Transformation of Linear Spaces and Linear Operators by Inverse Reversion Citation Elconin, Victor (1952) Transformation of Linear Spaces and Linear Operators by Inverse Reversion. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/RV02-JW45. https://resolver.caltech.edu/CaltechTHESIS:10112017-134914663 Abstract This thesis develops a new method for transforming and extending the classes of operators and operands which appear in certain linear operations in such a way that restrictions on the ranges and domains of the operands and on the algebraic manipulation of the operators are reduced and removed. In particular, the method leads to a complete rationalization of the P operators and impulse 'functions' employed by Heaviside, Dirac and others in the analysis of certain linear systems. In this method, the operators A of a primary class K are, in effect, first reversed, forming A * , then inverted, forming A *-1 , the inverse reverse of A, and these operators are utilized to effect the remaining transformations and class extensions. The method is therefore epitomized by the phrase inverse reversion. 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): Michal, Aristotle D. Thesis Committee: Unknown, Unknown Defense Date: 1 January 1952 Record Number: CaltechTHESIS:10112017-134914663 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:10112017-134914663 DOI: 10.7907/RV02-JW45 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 10505 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 11 Oct 2017 21:36 Last Modified: 10 May 2023 23:48 Thesis Files Preview PDF - Final Version See Usage Policy. 19MB Repository Staff Only: item control page

TRANSFORMATION OF LINEAR SP ACES AND LiliJEAR OPERATORS BY INIJERSE RBVERS ION Thesis by Victor Elconin In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena 1 California 1 9 5 2 Acknowledgment The author gra:t.efully acknowledges the inspiration, assistance and good counsel he has received from Professor A. Do Michal, under whose guidance this thesis was prepared. Sunun~ This thesis develops a new method for transforming and extending the classes of operators and operands which appear in certain linear operations in such a way that restrictions on the ranges and domains of the operands and on the algebraic manipulation of the operators are reduced or removedo In particular, the method leads to a complete rationalization of the ~ operators and irrrpulse 1 functions' employed by Heaviside , Dirac and others in the analysis of certain linear systems. In this method, the operators are, in effect, first reversed, forming ing A A of a primary class k Jl A ~ then inverted, form- :it-/ , the inverse reverse of A , and these operators are utilized to effect the remaining transformations and class ex.. tensionso The method is therefore epitomized by the phrase inverse reversione Contents Section Title ~ Introduction 1 1 General. Definitions 5 2 Derivators g 3 Integrators 12 4 Reversion of Derivators 16 5 Commutation 21 6 Derigrators 26 7 The Method of Integrator Transforms 31 8 The Method of Inverse Reversion 34 9 Sectionally Continuous Matrices Shift and Jump Operators 41 10 Strong Limits of Matrix Functions 46 11 Weak Limits of Matrix Functions 49 12 Appendix 52 Bibliography 56 l Introduction Let F be an abelian additive group and L the class of additive operators A on subsets of F to ,F • A ·is com2lete, ~~trictfill, univocal according as (1) the range of A is F , (2) the domain of A is f , (.3) Af::: o implies /-::o for / in the domain of A ; otherwise A is respectively incomplete, restricted, equivocal. !