. The Eight Relations 9 5. The Inversion of the Transforma.tion(l'.fO) 6. Continuity and Frechet Differentiability of 6 11 / Functionals of Several Fun ct ions. 12 I The Frechet Differential of the Bordered Fredholm Determinant l? s. The Product Theorem and the Group Property 19 9. The Case of Continuous Functions 26 10. Transformations of Third Kind 2? 11. The Product Theorem and Group Property for Transformations of Third Kind 29 12. Symmetric Transformations of Third Kind 31 13. Appendix to Part I 34 Part II: Q,uac1.ratic Functional Forms on a Composite Range 1. Intro duct ion 2. Relation between Forms of this Type and 3. 39 40 Symmetric Transformations of Third Kind. 41 The Transformation Induced on the Coefficients 42 4. An Invariant of the Coefficients 44 5. A Special Case of the Previous 46 6. Forms ~uadratic in a Function and its First Derivative 46 ?• Application of Earlier Work to § 6 48 s. Specialization of Transformation (7.2) 50 Part III: Projective Transformations in Function Space 55 1. Introduction 56 2. Reduction of Dines' Form to Type Studied Earlier 56 3. The Infinitesimal Projective Transformation 59 4. The Finite Transformations Generated by an Infinitesimal Transformation (3.3) 60 5. The Family Generated Forms a Lie Group 63 6. Oene-3--ii si or.a t i Pal?~ III ~ Part I Linear Functional Equations in a Function and n Independent Variables. Introduction: In the firet part of this paper the - theory of the eet of n ~ 1 linear integral equations: ~(J'I) ==- ,;J(J<)+ . ~l .:: ~ i. ~ l .+l<{J<.,s)~ <~)dst-.,if, K/t1J3; . ...... . . +( /(l(s) ,3rs)ds \ t>fi 1<)- ;;}1 (C= t,2 1 -- 1 ,,.,..J ie developed from first principles, the results being given in explicit form that will be suitable for applications. We assume ance and for all that the functions J(lX-,~) 1 K/Cx), t<''{sJ are bounded and integrable in the Riemann sense on both X arid 5 on (q1 k-) . F0r most of the applications t he assumption of c0ntinuity ie also made. A consideration of thie is deferred to § 9. We shall use the convention of writing the continuous arguments above as indices of position in the alphabet after q, and shall imply Riemann integration on {t:41 .t.-) with respect to any such index which occurs aa both sub- and super-script in a single term. Indices before q shall take on integral Taluea 1 ton unless otherwise stated, and the repetition of such an index as sub- and super-script in any term shall imply summation from 1 ton. With this notation we may write the above as ( I, 1) ~ ~ =: ~ ~+ Kst:"Js + l<J -9_< ~ l ::- ~ l + (( ~ j S -+ (< j ~ '& If tX. and ~ pass oTer the composite range consisting of the continuous interval ({,/1 '-) fellowed by the discrete range (1,"1 we may write ( 1. 1) as (I. "2) ~ c,( ::- ~fX_+ t<~ l' ? The surmnation and integration conventions do not apply to subscripts of indices. DineJ' 1has indicated an explicit solution for the case n=1 , and has applied this. He gives the relat i ons (4.1 -. 8 ) below for that case in different notation. 2 Hildebrandi J indicates that the existence theorem for inversion, the product theorem and the group property follow from the fact that the bordered Fredholm determinants are actual Fredholm determinants. • We shall prove these by direct methods analogous to those of Fredholm's classic paperfJ/ Definitions: We define the function d,--->'-.. as s ---S...._ 1 follows: Also the >.'X- X-->'.._ 1. 8 I ~ • (. s; - - -s- ~ ) )(, - - - Y""' i B :s s, - - - s-.,,..,. .c::, ~ .-:, - - - )<,,._,, i a I, s; -- - $',.,,_ U)L.L.Dines: "Projective Transformations in Function Space!' Trans. Am. :Ms.th. Soc. V. 20(1919) p. 45. (i)T.H.Kildebrandt: 11 0n Bordered Fredholm Determinants!' Bull. Am.• Math. Soc. V. 26(1920) p. 400. (3) I. Fredholm: "Sur une Classe d 'Eaua.t ions Functionelles!' Acta Math. V. 2?(1903) p. 365. - 8 Let us define a:e $be , 11 f.:rn t o r Qi' as the cofactor o~ i9cA A -m- the egf ,,, t.o 1 ,cf as the determinant ls} +t<J f where ~j is the usual Kroneck.e r delta. w. ftn l.he1 define We not ice that Now define: first minore. Convergence of the, SeriE!s Defin~d_: The general term of each of the series in (2.4) is the repeated integral of a determinant of order mt n. be such that N. Now let M lf<t/J /f<i/1 /!<Jl /bJ+r<f / are a.11 less than Thie is possible since all the kernels are bounded, and there are only a finite number of them. Then by 9 Hadamard's lemma the general term is less in ab solute M ..,,.,_+"" { -f- -~r""'-1" ~/ where p = O or l according to the series ta.ken. value than I'. - ... - ~-+-) -r- The dominating series thus obtained converges for all value3 Gf !'-f , fer t h e ratio of the (m•+l)st term to the mth term is M ( -f -") ( /\1'\. +t /1,r,_ T"/V\. + I) ,i,,..~ " " ' ( /vh t-,'V\ + I\ Y:z_ ,.,,.,.. +- ) , ,,.,,....+""7 Jr,_( I + ~ ) , 1/-z J ,..,,_+ti l'-1 ( 4-oJ [(I+ ..L. fn-+I)}', Nd \A/ L ,,.,,_,.➔ ,,o %;-~ q, ....L. J-"t-t--, li + ..,.,_+..... J v"'L -_ L "° (lf-_l:h_ ) X ~~ /Yr- t-, e v,_ = / _J__ therefore t h e l imit of the ratio ie eY-i../'1(-t-- aJ !:;"°~+1)½ - O, Hence these series converge absolutely and uniformly in (o.,4) , and the def in it ione are justified. /\ )( 'I, - •• ')('1,,,- The Eight Relations: If we take .L..l s- 1<, ---,c......,..._ and expand by minors of the first row, we get /2 x I(, ---r........ - I<>< L x; -- - ?<,.,,,,._ - t< J< A~ "12 --- ~SXr·•-r--- s ){,---)t'_. x, 6)'l··-)<"""-i- --- +(-,} ,,,.,,. f(X 1,_ /j,1' -- - Y:-. /( 'K_ 1::. -I I<, - -- 'f-""' sr:,---Y,,.__I+ 1 s Y,---r--. Since the ,c~ are variables of integration we ma.y rearrange them, giTing _,6 x- ><, ---r,.,..,.._ ~ I<.%' 6..x,--- Y:...,,.._ _ $'~, ---X-,,... s Y 1 ---r........ ""'"' t<.x Lltr, --- r,___, + l <l\ ~ 1 1<, -T s )C, ---Y,.,._, 1 1 ~ If we multiply by-9 , a.n d sum frGm m =1 ta m =="° , we get (4,V D; -1:!; =- J<; (D-L:.) -I<-~ D} + f< JDj-A1) Using ( 2. 3) and tranepo sing, we get 1 =o ' 0 5x - DI< 5~ + I<!< .o -t5 -t- I< 1~ 0 5 If we expand similarly by cofactors 0f the first column we get -)I'..._ x,---x,...__ 10 As x r. --- }('\¥' above let ue expand A 1 "'• __ , y.,,_ by cofactore of the first row, getting ~ )( x. --,r--. _ I< Y ~-t: Y., - -,y,,_,__, _ I<~ L:,.~ x, --c Y,---Y- - -r _,...,,..,_. t >:,-··Y,,_. J 1 x'½-- ~ ),:,-- · ')(.-i-. Ta.king 1_ ~' ef both sidee, we have ,,.,_, ' - DI - .6.-~ :: I<! C> 1 + /( j ( D f - 0 <; t f-~ {) Recalling (2.3) and transpoeing: o! - D 1< r +- 1<.: o1 + 1<; Dt ==- o. ( 4. 3J Expanding L:Y,1s v.)(,, -- - -'le......... by the first column eimilarly, - ?'...,,.... . 1 J f Df - DI< j + @i t<s + D-R !<5 = o, we have . (4-. 0 Taking the eum of members ~f the (m • j)th row A~ 1'- -- - Y by cefactere of the first rew of ,L.)s ><: ___ ; : , we have 0 = [<.1£:;.'f.r=:-~-- -,,__,,__ /(1' ,6-iY,---Y.__, +('i,-i+l,,-iJ.LS~,,---r....,_ .5 Ta.king ~ ..,.._:a , ~ """ ' S Y, --· Y,..,.,,_., t"" -' "'p ~ y 1 - - - )(_._ ' and changing sign, we have D~ - 0 I< 1 + I< 4Dt + f< j D: = D l lf, rJ Similarly multiplying the (m + i)th column by cofactors of the first, we get Df- DK[+otr<}+o; 1<f=-o (11-,tJ) For our final expansion of this kind sum the elements of the (m + i)th row timee the cofactors ef the (m + j )th row ef Ll;,:~~;::_giTing ("~ A'f-,--- r:.....__ _ /<i ,...-t-_ )( ,---1-.., ( C l f<i) ~~ ~.- - -Y"-"bd ?<, -- -'/',.,_- ~ -t;-~j)(,---Y:,,,._,, + C fl+ Ji I )(,---'y...,...,_ Ta.king 2- ~ 1 , we haTe f j· D = ~;~ D ~- Cancelling /) (4. 1) - ( ~} + t< j) ( D j - ~ i) D ~J.:: /) ~; ~j and changing sign, we get D~ - 0 I<~ + /( i D} -t f< j D~- ==- e) . Similarly by columns it.L ~) l' 0 i( - D l< 1( + O-t:i I< i.1 + 0 ~ l<-'I1• -== o• We collect the results ef this paragraph int0 11 Unde:;: the conditions in § 1 and wi tl!, Theorem I: t h e de~initio n ~ in f 2, we h~~e the following :t~ions: (4-, f) D:- D 1<; + 1<: D1 + t<J D{ ::: D (Lf, 2) (4.J) (Lf, If-) (lf, rJ (4, 1,) (lf. 7) (4,g) Since the odd and even numbe r ed relati ons aesociate naturall y , we may refer to th em later as (4. odd) a.nd (4. even ) respective l y, The Invereio n of the Traneform~t io11 .( 0# Ol.:. May write (1 .1) as SK(~}:::"; , a. transformation f r om If S'ei i e the product SL (51,) ~SL, S ,, We J to i . , the kernel e of are as follawe: Ls>< + (< 5X + L 'f-t (< -t5 t l 1 l<-PS X X' I )( + L X Ki: + L X (< p' Gi·==-Li+<j t- -j ~ 1 t" '1( \rt S - j ' ' ' " r i - L 5i + Ks 4- L t: I<;+ L 1' f<; 1.T15' t i .f Gt~- :::: L j- + I<}+ Li t<J -+- L.R I< -i We will now co n eider t h e c ase D#= o. Aesuming that ( 1.1) has a eolution (j ), apply the transformat io n 5; 1 to ( 1. 