/ is ~ect on ;: if (1), (2), (.3) hold; then A has a unique inverse / A~r EL (t'.e. A~! =AA-(= I lo>- SuQJ;lOS§?. / E F:) KcL is q_qommutative ring con:taining thELunit element I of L and §.Bgh tha~ _every non-:~~ A~ I< ~tight)_~ verse A'~ L (i$e. AA Jff : :- f foy f E" ;:= · • Then A is com... plete, but possibly restricted or equivocal, and Aif is unrestricted and univocal, but possibly incomplete.) * all A for 0 I A- f"K , and Let ,(*be t he class of Ag :: KI: ~ , the class of all products AB~ for A1 i3 € ;t .. The .£Qm!11Utator of A/3 is (/-11 13)-= (/-llS)(_/.5'-VA) ~L it vanishes if /J ~ Then AA= D and ~t exists. In particular, let """' / A JfLJ ) A ::- l/-1 ,,A ~ I - / 1 17 ' ~ AA : I • Suppose further that (A 13 /f = ;f~ /3 jf kiJ is closed under addition and subtraction: *' An~ fA 1. fJ-z. R *~(A I /11 f.>1 - t~ ) t3i. ±A1..J3,)(f3,f5'-) 1f A,f3,J/-}1..13, €,{q d }\< and that '13 inclu.des. a,_Iing /(/ §~h,_j;hat (1) foL.,anX A Cl( A.13 <{J for some /$ ~ /( , (2) even_nQD":"Ze:ro element of /(/ ~~ . *' * and univocal, and (.3) (A, 13, ) (Al.. 13.,_ ) ::- A, ,4t.. il A, /.31 : A-d3, ~ ~ (/ (13, ~l. )°* (then I({ is commutative and contains k~ o) 2 Our main result is that under the foregoing suppositions F may be extended to an essentially unique abelian additive group F -::: ~ and A:/ may be extended to a unique field /(J = llJ/!rJ within the cla~s l of additive operators on F to F .. The classes T and !rJ are the ranges of A respectively, with AEK/ verse reverse of A ~ • f fF; X ck* Jf -( 'f and A 7f -I X , where ,Af-r is the in- Hence the process which generates F and is called inverse. rgversion (see Section $). Each non-zero element of Ki is perfect on F • In part-1 ticular A is perfect on ;: , whereas A< A' is merely complete on r (since A ~ exists). We can therefore reduce the algebraic restrictions and limitations on equations involving operators A E" K by replacing them with expressions involving their .Il§Afect counterparts A -}{-/ ># !his is the ess~:qt;bal idea_.Q!:,.;the 'Heaviside 1 calculus.. It is implemented by the following considerations: (1) The correspondence between A and A (Theorem l j): _ _ (A t8f* '=Alf '±13 (2) For any *-I / ahc ~ -( )*-r_ (A/3 0 is an ismorpbism - A " ')f..-/ "lf--f 13 f in the domain of /~ (see Theorems 3 :2. If. I ) : A~-f°-A · f = A*-1,,t ""·.~. Af =A'<-'r i-AJf 1 "" r will of course depend upon the inter-preta~ The form of A tion of f'; /(, k ~ lfc1 ,., • In Sections 1 "' 8 below, the relationship between A and A is developed (Sections 2, 3, 4), the form of (A, B}-=- A/3'K- ll!I is determined (Section 5), the classes ~ and /(/ are st,udied (Section 6), a useful' transformation' calculus is outlined (Section 7) and finaJJ.y the inverse reversion process is carried out (Section 8) for the case where F is the class of everywhere continuous matrices [ fUJ] on the real numbers to a complex Banach l., space (see Section 1), K syst.em i.. -1 K is the class of 'derivators' ,A= ao AP I a, JJ f '" ftl-,.,.; * is the class of integrators A ~characterized by the 1 1 1 0 0 A A Jff-::: f J At'/= lo lJ ;{)Alf/=lo 1 ' " J ID ~' A K!