1 ) where the kerne l s of s,<-r are ..:. T) r oJ - oJ 0 ◊)-/JI" Thie gives us (S-,t) Applying (5 .1 ) to the right side of ( 5.2) the S <i-f 12 kernels of the product s,tsl< are - ~ times the left hand Hence s/sK~s0 , the identity eides of (4. even). transformation, and we have Theorem II: If a bounded an~ integrable .so~utio~ (j) ef ( 1. 1) e;i st s when D_f: o, it i e UJli_g__u.~~~!14....i s given -v ~ r-::: ~ ~ i. 'f. D'/ - =- ,;i L• _ DX r / 115 71·~-;1 Di _s 11- 5 - ~ · ,:;, ti Di , D Substituting (5.3) in ( 1.1), we get D _, - ) (s, 4) ,,:; :=: S k. • <;K -"J Applying (5.1), the kernele of 5k·S,;' are - the left eides of (4. odd). 6 timee Hence S'k.-s,t==-So and (5.4) is This gives ue verified. Thesrem III: The eguation (1.11 has a unique solution when D#O,. given by (5.3). Centinuity and Fr6chet Differentiability o~ Functionals of Several Fun!l,ti9ne: A fun ct iona.l F:2t,--, 2-_[ Cf, 1 - - - 1 cf,,,.,_] ie said to be 1 in ear and homogeneous in the n functions c.f,· , if ( 6, I) Fz-, -- -~,,...,[A , Cf, t /\-<, ~ J - --J ').~ Cf--... -+ rt(,... i.r-,.,.] :: A, ;l, 4 J?,µ, -+ -- - +A-,">-.....t i3~,L(,,,J ...tihere the A'.s and B;are fun ct ions of the Now define F' il,---~,,,_,_ [ Cf(] ::: ( Theorem IV: . F z, --- 2~ [Cf,)---, cf/\,\]) 1" _ A necessary and sufficient conditian a,__] b e 1 i near an d h omogeneous 1n . the that F~.---c-[ff.,1 - - ~1 •-~ n functions %is that (6. 3) 1) F/·,--·-'t-.ue.:J be linear ,and homogeneou s in 2) f •,--· ?-[Cf,, --j ct~]* = . '1'"f =D(j :/:-i) Ir Ff·---~--- [ J. Cf IPL' 13 Suppose (6.1) to hold. Then let ~ j = fU.-;1 =O ( i Fl) pr, --- z"""' [;\Cf.: +;tt.: '-l"J ==Ai.\•+ U.: µ~• which proves 1) '>-/ :::0 Now let >.l::/J Then (jf:.L) 1 /v{j ==o {all / j Fi.r:, --~r,_ [ cfc] =Ai Ne W let ~ l :: f1 /l,,(. ~ - -: :. o { a. 11 i) """" F1 ,---i'-,_ [rf)•· ---cf,.,..]=: ? r1,= _?. F~,--x,_L<(] which proves 17 2). / F- 1 L.-f l Hence the condition is neceesary. Now if 1) and 2) h old, we have F 21 - -· ~-- [ r1, 'Pi +µ, 'P., - - -1 ;\,_cf""+ µ_ ~] ::: ,,,.._ ""- 2 F.r,---~--[>-,'f~·tftAi'h]==-? Ai~.:+,t,t,Gi: [::.1 l &. :1 Hence t h e cond i tion is alee eu fficient. vJ be a fu nctional of the p Let FZr' ---1,,.,.,.,__ [ y-rtvX C r l 1 ,; I xt:- Y;- funct io ns '( of two Tariablee, of t h e q functions f'f. of 0ne Tariable, and of the r i ndependent variab les Give each of the Y~ an increment ~ Y, such that >, is the largest of the set max I ~YI. Define AF- as (b,4) r~,---?,..._[Y;-t.+sY;)-t Y/+~r,: 'ypt-b11]-Fi,---'i-[r~: r~ Y.?] We say that P is a continuous fu n ctional of the Y~ a.t the point of p+-q+r-fold function space, (Yc:t- Yt Y;,) if for every € ::> o there exists a &' such that if >; ~ f; JAFI < t . We eay that F 1 B a continuous fun ct ion al in a region of the above space if it is continuous at every point ef t h e region. CS '(h ~ rJ The Fr,chet differential f F i!,---E,,_ [ Y~: Yt )} / i Y of F is defined t o h ave the properties: 14 1) It is linear a~d homo ge_r'!,§.@Us in the 2) / 6,=:. ~$Ff approaches zera with S Ys' '2 , The distinction should be noticed between F ae depending on the variablee }_p , and on the variable ~, - -- :ie J J """ • The }; are to be regarded as independent of the Y{'t and Y/, while the 2~ enter as a rule as rt y1: arguments of the Y-L or 1- • a,pd the latte~. of the i:r;;i e;e~eral depel!Hl ~ For this reason we regard F as a functional Y; and as a funct iou of the i'-.5 • If we take the differential of F defined in the differential calculus, it ie linear and h omogenous in dz1;--ir,U,._, and the likewise the vrt 1 [ v )( and 1 i are regarded as fixed. 'rf , On the other hand, the Fr~chet d i fferential of I= will not involve the c?fi's , but \Vill, as defined above, be linear and homogeneoue in b Yr~ ~ Yi ~ YY. bordered Fredholm first minor which is a fun ct ion of )< and .S , and a functional of the K-Ji. • The above definitions and remarks readily extend to the caae of a functional of n functions, each of an arbitrary number (including zero as above ) ef variables. i?1 -- ,-z, "[ ] Theorem V; Let F ""' 'r;, -- -.1 Y..,.... be a functional af then functions Yt· ~ each of._ an arbitrary number (including zertl of V¥:iables,__and a function of the m~l variables i1..!J,; --,-i- 1 ::>:. . Le.t F posse~ J-J. Fr,chet differential, ~ F 2: , - -- z:,,,,., " [ Yi, H 'J Y..... / ~ Y., ---, ~ 15 Then - sl 4-F ~. --- z,..., "?< LY., - -} Y. ,,.-J ol-;x= f .t-S' F ~, --- ?. -, y,"'._J SY., - - -, ?J Y,.,..] o<.--,c a. 'K [ Y,1 ---1 ~ Let . us call Then by Theorem IV, we h ave =%. $ , r:/,---z-.1 X ~ Fz,----z-,x [r;1--1Y,...fb'r;1· - -, S'Y,,.] Then · 1r rl'\. _(-t.S F'i,---z,....,><[Y,,---,Y-f~'r.,---,SY-] == l i [ ar::, ·-· 2-,)' "- "'- -( an d thie i ntegral is linear an d h omo geniu e in t h e bY~ . ,tr- i? - - - ~ A Sa.fT' ~, X [Y.,---1 Y,...]ot-r::: 1.(,..F~,--· r ..-, X [Y,t-~'r; ,---) Y..... +n"....]c-0"' 4 2-," [Y. --- Y""]dvt-:::;, [f F~,---~--,7<- ,,1 ,./' _ J-'F~.--· (), ) } «. V'-"/' But 4 F~.--, e,,._, )( = 5' F~,--- 2""' 11 [ ~ - --) Y- I$ Yi, ---1 f Y-] +&~I it/ where (} ➔o with max ( S ~/. Hence .4 ( -k t -r,..., x[ ?, -· Y., ---1 y_J ~ :.- Jff~.--- Since B-) 0 with max {SY., 2-_,, x [ V., --- 1 Y- / I, l!IO does i ,e,B ~ ~ 'r\ - - -., SY'>'-] h -t ~,n~ifir-4 ~ , and the Ct t h eorem is pr0ved. N0w J.et F ;,-·· ~,,,,.._ [ Y., ---, Y""J (p=. 1, 2 1 -- - b e a set ) of f u ncti onals as b efore. Define 1 [ t ~ "" S~---~ -. Y, - J Y.J = L. F~--- ~- [ 'r', - -- Y""J p p= I I I I If ~ S p exists, we call it sil,---'i-['r,, --·; t]. this case define Ri,;-l,_['(,,---,Y,.,_7 as S-Sp • 1 In 16 (jQ c::; F"~ ••• i',_ We eay that the series ~ ~ L~ ··i t;,,~7 converges to 1"=-1 r the value 5?,---?..._ ['r;, --i ~] uniformly for a region T of n-fold function space and m-dimensional space if there is an N such that whenever P>N , /R.p/ <(. for every point ( Yi>-i "- (~,---~ in T. Theorem VI! of functionals of the n functione )1 , each of an arbitrary number (including zero) of variables, also func:_ tione of the m va:;:,!._ables ~,--- ~---, and if the Fre'chet brA~--i!---(r.>•1Y-"'/H> -·c ~ ~]exists far each -f' different~al cP_ and if the eeries f;, converges uniformly for a regio,!! ~f T, then if the seriee ~ s ~;---~""' [r., -- 1 Y~ r r ½---, r ~1 =- (6,1,J l F,! & 1 • •• i',.,,_ ( ~ - --J Y- I ~ r,I ---; sy,._ J converges un iformly for T ~nd for ~~ boundedL th~s. eerie~ is the Frtchet differential Gf S r---~-[Y. I I - - I t$ be L 1"=1 I J "" Define c&'~i to be f>F;,---i-.[r,,--jr,,,./1-;o,,'r(A··;O] and Proof: ~ S'~ vJ-~ pil,--r--[Y. ___ y:] to -1 I - 6' 5 ~.--- ~,...,,.. [ Y, 1 -- · 1 Y"" I 0 1 - - ) 01 b Ye: 1 01 - - · , o], ~ Then oD &'S~,---~--[Y., ---1 Y-. ($;Y,,--1fY-] = ~ &"'~ ~ o6 ""- f- ,...._ =- P=I 2.. ?- SF"-,.,. i. =.? L ~ F-,,,,: == ~ S Si: t=I p=t (::: J I f=1 Hence ~ .S is linear and homogeneous in the s--s = lL-s r,, + Rr =Ssp + 1Rp $;' Yt' • I f,=' . defined ae above and 'Rp , where \, ie defined by this equation, has the properties of the Rp defined above • ..65 - SS ::::: ASP + A Rp - $ 5p - 1Rp Now by definition of ~~~ for every e~o , there is a S , such that / ASp ~~.Sp( "- ~ for all finite P , whenever l '7 {&Yt./=>-,<( ~ Rp =- R/·--·z,,..,.[ Y,+~ r,>---; Y-+fY.....] - RP[r:,--·1Y-] max Since the c0nvergence is uniform, there is an M euch that / Rp[ Yc 1- ~Yi.]/ and / Rp [ Yi] I are each le se than ~ when P> N;, and similarly there is an /t/2. such that 1 / Rpl c::: ~ p > Ni. • Now taking P greater than both IV, and 1\/2.. , we if have I 4 s ..:. &5 / .( / 4 se 1? -s S"e ) + / f< e [ ~ f- ~ ~J / +/ RP 1 '<c:) / + I !!f / < ~ ~ - 1 1 7 ., 6 Si-,--•i,_ [½-··1 Y,.../ ~r,, --1 fYo..] eat i sf ies Hence the both conditions and is the Frtchet differential of S . Theorems V and VI are brought in to show that the procese of taking the Frichet differential is commutative with integration and summation to infinity, under certain conditions. The Frtchet Differential of the Bordered Fredholm Determinant: Theorem VII: The Frichet differential of D (§2.) is given by : ( 7. () S I) =- Db I<~ - DJ ~ I<; - 0; - ~ k j - D;, 6 t<( - Dj·bKf + D&I<} b.,'1, - ·~ ):",.,,,,.... ~, - - - s ------ is a function of the kernels in the ordinary sense,. ~ Hence its Frtchet differential is the P,f&ntheses here indicate omission of the index contained. 18 Setting S'i: )(._: , integrating, and rearranging the variables of integration, we have ("' JI. ,C, -" ,C...._ - () ~ ')(, --- ?\'....,, - /')N\ +~ ,6/.s '¥,--IC, --- Y---...., (' 1($'· Y~-, o J -'=' Now take 2. ~.::o A ~' --- ,c-__ , <' 1t _,,,,,,__ ,,..,,.,._- 1) .4-1 ><, -- · Y..-., 01'5 ( -1--/vV'.. 1:/t- X, - ·- '?(,,_ .7.. <' {( i-5 Y, - - - }(......._.2_ 0 t:/ · ')(, - -· ')(,,,...' b l<l -1- l:~:J )( x._ ~ (( l. I>;--· r,...,..., s L >;- -· Y....... u 71 I - •• ;;_-,• of both eides, and it is eaeily eeen that the convergence is uniform if the !<'3 are all bounded. Hence by Theorem VI, f h D-::: D S1,f - D~ ~ I<; - D & I<~-- D~-SI, _l -+ ( D~ f- D {) SI< j • which is the same as (?.1). Corollary:_ (1.1) Di vidi~g 12,y Dwe have if DI: o ,., 19 § 8 The Product Theorem and t h e Group Pr operty: If 5i(" is t h e :product tran sformat io n Si< •S," the kernels of St are given by ( 5.1) as follows: ('6, 11) i = Ki I<# -t K; I< ; • i- /( j /<;- (<?, 12 ) /( ('{, I 3) /( 5l ('75.I'+) J(l_ = i<l + kl· ~(<(l<i: +R L /(.() · -t = ,~ ~ + !< i + 1<.: I<; + R ~ I( ] 1 1 J i -t ..R 1 If D an d /) ' t h e determinants of SK and s,~ are both not zero, we can solve these for the kernels of 5'1< and Sk • From (8.11) and (8.13) we have /< { = ( (< : - /( ; ) - I< Ls, ::: ( QJ ((( 5~ - t<J) - ¥ { I< J- R;J f<; - k ~ ) - -~n~- (R;- i< ~ - B0 j CKs Rs.J P- f\ From (8.12) and (8.14) we have (r,L1-) kx. ~ (1<;--1<-;)- ot ci<) -i<}) - ~J U</-R1) 1 0 k~- = (1<}- i<}) - ~~ (f<;~-R; )- ~~ {R}-i<;) Using Theorem XI below, which i s i n dependent of we have from (8 .1 1 ) and (8.12) R;= CK;- 1<[) +(k.:- Rt )f--g)+Ci<J-t<))(- ~f) k; ~ (KJ-~1)-f (t<t-K{)(- ~) + (!<:- !<{)(- g~ From (8 .13) and (8.14) we have (q, 33) i</ =: (/<} -/< 5t)+ (t<}-t<~)(- ~~)+(t<j - i<))(- ~ R;- -= ( t<}- t<J) + (K~ - I< iJ (- ~) + <RJ-1<)°) c- ~) Theorem VIII: fonnation S'l<=-S,cs,< The determinant D of the transe quals D·D . 20 We shall prove this for the ca se when D and 75 a r e b ot h not z e r o. I f e1"th er or b oth van i sh, the theorem will f ol l ow f rom the c ont inu it y of t h e a s f u n ct io nal s of t he k e rn e ls. We have by Corol lar y to Th eorem XVI b elow S :5.__ DD D15 5 D- DDS 75- DD£" D - From ( 5.1) we have Si<;~ Si<~ +St({+ R~ 6 t<; +- k}~sRt +RJ ~t({ + t<1 sr?J • c ft'1 +7<. " 5f(.f. + k f. <' R x (-<-r 6I<;-r: + I< -t:,o -t.P ;J ,o .f sK; -::: s /(; + sI< i + I<~ SI<; t- I<i ~ ki + I<} SI< f + I</ ~ Rj ~ I< l :::- '6 I< j + S (< l + y: s ~ Ki ~ b RJ + s /(} -+ R~ b1<) + 1<) &>i<4. + 1<5 skj + 1</ fir<_,' Substit u t i ng fr om (8.4) and (7.1 ) we have (<B', 5"J D i5 b D- DD SD - fj D~ D Do! [Ci<; +s~ +f{s-1<} + r<}&"t<} -t- R~skJ +r<f si<J] - DDE! [tr;+ ~1<1 +i<!SI<~ + t<;sRi +- R;~t<] +t<; rR[j - DD D~ [ S t?j +S kj + 1<t St<; ·H K} &1<! + Rj bl(j +Kf ~ 'Rj_] -1)00~· [Si<~ -4-$1<7-t i<l61<} + t<}St<f-+R~ &,<1· +1<}~1"<J] 1 , , .P - OD 75 f' [SR t. -+St<~- + t<! rr<} + I<}&' I<~ -t-f<y Sf< j · + I< i $1<)] -[ 1 ' -i. 4- D15 D ~ Ri + s {( i + Rt 8 I< 1 -t (< ~ ~ R~ + I< 2 sI< f + kt bi?;J -DD [ 75 8 i<1 - D~ 8 i< 1 ._ D ~\ s/(;- - 75 i s- I< f - i5 { ~ /( ;: + D8 {< u - DD [ DS-1<: - D~ S' t< 1 - Di' s l<j - D5j· ~/( {- Di' S'(, ;- -+ D st<[);:::_ + s- r<! r 15 ~ R~ - Do ot -o o!~ R~ - no _D~- Ri +Jo D+J D +f R'J [ o o o1< ~ - o oni - D ti ;s~ K:. - oo Et K;• +Do "~J -+ s,,s. [- DD DJ -DDDii<;-DPD{i</+DI5DR{+DDD1] • H f < ~ [ - D O OJ - D D if i<i' - D D t 01 I< +0 D 0 { + - 0 D f J K{J 21 + ~ ,< t [-Do E; + Do B"R! - o Bot R; -o oi~ x;-+ i 5 Ds;J + sRj [- D no;- + D ool<J - OD =s-; /{~ -DooJ /(j -+ oD o;J + & I< i[ D 6)) - DD i5] + ~ Rf [DD D- D5 DJ + s I<~ (- D15 DI - 0 i5 0i /< f - D OD"~ Ri-+- l) lY OI< t + D DD J L -+ ~ R} [ - D V no: I< { - Di5 o{ - D oD { I< f + DOO1< ( 1- DDT5[] In the coefficients of t h e variations of the K's -,s by (8 . 3). eliminate the !< of S' !<} , after removing This gi ve.s for that D () l t) 1< ~ D\( : - I< ~ D: + I< ~ D: - K~ 0 ~ + J<: ~J -o:[oK1-oi<~ - 1< ~Di+ t<~ D~-1<;0{+1<1 ot] -Dj [O ~!'- D 1<{-Kl Di + !<~ D~-!<j D{ -t t<J D{] t) .c. == 5 S ::- - D.D t t D Dt = - D [ C! - D~~ + D~ I<~ t- D; K{] tD[Df - 01<t + 1< 5v< Dt -\-I<~ D!] D; I<~ 1 D~· 1<i] n1<i: + 1<~ D{ + K} o{] ' -i- D~(O~ -OKS((.-+ - b_) ~ [D 4 - t +D!C6:-~t<-;+ot1<1 -+o:-~iJ - !)_; [ 0: - 1) I<: + K ~ D~ + /<'. ; - l) ;' ] =: 0 bl<} i s D times + 5 r D I<; - D /( i - R~ D: + I< t o1- l<j of +-Kio: J _ Th e , ~oeffi~ i e::_t, of - =-,ic Dt D !\=::- ts - 1) Ksct: - I<===-t-u D ~s +- I(i- Dsu - ""i:: 1<51 DJ5 -+- I<,1: DsRJ I,( -D ic D1<}- D1<1-~~ Dj-+ 1<~Dt-i<1D~+r,; of] -() c;· 1--0 01s = 0 [ D1- DI< j + I< l [) i + /(j bf] -0 D - Dk + I< t I)i. + I<y Dp J "' ,11' [ vi v.. ·(1" s S IA lA 5 IA 5 _D~ [ D~ - DI<} + 1<~ b~ -+ 1<j of] L o! ' ' + o~ [Dt - D~-1~ + P-t +Bl,<~ -+ D! L~j - oi<i + f>{ R; + i{ k}] -D [Df- oKi+ Of fr; +"6{ RiJ == o 22 The coefficient of 15[ DK;-- bki - 1<_: oJ + t<i o} ~1<f is D times -r<J o;-+ t<1o 1) - + t -t t..\ "t- D'-1 ::, t ..I "t RJ -D~[Di<.1.D/Kj-ku D/+ f<.v.. f-/<,1D-; +-f<~Dj - 0; [OI< j - Dk i - I< ~ 0} -+ I<~ D'J - t< 10j·-+ I< f o)] _ . The coefficient of Sl<j is D times DLD I( t - DK { - (< { 0 ~ + /( { 01 ... I< j D1 + I<$' D{J -n;[or< f- DI<~ -i<~ o1 + t<~o1 -~j o{ + 1<10~] - o1c~ P-1- D1< : - 1<~ D~ + I< { Dr - Ki f) t + 1<i otJ -1 In the coefficients of the variations of the K\ eliminate the D times k1 by (8.2). The coefficient of SR} is 23 -~s :c-s =[ -S D D-1: - D-J<- 5 +El i<lA + D - D Dt + D Dt ~ 5 -t: v. -t ~ ,"<t] - D~ [Bt - l5 R~ +15 51 Rt-+ D; i<l] -s -{.,\ riS: 1- 1 - Bt [i5~- - Di<i -+ D~ !< 1 4- v.f <i J -15 [ Ds-t- - Di'<=~ -t- 1<~ o~ +t<; D~] + o: [ o~ -D ~~ +f<~n;+t<l rs{] of S' /<\a Dtimes ___ . The 'coefficient ,; D[ D I<: - D Ri - f5 ~ I<; + 15 t t<~t: - Dj' K: + oJR;] - 5~ [ i5 1< i - 15 i< t - r5 { 1< ~ -+- Dt i<: - D1t: -+ 75 j r7 ~J -o1co K:· - oK1- o? K1+ D{ R1- ol 1<i + n{ RJJ -Bo{ +551. == D[ Bi- 15 Rf+ I>J' i<; t-r5) i<;J -n~ [ 5~ - o Rt + D{ Rt+ 5i· R~] - D{ ( oj - r, R~ + i5{ RJ + Df° i< j J -BLoi- o1<{ +Kt i~ + Rj o{J + l5 i [D~ - oi< ~ + r<; o; ~ t<1 o{J + TS j r O: - o~ i + f< t i ~ -+ T<1 o;J = o 24 - D-t.[D 1~~- 75 K~ - 5~ t<~ + B; K~ -5; t<!-+ B;t<t] j ~ D~ [ i5 i<~ _ 15 -· ' ,<~ - B~ K~ + ot t<'J - l5i i<J + 75 ~ i<1J J -:::::s ::- 5 -DO ; · +OD-=J oro1- ;s 1<;-+ i5t R} + 01 R:J -c,1[o~ - o i<~ ~ o~ K~ + BJ «{J _DJ°n [5; _ D /(-s1 + D-s t<-,_,..R + D-si •f<l] .t IA -D [Dj - t> R1 + t<~ 'bi + t<J oJJ +o~EB;- oi<}+ l< : o}-t1<1 o;J i D~[D/- ol<j +t<t i ;. 1-1<1 o}J := 0 Hence the right hand side of (8 .§") vanishes, and we have Referring to Theorem XVIII below, we h ave D=CDD where C is a constant. 25 If we make all the kernels of SK and .SR zero, those of 5; also are zero, and Hence (i.b) C =I and D== D,D Theorem IX: tyge D:::. D= ~=I . The set of all transformations of (1.1), whose bordere d Fredholm deterrni~9At.A. ar~ not zero fonn a grouR• The product of two su ch tra.nsfonnations ( D)s un- restricted) is a third, with kernels given by (5.1). By Theorem VIII the product of two members of the set with determinants not zero is also a member. This shows the transitive property. It is readily verified t h at such transformations are associative. The identity transformation$ S:, exists, whose kernels are all zero. Its determinant is unity. The unique inverse for Dloexists, and is given explicitly by Theorem III. 26 The Case of Continuous Functions;_ from (1.1) that if I<; and We see readily l<J are continuous functions of 'X, then the tran sfo rmat ion Sk upon (-j~ ") 1) where j" is continuous, gives(~fl)where 1/r: is continuous, If in addition 1<1 is continuous, the adjoint transformation, 'Zs:::: ~s + l<;~x +KJ z!-L• zj ~ ~ + k j ?x + f(} il Fs app lied . to (~.s, ~,) vi.here--1is continuous, will give (::z.., 1 :Y.j) where Z 5 is continuous. Theorem X: X X ' If the kernels l<s, I< t,', !<~ are con- tinuous in both x an_d 5 {a..~ xis~+) and if Df:o then: 1) If b j 110:=- 5t> (fj:71.rmere ;Z" i s_ co.nt inuq_y_s in (C1c4 -J. then ::i" in the solution (j;,jL).= S;' { ~~_., l)is cont.inuous in C'1,"-la. 2) - ----- If (;?,.,',!.,t1.=.S",J¥sliJNhere Zs is continl.!o.us in r~,4). ) then r 5 in the _§,olution(~,?J)=: --,( s,< 'Es, 23it is continuous in (g,i-J, The integrated algebraic determinants used in defining the kernels of <'_, ~k will be continuous, as they are the integrals of polynomial a in the kernels of s,< . By 9 3, the infinite series of these integrated determinants converge uniformly in {~,k). so that their sums, w.hich, -, after di vi ding by the constant -0, are the kernels of SK are continuous in {41 4 1 . Hence /jy.="1~ oJ,s_~•-1i is D 0 continuous. For 2) we see that by interchanging sub-and superscripts, which is a change in notation only, a system of type we have (1.1) with same restrictions as in 1), 27 ""'" O which is seen to so tha.t 2) follows from l), if Di: D:/: D by Theorem XI. follow from 1._±: Di:o for the system (1.1}. 1 then !fl~.2£~1.n..XI: (9.1) ha.s the same bordered Freill~olm the system determinant and the unique solution is giv~n by - oi D')( -0 5 % i DJ( ::7 t ~_.,l ~ ~ .:::.i. z: )( - ~:f }!! i. er o o This may be stated more concisely: "es :::- 'i!-c ~ ~ ~,.... 