/o = 1 (a constructive definition is given in Section 3) and Aj is the "X class of 'derigrators 1 A 13 of non-negative rank., It is shown that the isomorphism between A and l~-f provides a satisfactory rationalization of the methods employed by Heaviside for ltunped linear systems, without rAef f cc transformations of the operands. Thus jf- ( Heaviside's mystical operator !' becomes .k9 , the inverse reverse of the elementary derivator, and Dirac 1s 'function' 7(-1 r becomes JD [ t] , where [ t J is the matrix of the unit function. In Section 9, the essential steps of the inverse reversion process are given for the case where F is the class of sectionally continuous matrices [ f(tJ] , with fa)-= o for all sufficiently y small t , K is the class of derivators A , as above , fi: is the K ~f class of integrators A characterized by the equation AA =O and the condition that Akf and its first (n -(} derivatives be every~ where continuous and vanish for all sufficiently small t , and tfJ is the class of derigrators, as above, of non~negative ranko 'Strong' and 'weak ' limits, derivatives and integrals of sectionally continuous matrix functions [ fx ( c) J()} x are introduced in Sections 10 and 11, and the representation of elements of and ~ 1:J as infinite superposition integrals is given. These sections barely outline the calc'UJ.us of such matrix functions; the details will 4 have to be presented elsewhereo ')(' Various other in·terpretations of G A-:, K, lrJ have been studiedo Thus F may be taken as the class of sectionally continuous matrices [ f(f)] with f {6) -- o for i <o (Heaviside case), k and .k'.J as above, and A * characterized by the equation A ;4Xf. =- O and the condition that A~! and its first (}i-1) deriva"' tives be continuous for t ,;;.o and vanish for i < o (and hence for t =o ) o Or /= may be taJcen as the class of matrices [ K>\J](essentially sequences), where .fl~/ is a function on the non-negative integers to a linear space, hi A= q 0 E f tf,E k }!'" system A ,4X;!::: f \,,'I A::' as the class of operators -f ''• f t?i., , T/ 1l where E(/lli)]-=L.J (•-d-IJ)1 as the class of operators A~ characterized by the I J?/:: lo 0 J L= fl : 0 )i - f , OJ ' " 1 E I ffo =OJ and /(J as the class of operators A Blf. of rank >r o (a valid interpretation also results if £ is replaced by A , where L1 [ f (~JJ =[fCli 4-t)- .Pl.i1J . With this latter interpretation (or others which are less familiar but more convenient)~ the method of inverse reversion and the related concepts and techniques of reversion, commutation and 'integrator' transformation have proved very useful in the theory of linear difference equations, including fractional differencing and summation.. The method may also be applied to cer- tain 'differentiators' of the Frechet and G~teau.~ type in arc-wise connected spaces, the right reverses then being given by appropriate line integrals. elsewhere. These and other applications will be presented 5 1. General_Jlefinitions and Not§..tjons Unless the contrary is explicitly indicated, F(~J is the class of r everywhere continuous functions S/ a complex Banach space of /i' ~ ( :L) on the real numbers /C to the subclass of F ( L) whose elements are 77- fold continuously derivable, /.;*(:£J the subclass of ~ ( L-) whose elements, together 'With their first (-n-1 ) derivatives q,.: { /= 0 ). cprcl<P-n(~ and x-J are the COrTeSponding classes of .functions 7 of ;l on /? to C , the complex number Vanish initially (at The notations class. \.