0 The reciprocal of the adjoint is the adjoint of the reciprocal. By interchanging rows and columns in the determinants which are used to define (), it readily follows -"\. that D= D. Hence if D=to, a unique solution exists . /\... ?--I( .-v Applying 51<to(is,~.1) in (9.2) we have (rj 1 rj)=S!(.J'< r~i";) . ----- "':.1 The kernels of the product transformation ~(' s,< given by are -6 times the left ha.nd sides of (4. even). Hence (9.2) is the solution. § 10 Transformations of the Third Kind: Consider a transformation of the third kind on our composite range, as follows: 1J ~ =- Kl< ~ f +- 1<. f j" s t- /( ;- -j ;: (Io• 1J -1 i ~ ~ 9 5 + j' 7 ii If /< 1-==- I and if I<} is the old I< l ( 1 • 1 ) p 1 u s the K f( (' Kronecker delta l'. Si , this reduces to (1.1), which we shall refer to as a transformation of second kind. If !<)(= o, this is a transformation of first kind, with which we are not concerned. what follows that In fact we assume for ,,x does not vanish in (C111r) • This is a generalization to the composite range 28 of the Fredholm transfonnation of t h ird kind used in an unpublished paper by A. D. Michal and T. s. P e terson.<•) The product of two such transformations, s/.s; is ] also a transformation of third kind SGt , whose kernels a re g i ven by G/ = L 'f t< .X (Io. l 1 ) (t O. l 't) ,,.. X - v1s- L 'f. (( sx + L sx t/S) + L x-t I<st + L f_, I< sI j = L"' I< i + L{ k} + L ~ K f· (I u. l 3) Ct (,b . l'f) e,f;: Li k(si+ L~ (<; + L~ t<J (10.2s--) i. Lt-t (( li + Lt'/..P c, i::: ,P ( i Letting /<Y;; 'f==- Cf~ in (1 0.1 ) we have ~ x-= ~ Y+ -1, _ (10 . 1) l<V,<"s ?j>s + k ij/ ,<i;,<s z-,s + t< j -i l which is a transformati on of second kind from (if~i9 to r1r~/) . If the D)S are the bordere d F r edholm deter/(}t'.1,_(s J /( !) Di= O , ~,, ~ and if t ( I< l Y,ffJ I< f we have t he un i qu e inve rse of (10.3), minant and fir s t minors of Cf 'f = ~ ~- Pl .M 5 - D;- ,.,, i 0 ff D "' r t' Dt' • ' i' ,..i i .,.; Ost ~ f? f ; ' . i ..,,,, 1O i'l = ,-, - ~ ~5 - 75i-:} = - 0 ,:J + i) <J ff Dividing the upper equation by /(~ , we have the unique inverse of (10.1) (I0.4) ~ "f== 11¾~ - D~/1< x?5 - -i D o( - 71 D,-l o,f,c~- 'di ot ' ~ ~ ~ 5 + 0 £; i -7 -; whicb is likewise a transfonnation of t hi rd kind. The relations derived from § 4 for this case are: (1u,>IJ l<(s10; - 0 IC,t: -+ l<~-t I< Jo} + 1<1-s1 t<J D{ (llJ.S- 2) o: k(S) -DI<; + D.:{ t<1 + D; I<{::- 0 -; (IJ Reference on page 40. =o 29 ( ((), 5" 3) D i + (,J (D1· - D~:) f- ff D} == o (to, 5 Lf) - D 1< 11r _ + 0 -tY t< 1t, + D,.,; /, 1-P_ = o (10 , s-!i) Di I< {5) + /< (S) /< t' Di + /( (s) /( t Df =- {) s t<t- t s J ..5 oi KcsJ + co}-DS)J1<: +DJ K1 == o ({O,!i1) I< >'j C 01~ - DSt~j +Ici• D-Ji:_ = - O~ ; 1- { oj - 0 &)) I<:-+ oi k; ::: - D b ;°§ 11 The Pro duct Theorem an d Gr oup P r,ope,r,ty f o r Trans:f'.ormatioE~ of Third Kin d : Th eorem XII: The determinant of t he transformation SG, dete,rpiined by (10.2) is th e pro duct of the de,terminanta of SK. an.._d ~L- • Th at is: (k/Gt<~) c;.,~J (/1./) D [ c;\; ,;/" J 6,; i (11,2.I) cr;ds-J = /< 1/(, CSJ L½_s t { 11 ,2_ 1 J Cft j' (11' L 3) (,.,s lcPJ (1 1.2.lf\r.,, Gi( 1 't LX L¥ {< L 1/,5 ½_{s) + ~ t- ~s) L + L Pr..,. /<f?s!i,<(SJ){lL S !< ;- L >< + L~/2.t- I< j Lt + L ) K : • =: ' l 't i ' = l kc.o + LYt-t- I<3/f<lSJ L1. + L ~ I< () 5 - L~/ / Lt: G/ I ½. ~ 71<<s} /<~1 Lt+Lit<f, I Hence by Theorem VIII, we have ~~/c,U> ~j] - [L~/ L(s) L 0 , t -D • , D [ c..;( c,(~J K:·l [l<!/,l J = D L VL('I L fa •D 1<,i/!(l'J i<:J From (1 0. 2) we have l 5 [L~(LCsJ L ;] L'f/. 1J, [ K1/~(sJ rIL 5 1=.~/ L(S) L~' l l<s/f< Csl )1_s k~. L] , t I< l Referring to § 2, we have, with an obvious extension of notation 30 Cancelling the L "/Ci, multiplying by ±-! and sum.ming and hence (11.1) Theorem XIII: The transformations of third kind (10.1) where l< xdoes not vanish il1..,{:!d·J ..1_and whe~ D I< f,l1<(f) /(1,J I<~/l<f,J t< L does no.t vanish 2 form a group, __ of whj,_9.h_ the group of transformations of sEt£_ond kind (l.Jj~t_th D.f D, is, a sub-group, Firstly, all the transformations of second kind D# D form a group and they belong to this set, so if this set is a group the group of second kind is a sub-group. The transitive property is given above; by Theorem XII the determinant of the product is not zero, and by (10.21) C/40 in (t:t1 +-) . The unique inverse is given above. The identity transformation is a me~ber of the set and the associative property is r e adily verifi~d. /° 31 §;pnmetric Transforma_ttons of. Third JU.nd: In ap- plications, we shall have occasion to use transformations of type (10.1) where the kernels are continuous and the following equations hold: (12i1) K~ =- 1(5\ t<f = !<( • I<(~ I< If (12.1) hold, the transformation will be called r symmetric. The inverse (10.4) is not in symmetric no- tation, and it may be an advantage to have it so. In (10.1) let 'f¼ Vt<x;;~ f~[0-.Jj¥ #x , giving the upper equat i on by ~ )1'<, .- - 1L Cf r< + 1<1/U 1<r1<s- f s+ -j i f( i/U/C/< !< ;- ~ -;' /< }Jv'i<~ 4' s + _ and divide If kY is negative, imag inaries enter. ( /2, ?) The unique solution of (12.2) is l V 'f _ 't' J',(>' - OV1<> 'f'J - Q1• ,._, ·/ .., -- 0 D D D <7 - D1lt<S -rs + D&}-o;- ~ ~ -1 c = where D and its minors refer to , ( 1<;/~ ,<i /vi<r.s> This is a symmetric matrix if (10.1) is symmetric. - Expressing ( 12. 3) in terms of /'J ~ and ~ ){ , and di- viding the upper equation by J /<J< , we have ~"::: ~71<>' _ ---,,,.:._ D~/ifi<,;~'ds _ - t O - DJ/@,:;S + D _;;l = ' D1/ .Ji<lf1'/ -----,y• .Q&~-Di" -"'1~ 0 By interchanging rows and columns in the deteri.n minants used in de~ing the bordered Fredholm minors it is easily seen t h at (12.4) is symmetric if (10.1) is. 32 Since (10.4) and (12.4) are both unique inverses of (10.1) they must be the same. Hence (12.4) is a )( real s olution, even if K is n e gative, in spite of the f act that it is apparently imaginary in this case. We shall show that the determinant of each is the same; A Y. --- V-t h a t .t..i ; , - - - ~ - thi s will follow from the fact for each is the same. I< y[~s) /( ; ·) For the matrix ( we h ave .k !l,<<s J I< j I<"", --- ~ 1< JC f • I<!---_, - /( Y..,_ )('..,_ I< x:w- I<)(........ 1 I< l --"--1<1t, t< l. k )', k}, 1 /( )I,........ I )( l<'K,___ _ I kc.)( I - - - For t h e matrix A..,._' --- r""" l .(~,. ~ 4., 7(1')... ::;::_ 'I(. --I G( t< ~, )r' Vl<Y, /(~' -- I< ~, /,< f,"-I< Y-- I t<~,l ~. I )(..,_ I< 'k I /(t-1<):' ' I< L -!.!. ~f<", '- - /( Y:,.,.,,._ ,( Y-:- v!Z -,t'f- w-- ,:,,,.,.._ 1 ~ ' l l<ic,.,... - - - l[Rr~ I< t. ~ 33 t< ",c,, - - - I< ~ , ,._ ' I ' I( ~-;--.- - ~. t I< " - -- A J', --- ')(,__ Hence ~ xI __ _ "'r- ,.,,,_ and likewise D are the same for both. The notation in this paragraph, while retaining symmetry, has the disadvantage of bringing in an apparent irrationality. 34 § 13 Appendix to Part I: In order to make the work of § 6, ? , 8 rigorous we must :prove the following theorems. All the functionals referred to are func- tionals of n functions each of an arbitrary number The notation of § 6 is retained through- of variables. out. Theorem XIV: The Frlchet differential of the sum of two functionals having Fr~chet different~als is the sum of the Frich et differentials of the funct ional s. More briefly: 'o(F+G,) = SF+ S 01 Requirement 1) for the Frlchet differential is seen to be satisfied immediately. For 2), we have, where 1: max t,Yc} / 4( F +1r> -;,m I != j +i;.0J AF-:,,; F I + IAG-~< .., I which approaehes zero with ~ . Corollary: ~he Frlchet differential ~f the sum / of a finite number of functionals having Frech,et differentials is the sum of the Frichet differentials of the functionals. Theorem XV: The Fre'chet differential of the product of two functiona~s having Fr6chet differentials is the sum of each times the Fr6chet differential of the other. More briefly: (I J, 2) 35 Requirement 1) is easily see n to be satisfied. For 2), we have '6. { /:' e,) -=- t b. G t-- Cr L F- -+AF·~ G, For /~/. f4Gi(. we may write AF= ==~i::+-61 where €: ➔ o '1 ( ~ / ::- / 4{/ + I GI, wi th '} , If we can show t h at I ~ / is bo u nded, then / C),.,.,,F/ is bounded and s ince 01 i s continuous, ftF'f,fAGif approaches zero and 2) is s atisfi e d. LeIIJ}lla: With F and S°F defined as in f 6, f 5.{l is bounde d. From Theorem IV this will follow if we can show that t&;c/is bou nded, where In SF~ l et SFi is ~ Yi: be 6 l~ where Li. of the variables that C&F)'o'r',i==o, l i i is a fixed function Yi depends on, such that max /J.../ == 11 and ~ is an independent variable greater than zero. Then $ Fi [ Yj / ~ Y.] == G S Fi[~ IL], since the differential is linear and homogeneous. ~:::. ~ (t;;'t;·/ "?- 6- ~ l L ( =:G Hen c e / ~~/£/ff Fi[Yj l LJ which is independent of >-J, Corollary: - The Fr{chet differential of the product of a finite number of functionals having Fr6chet differentials ~s thE:_ sum of the Fre'chet. differential of each , times the product of all the ot h ers. 3? Frichet differential of a rat ion al fun ct ion of a finite number of functionals is simply isomorphic to the process of calculating the ordinary differential of a rational function of a finite number of ifl&9i!i!:.. '1.pt. variables. Theorem XVII: The Fr{chet differential of the logarithm of a functional having a.Fr~chet differential, taken at a po int where the functional does not vanish, is the Frichet differential of the functional divided by the functional. More briefly S½F = €F (IJ. 4-) I- Requirement 1) is seen to be satisfied. For 2) we have 1:. ½ F = ~ ( F+~f=)-~ F = ½ ( t+o/°1 Now assume /~Ff ~ (P:(. Since f # o this is legitimate as above. I}~ .6 p-J _ .6 F (-6 FJ'~ (I+ p: - T - '2.F1- t- terms of higher order, Also 4-~= SF+0 '>J where ~ ➔ o with 1. Hence ,6 ~J F = S F- t ~ _ (4 PP- [ l _ Ll f t __ -J ~ ~ f= F1. i '3F The series in brackets converges for /4/ll I , hence is bounded, say less than M . 1A0~~v1~ ltlt/¥1 1LIF./•M As above /~/~/VI+ 10 / and this is bounded. Then since rt 1_and , ;:1 approach zero with ~ the 38 Theorem is proved. Theorem XVIII: If the Fr6chet diff erential of a functional exists an d is zero identically in the arguments and the variations of the arguments, then the fun ctional is a co n stant funct i onal. Let Gateaux' s (r) formula gives us d_ rr Y~, ✓'])~=o = S' F [Yd~ YJ Hence 1 has a derivative with respect to ;:l at ).=-0, Now d_ t Yi + ~ rh] =,,A F [ Yi- +) H'c: r µ ~ Yc J ' = ~ r [ K· + >. f; r, t s Yt] o() 7P d) C,_ [ r 1/). "CQ Hence ~ J ['(.,)] exists for all ,) and is given by $' F [ Yt· +~ ~ re f S r;:J Since the derivative exists, we may apply the Theorem of the Mean of the differential cal culus giving But i [ re: 1,J - J [ Yd OJ = F [ ~- -+- ~ rl] - F r Yl] = 21 F Hence under the hypo thesis J 6-F=D , and r- is a constant functional. Paul Levy: "Analyse Jo net ionnelle!' ( 1922) p. 100. Part II ~uadratic Functional Forms on a Composite Rangef' J (/) Part II includes some unpublished work of Professor A. D. Michal. 40 Introduction: Q,µ adratic functional forms of the L-1-,r~--, (x,s)-:; (x)~ (~J c,0<ds + .[, (~} [;7r~il~ type have been considered. (/J • Us ing the convention of letting a repeated sub- and super-script in the same tenn stand for Riemann integration with respect to that index, we may write this as: r s 1, # Y:5 ~ ~ + 7Y lr,,y x)"2.. In this work we shall consider forms quadratic on the range {~f--,iJ where ~)' is a function continuous in (4_,J-J . and. f; l is a set of n independent variables. We may write such a form as 1 2 ,, U(-:::: ~x-s 1JY:~S -f?-,:{:;Y) + ,.._/.,:t· ~ 1 i. 1 +;Jt'J ~ ;:;1 The indices >:, ~. t- etc. stand for continuous • variabl e s in ('1,4) while l,i J etc. a.re discrete, running from 1 ton as in ~ l, Part I, and are to be summed from 1 ton when repeated once as a subscript and once as a supers cript in the same term. A parenthesis about any index suspends the summation or integration convention with respect to that index. We assume that the i~ are continuous, and that ? / /: o in {<-f, 4) • Hence j,,_ is of one sign in {1-1-J • (U A. D.Michal: "Affineiy Connected Function Space Manifo1:6~ Am. Jour. Math. V. 50(1928) p. 473. T. S.Peterson: "A CJ. ass of Invariant Functionals of Q.uadratic Functional Fonns~ Arn. Jour Math . V. 51(1929) p. 417. A.D.Michal and T.S.Peterson: "The Invariant Theory of Functional Forms under the Group of Linear Transformations of Third Kind~ To be published soon. 41 §2 ~tio!l between Forms of this Type. an_§ Symmetric Transformations of Third Kind • . C.onsider a transformation of third kind as follows: -z'.x = Jex; 1J r -t- ;'Ks 1f1 5 -+ 7 x j '-;1 i 5 • :J~i ~ +/c'/71 in fo,., m c1.I) We may assume without loss of generalityAthat ~xs and :Jc'/ are symmetric. Then transformation (2.1) is a symmetric transformation. In terms of the ~ 15 and ) ~ S we may write ( 1.1) as Hence we see a one-to-one reciprocal correspondence between such transformations of the third kind and quadratic forms of this type. Now consider what happens to Q when the ~~ undergo a bordered Fredholm transformation of third kind with continuous kernels. -;J J< = I<~~ Y + I< . t /J + f< -i ~ -; , 5 /( j 3 .5 + /<;: -j -;J We have (} :::- ~ r: "P-x +- ~ t'z i. :::- l<y:-,j"r-" + z'{ t<;.j 5+ l-.,,. Ki7i+2t' /<~ j 5 + Ez l<) i;J;; where the :i's are given in tenns of the --; 's by the product transformation of ( 2.1) and ( 2. 3). Rearranging, we have (2,LfJ {r ~ j 5 ( /< 5 -zs + l<sy=z:x + t</~i) +-1 -i( I<} 7 x + t<}=z"c) 42 If we now call (ft i;J t he transform of the :zls by the adjoint of ( 2. 3) i. e. (:2 5) I = ~5 -- 'i?j =- /({SJ E5 )( + I< .s -rx + I<} =zl )( I< I 27< + !-, 1t' z·L } We have (~' J,) f{::: ,-:J.- 5;Is + 7-; ~i But this is a form of type (2.2) wh ere the trans- formation of third kind represe nting it, is the product of three transformations of third kind, and may be written symbolically ff ::: I<- CJ • I< ( ~ . 1) the tilda den oting the adj oint transformation. The Tran sformation Induced on the Coefficien}s: In order that Q , wh ich is defined to be }Ks-17i 5 +:JxC3YP-- + 1;:t~i~Y"9L+ ;/L'~ ~t;y;i shall equal q , t h at is, in order that Q. be an absolute quadratic form under (2.3) we must have (], I) 0 =1Y:}JS(fKs- ;l<s) +(jx/2 (7)(-~s ) +2-JJ~i (}~i -fxJ +- jcj1(Jtj- j tj) wh ere the f are t h e coefficients we get by expanding t h e ~>s in ( 2.6) and collecting t erms. Theorem I: X The necessary and sufficient condition , , • 1 (s . 2.) Jxs./';J -"J 5f,,f~(~Y)L+.,,,f~<.1J)("#l ..L,(l·i~ 1'J'if; ~°' ) =,ls) f,' j .:-jt vanish id,ent i cally for all continuous ,;x ard all ~l· , when the J~ are continuous, i s that the -1s all vanish identically. The sufficiency of the condition is obvious. To 43 show the necessity, that is, that if the form vanishes for all c~r~9as stated, the Pl are all zero, we will show that the }~ are all zero if the form vanish for 4 a11 fJr1Y as stated such that ~ :: ~ 1=-o. In this case first set all th e ~t::: o . Then by Lemma 2 (r) of 11'. ichal a n d Peterson's paper ( 1. c. ) , ,/ ><-.s and f~ are identically zero. Hence the fonn becomes (/Yi~x +Rif";J)~C=o Since the ~ i a re indepen dent, we have .1 )( ,· ~ y: .+ Jt'l 7 ·i = o Now set ~ X= O , and by the i ndependence of the -:;, ' the Jt.j are zero, 1 eav i ng only __Ry:t· ~)(' to which we apply the fundamental lemma of the Calculus of Variations. Applying Theorem I to ( 3.1) and writing the ¥ ~ out in ful l , we have: Theorem II: The necessary and sufficient con• dition tha~_the form (11 1) be an absolute fo;rni under (2. 3) is -~pat ~c?efficients transfonn as f ollows: complete statement o-f this is as follows: "Lem.ma 2. A necessary and sufficient condition that (l) A yol-/J ~o( 1 IJ ~ ~ { 1J°') 2 :::: {> }' "r'(l(/.J ;:: 'f;;tX. J be true for all continuous functions ~ t' , 1- A=-~"'- ::-o is that 'tt1==- 0 > ~ = 0 !' for which 44 (3.31) :J-x;:::: f!x:s f<WK{s)+-jyu.l<{)')t<1+Jisl<(S)f<1 +-,;lt:u t<IkstA -f-JJI l<(Yfkf +;Is {<(Sit<~+ 1-4 t<} t<; +:;x.,fl(rx;l<Jf-,js-Rf((sJ t<t (3. 3 l) (J, nJ + <Jt:1 K\ 1<; +-j-tP I<: l<t +J.~-- l<f /(;: 9-K = :Jx ( (( (,:!) 2... i(,xJ 1 t: J ~.: - ir-t t<f.KI 1<1 + /-t1,1. I< x- t<'!_ f- J>< K I< { +;4 I< YI< l -4- l-t .R t<;, /( f -f J-1;, I<} I<: + J-t p tcO<I f< { +-j('~ !(; !< i 4 (s . ~ J j,,j =: #-tv. 1<1 t<} t J-t /<;t<) t J-tJ ((; f<i +J-u l<i kf +J(J-. l<f k'J' It is readily verified that the transform.at ion of third kind represented by the matrlx , x ( 1-,;s ~:><j) ~SL :}L' j) is symmetric, and that its kernels are continuous if I I the j5 and I< 5 are. An Invariant of the Coefficients: The bordered Fredholm determinant of the transformation of third kind (2.3) is the bordered Fredholm determinant of the transformation of of second kind with matrix: /< ~Kx,~sJ ( t<y{K[s) (< ~tfk,..-) (( ~/ Hence applying Theorem XII, Part I twice, we have: (1-t.1) D[J(5/~ ~st/Jj; ~,;l~l = D[k:~s ~ cj Theorem III: J k i//l(~ I<~ •D (J~~Y~5 11//jj ,J) r~l~!(~f/ J 1~tVv7.s J L'<mcz, I<, ~t'j The bordered Fredholm determinant (4,2-J is an invarJ.ap~ of w,eight two unde,L the ;r.ansformat..1.9..!l 40 _, and the same holds true for the j 5 • ?rj] J Cj determinant is an invariant of weight two. If we let 1x-5 :::: J- l< t = O (4.2) is independent ofJx. , then the determinant Let t ing this be zero al so, Q becomes the ordinary algebraic quadratic form. If we let I<~,, I<:= (<},= t</ = o , then ( 2. 