(::t...c ct_// 'I will be used for the value of ~ Tf;C=a.. at ,L- =a... . The mark [I] is the matrix of the function /er-/ " F is the class of matrices corresponding to the functions ~ /-[;:t-) and similarly for r;; ./ r=; -t(- o Equal matrices correspond to equal functions: J for The sum [~J -f lJ]and difference [I]- [ /] of [r.l [ 7} [I ] = [ 'J ] every j , etc. is equivalent to / /C 'j Le" to ,/- :::- and the product of [f} by o< z C are defined as the matrices respect- ively equal to [./ -1-J J/ [ :f - J J and [.,,; f] is the matrix equal to[or] and aE o • When considering functions with dif- ferent arguments, the more complete notation [ / Jt Thus [ il (;t-1)];6 = [ Y.. or [f 2 - S/ v[f] ~] 1:- • JJc;.._ £!'.. [I} at a is Vf f is conveniento The value • In practice, contrary to the usage above, single letters 'Wi.11 be used for matrices and the same letters with the argument in parentheses for the corresponding function: / An operator on a subclass =: [/Le}} K of a spa.ce1;: t.o a subclass K of a space E' is a mark f1 such that A-:k is a function of 7c on K to 1 K{ 6 The domain of A in "E' and rrui.ge of A in L ' are those of A -y: o Operators A/ !3 defined in a space Z' are egual on l{<E'H /1 J: := 6 J::' for K-t: K. If =13 if they have the same domain in If 13 is on t<c[; ko that domain. t<' c 'E.' L: and are equal on I and /1 is on K' <- 6 to I<'' C ~ " , then the product /1 .8 is the operator defined by fl B) ~ -= /l ( 1.3 v) F'O/.I? "Jc c 1<. We shall be mainly concerned with linear operators, i.e. additive homogeneous operators on linear manifolds of linear spaces to linear spaces. The null domain #A- of a. linear operator in a linear space /l is the linear manifold of all xz.A such that /-1 y =o • A set r\ <ll.§]2fills ?\.._' with c(oic R) if every element of A is a finite linear combination with coefficients in C (o..e A;bf elements of f"\ ; I if moreover every such linear combination belongs to A: , then r\ is a basis with C (oR I?.) of i\. 1 • For linear operators ~ 13 defined in a linear- spacer-A ., the sum /J r.13 and difference /1-8of /-1 and 8 and the product of 11 by ~ ~ C (oa R) are the linear operators respective- (/7- :/- /.3) / ly defined by == /4 / r- L3-0 (/-7-13J / = /1- r- .8;5 c4)I /).::: (o(A)/. The operators .,,6)1 :( ~ are defined as follows: ~ f =f ;i ((,LJJ Fo/C r ~ F, f ~r CJR <£,, ; I ;f' =: [ .i~r•J ,/._ r 1 <f ( /.., ,t- <>t< ~ .f = {<Z ,,,(,/' , is replaced by /_~after Section Q); Po~ /~FOi(? ~. ./(,.{!J7 A for -o.. These operators belong to the class L of all categoric~ linear operators with ra.~ges and domains in r:" , i.e. opera- tors which may be defined without specifying S • A correspondence P{ le/ :f J between classes X; Y is uni- vocal in Y i i for any 1(,, ;vl.. t X and any 'J:~ 'r', ~1)(1/,i}}ind /::C-'11:.'L;/j)implies -V.1 = VL . The correspondence is biunivocal if it is univocal in""K and in '/ • 7 If A is on operator between A ~ -= ~ is a correspondence between K, K ' • dence is univocal in ")( K' to K • I l<cZ and I KcZ' , then If this correspon~ 1 then A has a unique right reverse A on rr Ax-=~ is equivocal in -x (i.e. not universal), then a right reverse exists but is not unique (a supplemental condi• tion may then' be applied to make it unique)~ Similar remarks apply to univocalltlity in 1J. and left reverses *A on K to /'( • If I _, ~ ff AX= 1;f is bitmivocal, then a mique inverse A =-A= Aexists I on I< to K • 8 Derivat~ 2. be the coefficients of a P The coeffj.cient segue nee of P is the sequence e .f = £$ 1, •,,, hi for .'1 - polynomials ?, Q efficient sequence. and a.le = o /fl- polynomial [ ak for k ;> ~ f such • Two are cogredie,.nt: r' rv Q if they have the same coClearly cogredience is an equivalence relation .D- polynomials are (reflexive, symmetric, transitive) and cogredient equal (as elements of L ). For _an:v two JiO _polynomials /', ~ the following state- rnents are eguivalent: (1) p :::- Q (2) l' and Q ye cogredient (.3) Let [ a1c 1) f b~ ~ IC= ~ C'k 1t9k , where be the coefficient sequences of ~ Q and let C'k ~ Cli: - /,It ~ Jt' 1 • Then IC= a on A ,_o() 'r->t v..,. JC [ t,, J= lo :: o, so that R = c, l:) f cl.. ;f)"l...f , ,, Hence 4 ,-hV...,, R. [tj =- (' -= o , so that R=-C'z.~f. "' . ,,,_,. c<> h+ I hence I 0 I Continuing in this way, Ck : o for k ==- ~ 1, .. • 1 i,.e o (2) fol- Of course (2) implies (.3), and (.3) implies (1), since lows. within the domains of f', Q for all sufficiently large A,_derivator A is an element of L. equal to a say P • >-i fh ~ is • ..40- .12.Qlynomi~, The coefficients, ~r..fill and coefficient sequence of A are those of P ; this is unambiguous since, by the preceding theorem, any other ~- polynomial equal to A is cogredient with P • In particular, 9 every 1)- polynomial is a derivator. Derivators are cogredient if they have the same coefficient sequence. A sequence [ a1e] of complex numbers is ~k..Bull if al- tlk -= o most all of its terms vanish, i.e. large for all sufficiently k • For each nearly null sequence [ e< k f , there exists a IJ- polynomial P whose coefficient sequence is f a k ~ • Let 2.~ Clk iJ ~ be an element of I.. is a derivator and [ ak. equal to !' • ~ ale IS~ Then j is its coefficient sequence o Conversely, the coefficient sequence of every derivator is nearly null. A ~ Lk, Cfle. A9 k is a bi univocal correspondence between derivators A and nearly null sequences Let A, /3 {erk f • A ± 13 and alA == 0(o( a,e)~k of A f/3 1 A-B ,A!3, <¥A k J From the algebra of L , it follows • = 2~ ( C(k t .b-'J ~~ [ ~ a.e /,k-.e J, [ c akt f ai:f f i.,~ be the be any derivators, and let coefficient sequences of A J B that Thus A 13 -== 21e ( t C/-f bfe-.t)J[) k , and hence that the coefficient sequences {e<k-J,kf are respectively [ak.f6,tfJ • The class /\ (D) of all derivators is there- fore a connnutative ring <: L which is isomorphic, through the cogredience correspondence ~ -t Lk qk !) ......v .t L1c qk e with the abstract ring /( ( e) of polynomials in the 'indeterminate 1 8 E C o Through this isomorphism, all the factorization and distri- bution theorems for 1((6) are applicable to if with coefficients K(l>) • ao is the leading coefficient and o(, o( <.. I zeros of a polynomial !Ye) E /((8) I , then I I , / In particular, c( >i..i the P(D} = t(0 (LJ-4,)(f)-o(.J ,,, tp.d,.,, is the factorization> unique except for the order of tne o/ 's , of the iJ- polynomial P(D) cogredient. with f(e) into 10 Moreover, D- o( = €,,( ,D €'"_c( , lJ - polynomials of degree ~ / • i .. e., the §im12le., derivator ~ ~ (o-o{) is the transform (in the grouptheoretic sense) of the elementary derivator D by k k. Do( ::- E.o( D €-At f:o( , whence Vie summarize the fore going in • THEOREM 2 ao =f o, m Y,, ti are the zeros in c of Pf e) are the distinct zeros with multiplicitiera Jn, 1 ln 1 , Moreover, if PrD):::: 60 (D/,),., {!>;19,,) then ,(31. .1,,. (3 .,_ ~' is a permutation of J C( 0 ,, ' , 6 :;- 0 cf, 1 o(7..., ,, , 1 I ".I r-y- ),,,._ • ~ qi'"" , Thus a derivator A of degree >., is characterized either by its a 0 ).,., + I Coefficients and ~ice§ o(, o( 1. ' , ' •, 4'0 I Q 1 , , 11 I q ~""' Or by its mod.uJ.e o/"- where the first, second.1 ', , , li. 