3) becomes a linear algebraic transformation on the In this case Theorem III reduces to the corresponding theorem in th'i theory of algebraic forms. If we let J it'::: Ii;-=- O, Ji=- 11 Q. becomes a form of type previously studied '' · f( '{-=: ,, the Letting t<;-=r)) !<} ::-!</=-~ transformation reduces to the ordinary Fredholm type, and Theorems 2. I and 2. II of Michal's paper are special cases of Theorems II and III of this part respectively. If we do not restrict j 1 and I< x t o be unity, but merely to not vanishing in (0i,k) , we have another case considered{2l Here i t is the symmetri c (fJ A.D.:Micha.l: (1.c.) p. 4?8. (2.) A.D.Michal and T. S. Peterson ( 1. c. ). 46 form of the determinant involving square roots that is obtained as the invariant rather than the rational fore (4.2). Special Case of the Pre~ious: If n•l, equations ( 3. 3) b ecome ?, 'J(5 ::: i 1(5 /((x} !((SJ+ J '/<IA. ((~) f( 5lA +J-ts !((SJ((;, -f JI t-CA. /<;; K/ (S', I I) +J 1-- /<Cx) I<; t- J-s l<s;(<; + J-t t<; I<J + ;l x t 1/x/ f<.s' 4-Js, !<cs;/( ~ +Jt I J<t Kj + J--1. 1 K; I<~ 1- J,, /( !, t<l (5', I 2.J <) X ..:: J,_(f<r.xJ)'- ( S-, 13) ?J.., I (/ " = ,1'1 )(-f ,J.AJ I< 1f t /q-' IA. ((-t)( I< ti.., + (/-t7J( /(if) I< Ix +/ q-£ I< "i I( 1: f 1: f 'J 'f I f((x) f( : + J-t I !<; k : f- J-t I f(; k ~ T J l 1 /< ~ I< : :J ,, :: /i:tJ.. I<~ K~ tJ-t ((<~ )'+ 1- :J-1 , K; I<: t- J 11 ( !<:}"Forms Quadratic in a Function a.r1;d its D~rivative: Let ..<..<.l' be a function with a continuous derivative and 1 e t ~ )(' b e ~ faJ)( d'}{' The quadratic fonn with continuous coefficients 5 (6. o t} ~ A?(, ,MJ~ + 2. a"s 11 + c~~ -"i- x-7s +- A)(., NI/""1'2.. + 2 f.?,x µr-Y-1)(+ Cx(-1-,:/-- ~ AKs-=-A n- , c)(5.:::: CSX ./4,l!r/( is not an invariantive form. That is 6l may be zero for all d wi thout having al l the coefficients vanish. For example, i f ( Y::S ==A/< = : B X ~c ~={JI A ' K s : : , , f ' f t ; , 2 B xs :: l 1-1s, where i 5 is a fun ct ion such that -t4.::: ,,/,_--&- ~ 0 , we have ( ) =-- 1 ,j,_~ 'f. ,{,v 'I-, ,v.r + s I X ,I., 5 /l.AJf? 5 =- {-')( .MJ >' ( _/.. ~ MJ- S .f- ,I,._ i A•/ s_) =. {-x _,wl<( /,_,(,.A.Ar(.__,/._,:,, µr q_j ::::: {) We may make Ar.s symmetric here by letting -{-,==~~ . Now Ml'" )(::::- .AH •~+ J.-"IX d. s , and if we write Y f o r 5 '4 and define the step-functions A,,., IA.. , ( 47 • we may write ( , . 2.J .A,,<J-i<-:::- G ~y +-f t --:;z s- substituting (6. 2) in (6.1) we have 0::: A 1<5 ( G,.,: y + E".f -1i:) ( c/ y -4-- c~ -;,v-. J+2 Bxs ( c;./ .,.,+ E:-70~ 5 +- C><s ~Y~s +A ,d&1-r:Y+ E J,ft)(Gt',:Y-t- E: -:; 1+ 2 By{ e,-Y:r + E.{ ~~ >:+ C,d~x J?.. Collecting terms, we have Q = (A-t~ F: +2 f31:s E! +Cys +A-t F: E; + 2 F! (6.J) &)( F; )-:; -,:'ii 5 +C-,: ("'; 'fJ-i.. + (A-ft.A-c;'=_F~ t- A-lc... ~ F"°; +2 &-i.x~++2A--l~tIJ 11 +2 'sx G,."')"'J -y: Y + (A-tu. crt ~ « f A-i ( &--r°""P) Y L Witho u t loss of generality we may take Axs and Cx 5 to be symmetric. If we no t ice that ! + B., E {) 7 .,1J s 1 ( ls-ts E -=- ( (Jt:s E and that ! + il-t ,r Cj + &,r E: t B-5 E()~ YJS 1 i<{ A-t £A c;-t f :; -t- A-t- u. e, ~Et ) ~ 2-A-tlA c;; ({ E ; ~ j we may rewrite ( 6.3) as a form of type (1.1) with n:l and coefficients as follows: J+ (,.41) 1 XS = A -il\ E:~ E (l,, LI 2) Jr= C;x r, If 3) I (6 ,'f 4) B-fs E! t '3t-,x F} +Cxs + A-4: E; f ; t- Bx E' ~!+ 8s f<f t tj '/ I = A i (-( C 4 E + fs-t y Gt -t- -f- A-t ~ i 8 ~ 11 =- A-tu. c/ Ct lA +-11-t CGt ~J 1. t+ {J x- &t V!J These coefficients are continuous if those of (6.1) are; for example: )' 4 i3 t 5 E ~ :: 1 o ' '31 5 uU t- i B-t-s d tr:= ;x (A_ which is continuous in X and in 5 • Similarly all others which have r:;t ntegrated are continuous, with the possible exception of the term A-tF~E1 . This term is equal to f~-t~t where 'i-- is the greater of the pair r 48 (~s ) , and is continuous, since ~ is a continuous function of the two variables x and S • 'rhis accounts for all the terms except B~E~' and Bs E~ , which are both di scant inuous. However their sum is continuous and is equal to B.;. where i is defined above. By us~ng i , and by using the convention of a bracket about an index to denote integration over with respect to that index, we may get rid of the and rewrite (6.4) as follows: (C.s-J) #xs= A+u_ c; E1+-B+;,,:E";+-T3-tsf! +-Cx-5-f-A-tE: + g~ (, ' 5" 2) 7 K ::: Cy ( 0 ,S-3) (~.slfJ §? d'ft ii( = A[-fJ~E ~ + 15[-t] X +At:-£ ; + Gx .:: A[-Ofl,(J + flr1:J Application of Earlier Work to §6; .. By Theorem I, we have: Theorem IV: The necessary and suf~~cient con- di t ion,_that the form L6.1) vani ah for all .Mr.xwi th continuous derivatives is that the foll~wing relations hold among its coefficients: A-il,(E : ( 7, I) EJu + 13-t.><E ; +-&-ts E!+C,:s+A-i E"%, +b'z-::=- O C.,: = O Ar-tJlA g ~ +- Br+J;,c + A-c E; + isx ::: o Ac+:i[lAJ + 11 [ -{J =- o It might be interesting to show that the coefficients of the example given above satisfy (7.1). Substituting J~,.,{~ for A><s , ½t~ t 5 for the other coefficients, we have: 8'¥:s , and zero for 49 J I r-, 1 I /f\.-£ r X .trA I £A I ,-...1 I -l /I B s +- 2 "-+ t: y-1.5 +2 "'-t £5 -1.y ~ O 0 ::::- 0 -t~-fJ /4: E ~ * Now + { /4 ~-tJ *J< == o /Ci:] -i_ /[u] =- 0 J =;:_ -; ~ c?L t = ~ (4) - -t.. ( ~) =- ~ -;I( [-f) i: + -A1r - --1... ~ 0 --1_ <t E 1 ==- .-1... e,{t .:::- y 4 The first of these equations becomes [ -f;,c] [- i s] + i [--i_,-:]4-5 + t [-~ s] 4._ 7 = D which is true, and the others reduce to zero, so that ( 7. 1 ) is s atisfied. Now if we apply a transformation ~ ><= !<;;~ ~ + !<{ ,35 + K,Y y I< 5I ;::; ,, S + {( , I y y =we se e that the to (5.1). ~.5 defined by ( 6. 5) transform accordin g Sin ce t h e set (6.5) is not in general uniqu ely solvable for the original coefficients, t h is does not give us a definition for A-:s , etc., so t h at we cannot consider a law of transfo rm ation for the original coeff i cient, without further restrictions . If C>-io in(Cf,4}we may write the invariant (4.2) in terms of the original coefficients, In terms of these it is A-l « E'; E j +~is E; + &,u E; + Cx:s +-At_ f ! +Cz- Cs i::-u i AL+JuC:5 i-G[i:]S +fl-! f's+(] )< C .s 50 §8 Special i~~t}.on of Transformation (7, 2): We assumed that the transformation (7.2) was general. In various particular cases the k'.s may be related. It may turn out naturally that the 1;~ and Y transfonn independently, that is In this case the determinant of our transformation 11 D trK ' : : ] :: I<: D [ l<Y/<1'] becomes If in particular we consider functions;W~that all have the same value at tJ.. 1 /(: ::: f • and ouJ' determinant is D [ t<:lt<(s)] Perhaps the most interesting specialization of (7.2) is when it is derived from an ordinary Fredholm transformation of second kind on .{Ar>:,' (<t. 2.J ,,A,<.,J< If == ,,,(Ar Y + K: .zv-s D[k;] -:f: o then to each,,w><, corresponds a unique ,A.Al",., so that to each {"5fY)corresponda a unique (1 ~ Y), and we expect the determinant of ( 7. 2) derived from (8.2) to be different from zero, when is. O[k:J We shall prove the following theorem: Theorem V: The bordered Fredholm determinant of the transformation (7.2) detenni:u,ed b_y ls._gj, where the kernel has a continuous derivative in ~,~)with resRe~t to~, equals t~e ordinary Fredholm detennin1¥1t of (8.2). We must first show that (8.2) does determine a 51 transformation of type (?.2). giving: To do this, set Y= Y + I<; Mrs = Y t- f< 5~ ( )(= ~, c,5 Y + E f j j Differentiate (8. 2) with respect to X . If we denote by a prime before a function of a superscript and a sub script, partial differentiation with respect to the superscrint, we have: ,,.c;, ')( _ -~ \ k~ -.s (I ~ + J /1.N' = 1 ,.:+ , I<; c;:;5 f + ' (< I E J-J-! Collecting these, we have the transformation of type (?.2 ) determined by (8.2) as follows: ~ )( = 3 7 + ['t<{ t/J 'r' = + [ '((&]] y ft<! & JJ ~ s+ [ Ii t ( c~D P ~5 'rhe :ord[~:e/:r~:~ dJetenninant of this is !< f Ei ; I t- I, f.t J Referring to the definition of D , we have with notation of Part I ~ 2: Ll. _., __ , x~ - s,-, -s,,..__ - I 'I<(t:J I(_, I ,e.. l J SI' r< vH 1 - - 1".,,,.,, -ti C: -t 1 52 We may split this into two determinants as follows: and integrating, we have formula we have · A S1 - , -S~ 'Ps.I - - - /h-\ s ::: , !<(51 _ _ _ ,·,<s. ti. , i I l r,..,... '{<S, ,~ ) '!<~-- - '(,~:. 'f<~~ l< i,~-- I<~""' /<i -- J~ ( &(_ ~ i, 1<-t - l<-tI t t<i ' - (< "t:,,,,_,._ ~ t -1,, I t \ ·t-...,,,,.__ q /< t:- - Kt', - t a. - I<t:,,,,_ - I<-t t /1,v- + 53 i ,,__ r<i1:, I I I 1 (t, _ _ _ I< i, i I 'I I I t<-t-..._ l<-t._ c,,,,._,_ t I< i-. l<t The first determinant may be split up into 2.. ~ detenninants, of which all but m + 1 have two proportional columns. Omitting these, which vanish, and rearranging the variables of A s, - -- s,___ . _ 1.,- s, ---5' ,__ - J o{ t z c:( + /2-v-J--»l-1) f (-1_ 4-d i i ~ t< ~ I< t"'½--f ll kt,. t;,,,,,_ _ f I 'f I<tA1-_, t-.,,,..._, 54 A;,=~~~~ refer to the kernel I</. where Multiplying by ~! and summing from m:l to o0 , C, .Y. -¥ [) =: 0 + [ ~ ~ DNJ - D (< r!J -t- I< i D~1 4(4 ] Taking the first of Fredholm's relations between D and its first minor, setting X=="'- , and integrating with respect to 5 on (o,, -1,..) we have :J(- ~ --¥ D~] - D t< i ] + I<1 0 [t] (~I 5) 0 Hence D= '5 and our theorem is proved. I( It may be useful to note that the kernel - Ds of D the reciprocal . transformation to (8.2) has a continuous partial derivative with respect to X under the as) /< X- sumption that ~ / =- 'l<sKis continuous in {o,, -<,-) • 't< s,c 'I<;, - - - ' l< ~ ........ K x-, I< x-, I<1 ~ , s ,.., I ,C-....._ which is continuous under the assumptions. Since 'f<; is continuous in ( C! 1 -tr) i t is bounded. Now if M be the greater of the bounds of / /({/and I 'I<{/ , we have by Hadamard's lemma the dominating series 2. (,,,.,__+, ) ¥ /'1 %-f- f(-e.-c<.) ,._ ~==o ,;___, ·r - /'V1-\ I X" d D5 0 J< 0 = z_ ...l..... ~, ~ ..6.)(s "'·x, ---1'-........ __ , £ • ,_ =- () , r )(--w-. As in Part I § 3, this converges uniformly in (t.1,-<.) for all M , so that &-(-.!g) in (ci,k)• exists and is continuous Part 'III Projective Transformations in Function Space. 56 Intro duct ion: . . In t h e paper of Dines ( 1. c. ) , h e considers the function-space analogue of the general projective transformation in n- space. Xl ::. a~- + -Ir t- xi + ~:p- ct' j -Yi -1-, d.. t-?~: ~ i 'i-j- If we consider the Fredholm transformation as the fun ction-space analogue of the transformation ¥( == Xc: + I --e,{' Y/ 6' =t we have that the above analogue is (I, I) - ·{{} fx)-::: rx f x) + 4> ( 'IC) cp fr) + ['Yr~, f) Cf rsJdS 0-+- £'Gcs) (fr,JdS In this paper we shall reduce the consideration of this transformation to that of a bordered Fredholm transformati on of the type considered above with m•l. This reduction is suggested by Hildebrandt (1. c. ). We shall get Dines' results somewhat simpler, and shall show that the one-parameter family of tran sformations generated by the infin i tesimal transformation is a group in the Lie sense. Reduction of Dines' Form to Tne Studied Earlier. Let us write ( 1.1) in a form more convenient for our use, and introduce t h e convention, used earlier, of letting a repeated sub- and super-script indicate Riemann integration on ~,4) . Then (1.1) becomes 'f (<)(qr+ I<: cf's + I< i ('J.., 1) (.f -=:I<~ Zf5-+ c</ The retention of the / ~ as indices will later emphasize the connect ion with bordered Fredholm transformat ions of third kind. Le t r1' 'f. JVtY -d - Y_ 'f'::~,c.f- ~y ~ , where the pairs ( 1;;~) (-j{ ~) 57 are homogeneous. Sub st i tut ing in ( 2.1), J~impl ifying and equating numerators and denominators, we have: AJ r; ==- /( r--9 y + 1<: ~ s +- Ki~ ~ ::: This is a bordered Fredholm transformation of third kind, which has been already studied and is entirely equivalent to (2.1). Referring to Part I § 10 we have the definition of the determinant of this transformation as I< f/1(CSJ D [ I< }/((CSJ (1, 3) I(? {( 1 ~-J As before, a parenthesis about any inde ;x suspends the integration convention with respect to that index. Expression (2.3) is the same as the B defined by Dines to be the determinant of the transformation. The following theorems given by Dines follow from Part I § 11. Theorem I: The deter~in~nt o ~ product of two proiective transformations is equal to the product of their determiJ_1ants, Theorem II: If the determin~.!11. of the transfor- mation Jbl} is different from. zer.Q...s. th~n.JJ..J..) h~!? 1 ~igue solu~ion for p in terms.of '! , namely (f' y =-- c(. 1( ~J + ;.3'{ -,:J cp1,r) + J: 1y 1( 11 r) ({) 1cs)d5 ~ + i f h; q {5JcA5 where cx;/3~ y: ~j f' are given in terms of tX,/1, r:; ~ €1 We shall restate the latter as follows: Theorem II! If the determin..§.p.t P!2t3J . .i.s not - :x zero th~r1-.Jl, 1) has a_uniqy~ 139lution ;fo~ :f io :terms. 58 first minors of (2.3). Dines gives two corollaries to Theorem II, the first, that the product of the determinants of a transformation and of its inverse is unity, following from the fact that the determinant of the identity transformation is unity. In our notation, the second becomes: I Corollar_y 2 : [ Di - 01 cps], [ tc,' +r<; Cf s] -= (2.f) For D'- D~ q>~ = DI- D D~ l<Scf + D~ I<~ ztS-4-D~ 1<1 1</ + t<l Q5 so that [t</ -f l(l ZfSJ[ o'- I)~ <fj = 1 D't<:-+ D t<l {f 5- D~t< 5 "Zf 5 - D~ f<_ju q 5 -D~ I<;:: 1 D 1<: -D t-if<(-Cf5 [D~l< 5 -D t<;-t D~f(t] 1 1 Referring to (2.6t) and (2.66) below, we get (2.5). The eight relations that Dines gives between Band its first minors, are in our notation: c1 ., ,> k{,$1 (1. . t.2) o; - o 1<; + ,<~-+ 1< r o1 +- (( Kr o} ~ a (SJ f))( ,~sJ - D l<r -+ D! K; + t); /(s ·=-o 5 5 t :- ('2. , i, 4) D; + K~ D + 1<1/f<-t D 0 - Df< 1I + D+y I<~I + D( I<: = 0 (1 ,' r;J - D K; + Ki ~ l<~ D; (1 , b " ) D~ t<(5J - D't<J +D~ t<] ~ o (-J., ~ l) - I<: D1 + 1< D = - 1) (1 , h~) - D' I< : -+ Di I<~ ::. - D (1, '13) 1 + /( <sJ k: D~ =- O 7 These are ~erived from (10.5) Part I by letting n:1. Dines proves, without stating it as a t h eorem that t h e set of tran sformations (1.1) whose determinants are not zero form a group. . This follows readily from Part I. The Infin itesill!al Projective T~ansformation: For the identity transfonnati on, /< 1 =- I< : ~ f I t< : =I< f -: : . I<~ -:::. o Hence for an infinitesimal transformation we can give each of these a small increment. Following Dines we take I</:: I for the infinitesimal transformation, which on account of the f ractional character is not less general. Hence our infinitesimal transformation will have kernels as follows: r<l<.::: L~St ,t_ ( 3' I) I<,= I + L )< b t I t< 1 ::: L ~ ~ , , , I<'s =L'so; and our infinitesimal transformation is t) (s. i.J I+ Ll cp 5 St- expanding the denominator and dropping powers of ~ thigher than the first, we have: Cf~= Cf Y + Si [ L XCfJ< + l ; Cfs-J- L ~- C(J Y L l 2'5_] A family of transformations ge nerated by continuous application of (3.3), 0.'-f) Cf'J{i)::: f(x(+)'fx + /C[ {-f) cpSt- (<; {{- J I<~ (--t) q>.S + I</ f-{;) -)( will then transform q,, into c('ftJWhere Cf (:tJ satisfies the integro-differential eQuation, (s.5) 0 t ~-f.,! =- Ly cf><l+J+Li cjJ (i:} +- L ;ft) -Cf7-tJ L1 if(~) with the initial condition 5 cp't(o) = cpx The term "regular infinitesimal projective trans"'-' formation" has been given to (3. 3) by Kowalen5ki. 60 The Finite Transformations Generated by an Infinitesim_§i,l Tr,ansf,o_rmation (3.3) . . The question now to be solved is; given the L~ (3.3), can we find a set of /()S depending on t- , such that Cf 7+) defined by ( 3. 4) will satisfy (3.5) a,nd the initial condition Z()'1to)-::: Cf}~ If we follow Dines and differentiate (3.4) with respect to )< -~ -t , express Cf in terms of <f(i-/ by ( 2. 4), eliminate the denominator by (2.5) and equate corresponding terms of the result to those of (3.5) we have four 9 t<' C-tJ\ - o equations. Adding a fifth to show that ¾ ~~: , we have five integro-differential equations in the five I<~ I Solving these for the derivatives of the t<s , we get: (4,tl) '/(; ~ L 'ff<}+ L; f<(sJ +L)(I,( !<~ +L~I<; 14, I 2.J \.• 'k (y ~- L Y I( "I -'1 L ll:."' f( l.fI + L ><I I< I1 {L.1-,11) 'I<~::- L! l<(S/-4-- L'v. I<~ (4. '" J ' I< ,1 :: L ~ K ~ (4.•~J 't<JC ='--,: Kx where the primes, now and henceforth, indicate differentiation with respect to -t • Barnett(') derives these equations much easier by using homogeneous co6rdinates as in (2.2). In our notation this is as follows. l( - Writing CfY::: ~/ct- J. CfY(t ) (4-,l.) = jy.~-~(t), (3.4) becomes /< r CfJ ,;;x+ I<: ~ 5 + I<,"~ {(; (-t) ,s + 1<: (t:) ~ (l) I. A. Barnett: "The T; ansforma.tions Generated by an Infinitesimal Projective Transformation in Function Space." Bull • .~. Math. Soc. XXXVI (1930) p 273. 61 and (3.5) may be written f x 3 :i<r+> ~ 1+) fLf JJS(-t-}j u) t-L rrJj<tili._ ~ ~t) Ll 1t~J ~ (+J '~ "{-l) - ,;;. ''/-1:J '.AJ (-f) ~ (tJ] ~ {t-J] 1 "Z_ ~ (+) Cancelling both denominators, letting = o we have, cancelling .JJY(-t} , ~ (+) = L ~ _;;Yi) (If. 31.J adding-1'"c+J times (4.32) to the previous we have, cancelling ;,J (+) 5 (4:31) ~~{-t) =: L 11'~K(t) +- L 1-7J N) + L~ 3 (-t) From (4. 2) and (4. 3) we can get ½"XjfJ, 'q(t-J in terms of(~rJ)• Differentiating (4.2) and replacing the left hand sides by this expression, we have (4-. 4 !) L X K x(l) 7" + L1 1< t (-£-),:/; +L" I< ( f-t) -{j +L} /,s(tJ-1 5 + L: t<s"1t+i,;y s -+ L l l<f(-tJ-1- + L ~ I<~ (-fJ ~s- + L-; !