'f{ elemen- I tary symmetric functions of the indices are respectively equal to ~ > a Clo ao 4 - t , • • • , (- )~ Q ~ ao 0 The derivator is normal i f is module -= I • /I TP'.J50REM 3 a,nq ~!'!I (1) f' E F,..,, 1 F, Do(, I', De:{, ~1 ~ ,, I J De(, 4'1. , ,, lJc(i.. p lire linear combinations with coeffiq:i&n~§ in (2) r ~ r, v~r, or- 1' ,, . ' r• D13r... ••• D r 1 For A ... I , k :- / i I (3) C .Q!: l>o(, Do('l.,, Dot.+. f I I l>A ~i... ,, . I ~I • k =DY -(o{, f af-i. f ", o(k) D k-:e f ,,, f (-) ;r, f o(1.", These equations determine ..O,,,recursively as a linear combination {f. 11 ~ / c/2 with coefficients depending only on If o<,..1 -<-...../ - - - ,,. ..< ~ /,.I ~ /e hence ~ ,Lf}/3~ J <><k - ./ - - - ,, ,d?->-> ./ - - - ~.1e P (;(} is a polynomial in .L- of degree -:rn with coefficients If then P = [ .P(:E/J is a polynomial matrix S -' and S< P and the coefficients of P and of <.?',( P are those of P (;(') ; the ex- m~ and ~ of B.t P are ..<.. ponomial of type ( oL:. 1 7n ) is The element~~ ex- and ( -<.1 Jn) • ~J... [ :i >'>-1 J • 717,<_ THEOREM 4 If where the r7j_ j 13;::; :t-; 0 I L:t- J] ~ f;Q! ~ .12.m ~//,.1--/ c>(---n 5 o; . ,<.I =o J.mlies ~ =-o Proof: 'I'he theorem is certainly true for implies I-? = O i.e. ~ /o = V/k - n / distinct com12le~c E«, p 77 =/ : Assume the theorem for c=-.,(, P; + c.,( z.. Pz. r - - - -t- ~kt-i fk.,_, ~ 2/ - - ~::::: • fg_r j :::: 0 2 / - - - 77l,.L. ./ which i~plies. ,// ,f!J k P, " =0 for ~a -n ~ k • Suppose == o • Then €13, P, --f- c;L Pi -f - - - 7- C~k ~ -::;: - Ptert where ;S,i.. : : : o<A. - o<k+-1 distinct and r 0 • for A-=/., z../ - --,, k ; greater than the degree of are Hence if A: k k P.e+ 1 ,,<J> !'<. 2:'1· ~l fj' = 2 I . c-./S;• 't' ~ . ;- = 0 .} , Cl> J the fl.A Clearly the derivative of an exponomial with non-zero exponent is an exponomial of the same type& where is an The degree exponential-polynomial matri..~ or simply an e~onqmial. (4) is /; ~""' r, - - -, ~..<, ..&...,,. - - - 4k :1- a linear combination of in - as linear combinations r --- -~ r , .I - are replaced by ~,/ / :f= / 4z equations (J) give ~ of / ( <T I is a polynomial matrix of the same degree as P/ • is 12 4> j From the inductive assumption, the 0 and so are the 0· vani sh, hence each is of degree for I == /.,, z/ - --_, /e . ~+/ From (4), =- 0 "7; • Suppose :E.;. I\,<. ~ = O I A,,.:. Z. C Then by the preceding theorem <\. ~ A • Hence the /'\.--<... vanish if each • ~ where the er--<- =- ~ 2_, - - J??. .5 are linearly inM • P<. are in C • Now suppose the coefficients of the = o / k. Thus non-zero exponomials with distinct exFonents and wit h coefficients in dependent with respect to o _,, :=::- where the are in S then by the preceding theorem, 0. P,._ (which equals a polynomial matri.>:: with coefficients in e quals zero for -<. ::: ~ z / - - -,, 71 1 and hence the VA- S ) vanish if each P...<.. ~ o • Thus non-zero exponomials with distinct exponents and with coefficients in C' Since are linearly independent with respect to S • 4,e { eo<'l k j = ~ ,,<9 £ [_;{ kj = o if J. /' -k./ the linear manifold N1 of the elementary exponomials [ e ~ ;C;t. k ] for j . == /.; 2/ - - J ;t. J• k::::: oJ /./ _ __ / 7?'1 1 is cert ainly -I within the null domain of the derivat or /1 =~ I 777 '71 ~/ ""z. ,r, .. - - - ~/'t -n?A, It will be shown below (see page :--- ) that they span this domain. Since they are linearly independent with respect to basis with S for the null domain of r:l . S , they form a Other bases with S for the nuJ.l domains of derivators (i.e. linear combinations of the elementary exponomials, with coefficients in C: , which are linearly independent \'V'i.th respect to C ) are given in Theorems I/ 3. and 2'f . Integrators Let _Lo<. = c-~ IC-.