<,' (-t-J ,:;- = 'l(,:(f)1JX+ 't<I l+J15 + 'l<;(-i)--:J-(lf, 4-2) Li 1<5(t-J "d5+ L ~ (< 511 (iJ ~, + l ~ t<," {+J ~ - '1<1 (-f:J,:J 5 + 'I<,' {i) -:J Equat ing coefficient s(IJ of 1J ~ ~sand ~ in ( 4. 4) gives us (4.1). The initial conditions are t<K(o)-== !1 t<.{(o)=OJ /(((()) =OJ /()(o}= OJ , !<,'to)=I, . Now let us assume that the l<sare expressible as power series in+ as follows: (4-.S) 0( oto -t ,,,___ I( 13 (+) =~a ~.' t<:[-.J ) where c/1;3 may be any set of indices for which the /< s are defined. Substituting in (4.1) and equating powers of~, we O)That this is legitimate is proved in the unpublished paper by Michal and Peterson already referred to. have the following recurrence formulas for the!<%[~ (4.6,) (~.h-zJ (4, b3) (Lf.~V (lf-, 65) k;[~+D:: L,<l<{[--J +LJ !<(5 1{-J-+ Lt /( 5u[--J + L{ t(; [--] 4 I<; [----+tJ ~ L 1 l<t[""'J + L~ t<, [""']-t L~ !<, [-,,,,..J 1 J<j [~ti] ::- L! /<fS)C--J + L ~ /< ~u [--] t<!C~+O :::L~l<~D-mJ ,<x [,.,,._-ti] =Ly:/( Xe~] with the initial conditions /( "'f o] =: K! [a]=- I J I<:[~]= !<,Y[o] =- l(j [o'J ~ O, It readily follows that the !<~are defined uniquely ) as functions of t, and that these /(5 formally satisfy the original condition,' namely that ifiJdefined by (3.4) satisfy (3.5) and the initial conditions cffo>=cf~ It remains to show that the infinite series (4.5) converge. Let L j Lt L~J Li all be less in absolute value than MJ then we will show by mathematical induction By (4.6) this j_s true for ~==-D, that/t<;c~/<(L/-h)"""(-t-o.)/W'-. Then if it be true fori1t1~At\ by (4.6) /k~['»-+Of<' LfM U-a.Jll<Ir-.J/ C::::(LfMI-U-aJ-♦+-' Hence series (4. 5) are each dominated by 2_ (4kf1u-"(.,...which converges for all H and -t-. 41(-::::o This .,i,..,, means that the formal solution (4.5) is an actual solution. It follows from either (4.14) or (4.65) and the I/~ initial conditions that" {+)=e L )(t- :/: 0 in (0.1 4. Further- more the determinant of the infinitesimal transformation (3.2) is D{H) ~ I-+ y,1L1L1:5'- t r.t J_ '{~t( o½< I ~t + ~ ol A-ML4/t ~ ; , , . . . . ~t: - I - ;r ::: I + L { ~ t + ~ ~~ ~ ~t Now D{-t+bt)=Dl-t), D (6-t) by the product theorem. rrt Hence D{t+Si)-D{tJ = _;_._ Dl+)[t+L~St+ ---] - D(f} = D (i) L~ St I 63 Dividing by 5t and passing to the limit as 'f;'t-,o we have dDli) d_ t Hence, since DC-t-J::;:. D(t:J l~ I D(o)==- elrtl o Hence we have, with Dines, Theorem III: The finite transformations generated by a r egular infinitesimal projective transformat,ion b Cf' J< = [L Xcp,X+ L: er+ Lr - Cf><L j cps] f;t- consutute a one-par8:W.,eter family of non-si neylar pro.j ective transformations. For an;y: value of the parameter t , the kem,_els are given •by (4.5), (4.6) and (4.?), and the determinant by (4.8). The Family Generated Fonns a Lie Group: We shall next need a set of formulas concerning the /d which we proceed to prove by mathematical indu ction. They are: (9,11) !<}' [ p J =- /<'t--J !<; [p-,..,,..] + l<fT~l/~5Jff-'-,,,,..;J +f<if-Jl<J [p-.-J+t<r[-..-J t<;Ip-~J ~-;12) (~, l't) K; [t,] ::- /( X[,.,.,._] /(~ [f----]-t- K;[,,,.,.._] {(,lf[f-~ t I<?'[,,,,.,_](<,' [ 1'-~ /<; [to] = I<~r~J l<'11£f ---J + /<~ [,,,,,_) !<~ [ f -"""] + t<: f,i,,-_J I<~f £P----J /<: [ P] =- /( ~ [""'] I<~ [ p---J + l<,1[--] f<, f p-,.,..J (r. (S) /(K[ p] (;-,13) 1 = l( l'[""'-] l<"[-p-~ For -f:::::: o 1 /V"'- ==- o , these become o-o o-o 0 -o 1::.1 l=-1 - ' J - 1 J ) -respectively, For P=', , ., _::- o -r=', ,,,,,,._=( 1<: [ I] = f< ; c0 1<{ [t] =1<;[ 1] 1< ; [ iJ = I< ; [ ,J I<~ [ ,] =I<~ [ t] J< ~ [ 1J ~ I</'" [ 0 1<l [ t] ; I<~ [ ,] t< : [ ,] ~ l< ,' [ 1] I< I, [ 1] ::- I< 11 L!] f<x [ 0 ~ I< x- CtJ 1<''[ 0 == (< y 1] 64 provided we remember the initial conditions (4.7). Now assume that the recurrence formulas are true for all values of -p , less than p. Then k ~[~ t<: fr----J + l<s'°'"f--.,,,,J l<lsJ[f?-"""J-+ t<~ r"""J /<' 5u[,p -,,,.J+t<(f,.,,,..Jt<; f/R-~J == Lx t<l<r,,.,,,-o k{ [f7-"-) + L): t<J[,,.,,._-D r51rr-"""J +L; J<sJc,.,,,,..-iJ l<csJrfP-~ + L: !<~ c'Yh-o 1<(stp-"""J +L; t<1 r~-D t<c Jcr-,,-.,.J + ttK:r---o,<J cp-~ 5 t< 4c~-o /,; tt>-,,._J + L~ t<~ t~-ut<:[p-,.,,..J +Lr l<fv.-l,,,,.._-Vl<;[f,,P- - J + L 'Ii l<f [ Nr- - t] k; [ ,p-,,,,.,,._J + L: 1<1 [~ -iJ l<i [,p-~ + L ~I<: r---u /<; f 1'7- ,,,,.,,_J -+L: ~L''{ /(~,.,,,._-i] f<f[p-~1 t-1<{[---D ~Ip-~) t- 1<tc---D l(sl{ [f--]+f<~[,.,...-!]t<;[p-,,,;JJ -+ L} [ t<<sr~- iJ // 'C1LM1J} + L ~ {I<; f ~- D 1< 1[ P _,,,,.,,J +t<""l,,,,,,.. -i] f(1 Cf-~7 -r 1<tr,,,._-!Jf</°£{/--] 15 5 f 1< 1c -w- - 1J /, l cr-"""J} t-L;f 1<~ f O1/ J[ r-~ + /<~ r,,,,., -u t<J'ft-""J +1<:c--0 ,~ rr~ ::: L ~ 1<; [1-D + Lj 1<< 1[p-j] + L: I<~ ((P-t] +Lf 1<; [f- -iJ-=:: /<./[p_J -½ flu,o 5 M- - 5 1 Similarly t<t'CN>\J /<; cr----'J + t<! r~ 1<~ C-14 -,v,..) + t<~ r---J r</ Ct-,,..,,.__J = L 1 t< I([""" -O I< ;fr-"""J + L., I< t [""' -u J< ~ [f-""'J + L~ l<'-l c--,,.,..-_D /(,; [ ~-,,.,,:;J + L ~ 1<;c ~ ,I] l<'(Cp-,_,,,_J + L: I< 'v- [~-D /<~ f p---.J +['><I<~[~ -G t<.' [p-""'J -+ 1-. ~ /<~ [~ -[] l<.1 [ y-,,,,__J + Lr!</ r~-iJ I(! [/e1_,,,,__] ::: L ~ { 1< x-C 0 I< .X [ r' -~'J -+ I< t C~ - D I< ,11 r r4- ~J t I< f [ ~ - DI<: CP- """'J] -t L~f'<~f"""-D I<; Cti-,w,.J-4- t<~ [,.,...-D t<;" [p-~:J +I<~ c-.-!J I<,' ft1---JJ '1M - -+Lt( l<~[-w--D l<;C-y-----J + l<,1 C~-i) k: Cf'-""Y'J] =- L~ I< 7r10- o+ L~ t< ,LI [ f/1- D -+ L ~ I<: cp- o = 1< r c1,J ½ ( 4-, "2.J Also I<] [~J Kl-S)f p--J + I<~ [--.J 1<Jrr---,,.J + I<; C"-"'J t<] [p-,,.,,,_J == L; /.(lS[ ~ -,] !< (St jP-~J + L ~ 1< ~[,n_ -t] 1<1 [p-~ + L ~ l<.:;IA [ 4 L ~ I< ~[~-D k[[10-"""'J + L~I<~~ -D 1<~ [p----J ::: ( . #- -[] 1<(5)[ f-~J w L ~ ( I< st* -a l<rsJ f p-~J} + L ~ [ I< t-1c,,,.. -D /<; [;----.J + f< .str r--o K Cf?-.--v--J 151 il<~[~-D l<;"[fP-,,,,_,_J t- I<~ t~-,) l<iCta----]= L~f< Cp-i]-tL~l<s1,1Lp-i] ::: /<l [p] ½ ( lf. bJ) 65 Also 1<: f--aj 1<: f r - ~ +l<~ r~J l<i1(r-/Vh] = L~ K~ [,v,,--,J l<{fp-,,,.,.) -t- L~ l<uf-»--1J !<~ f,-,-"""J+ L~ t<~[-,t] l<,"fto-,,,..;J :- L~ {I<~[, , , , - Q I<,' [ r-"""'J -t I< ~c ::- LL._ l<~[p-i] == 'Vh - I] l<; Cp-"""J + I<'!,. [~ -t] t,~Cp-,.,,...JJ !<,' [pJ 17(4. bit) Hence the formulas (5.1) are true. Let us write these as where the Greek letters indicate such and only such indices as appear explicitly in (5.1). The product of two transformations of type (3.4) with parameters ~ and t:i... is a projective transformation with kernels as follows: (5.31) P;u,,tiJ-= t< 1 c-t,J 1<1 (tlJ + 1<{1t,;t<cstt,.J -t-k~et-,)l<ir1:iJ +t<,Yr1:,J 1<.5't~2J (s.J2J P; lt,1-t 2) K"{t,J !</fi.J + l<~lt,) t,;{ti) +t<t'ctrJ t<.'(-t:1-) (5. 3J) p~ {t,) ti.) 1<1 (&,) !<(51U2) + I<~ (t-,J 1<~ {t) + 1<,' (-t,) t<; (tt) (5".3't) P/(i,1tL) 6~3 t) Px(-t,/tz.} - /< ~( +,J I< ~lt 2 J .:::. e (t, tiz.) L 1< - /<~(t,)!<Y(t)+t<!Lt-,)l<)(ti.) These formulas (5.3) may be written {}, 4J p~ ( t-,1 t i J =- K ~ N,) /(} ltz.} where it is important to notice that the Greek indices have exactly the same range as in (5.2). 66 11 = Pi ( t-'/ ti.) :::: P~ if ~ c,__ +, r "H1t1 0 --- (ci I< y [/W'J kJ ff'~~ ) Making use of the fact mentioned above that t~e range of the Greek indices in (5.4) is that in (5.2), we may apply the latter, giving .,o £_ IX c6 p« {t11 -+-J -=- .Z I<~ ~1--1 z._ cp,v,,, --t,~i/4-,.__:::. L tf,ttJP Kjc,J /3 /7! Hence the product transformation is a member of fJ, fl:::t> "'11:::o the family, with parameter ,t=o t-, ~ti_, and we may add to Theorem III of Dines, the following: Theorem III 1 : !he one-parametE}.l_ family; of TheoreILllI_!.§.__a one-parameter continuous group in the_ sense of Lie, and is in canonical form, The last phrase means that the parameter for the id.entity transformation is zero, and that that of the product is the sum of those of the factor of the product. Conversely, if we have a one-parameter continuous group of projective transformations, it may be reduced to canonical form. ( Jr, ') lb (f}.,..(f) ~ If it be so reduced to f<"/-t)cfx+ J(~(f) +-{</<(-f)<f 5 1<;£-tJ t- l<;t-tJ cps where t i s the canonical parameter, then replace t by 'bt and expand the l<>s, neglecting powers of bf-higher than the first. {f ~(6-f)=: Since c'p?6) = Cf~ , we have cp, + K/,«J C{lx~--t t- 1<; c,J s + + 1<: r. I_] q:r5rt o; l-+K/r,JSt-4-t<~UJCf'.s~t- , where l<,e, [ ,] is the coefficient of ~-t in the above expansion. Expanding the denominator and neglecting higher powers of St-: Cf yf t-=- cpx-t- St [c (((,] -1<:c,J) Cf)(+ 1<~ [,]_-ti<~ (,J <p 5 - Cf')(KiW <P~ which is an infinitesimal transformation of type (3.3). This transformation generates a one-para.meter continuous group which may be (5.6). Hence, it seems likely that any one-para.meter continuous group of projective transfonnations in canonical form is one generated by continuous application of an infinitesimal projective transf o nn at ion o f type ( 3. 3 ) •