< eL clearly t he domain in r:- of _Lo( , • 13 and hence of ~ »t is F . An j,ntegrator is an element /-i .t L equal To for some/ ,(.A./ ..c, / - - - / -< ~ <E C The module and order of /1 are / .u the o!' 5 • /-1 and 7?? ; is normal if its module = / . • the indi~ of f l are Any complex numbe~ ~ o will be regarded as an integrator of order zero, with module ~ , but without indices. It will be shovvn below that for ~ / -,....; -7 0 1 ,,£< Lor, hi. - - - L«?h =:, 71-<~, 1 ; -- -71"' if and only if,,£< =- J./ and the/3 '5 are a permutation of the ~ $ 15 so that the module and order of an integrator and the set of its in• dices are unambiguous (the set of indices o<'o / .,,1, _, - - - , o('h7 always contains exactly '7?7 elements, whereas the number of elements in the class of indices is the number of distinct indices). the first order are iaimgle. dex O and module / The integrator I is elementary. Integrators of of order I with in- The product of two integrators is an integrator; the order of the product is the sum of the orders of the factors; the indices of the product are those of the factors together. The module of the product is the product of the modules. THEOREI\11_2 The product 1113 of two integratot§ 11..,, 13 is commutative: lfl.3 =8/1 Proof: Suppose, without loss of generality, that ~1.3 are normal. Let 11 and 13 be replaced by equal products of simple i ntegrators. Since multiplication in L is associative, the theorem follows if 1'913 is unaltered when the elementary integrators are permuted& be the case if .L.~ ~ true: for any / ~ ~ as may be seen from = L/5 Io( - - for - .L~ -1,,.s - o( .I~ This will • But the latter is L-.L~ ..<-/-3 1 -'--,< ~ = [e o< k £ ~01-"'-J 7 . { ~ -/S/-rv-) wl, v- d r J either by integrating by parts or by reversing the order of integration. TBEORmJl 6 For a11L..distinct_ comnle* _nu.mbers o( _/ c<,_ ./ - - - / ,,,<??? [ ..<, I ~2. - - - L..<,..., -::-(. -7.x-, L L -f2-f- - - - +- --.:.°'~?">->~---c(, ..<z - -- ..('-,.,..,} (-<z...<3---..(?>,o<,) (o<.,..,.,o<,o(z - --,,(,.,.,_,) J where (-0 .,(2 - - -o/-,....,) =- (~, -..<z..) («; -c/5 ) - - - (d, -.,,.(~) .1 and the denominato:r.a_of the terms following the_j'irst ~re oq~~ined from i~t~ ~_irs t by cyclical permutation of the <><: s . , the theorem is true for 777 == 2. • Assume it is true for The theorem follows for efficient of 4~ f-1 7n =/U/ and 7-?? =k . Then hence for any 777 ~ z i f the coI equals It does, as may • be seen from the following Lemma 1 Proof: Egr any distlnct complex n_umbers For any polynomial /f;J on C to C of degree <>?? + /(.t._) ()o<~ ---./,,..,.,4)+ ---+/(o<'n,) (j~t --- ot,.,.,_,) I J (,,..(, ..(>- - - -..l'?n) (..:'._.,{ 3 - - -ol-,~1) (..f-,,.,D("I - - -,,.('-n.-1' J this is the Lagrange interpolation formula. In particular, the equation /(j::./{ot,) (}.t4---./.,.._,) holds for k =o,. t/ - - -./ -rn-1 with /(J).::: J k / and the result follows upon equating the coefficients of members.. The case k J = D completes the proof of the preceding theorem. ,C- Since _/..,,( r = [ L cz..<(£-rJ /(rJL r is true for m ~1 • in the left and right 777 - / 1} the ·t;heorem Assume it is true for 7n= /e. • == { /_,;!(:t--r)k-t .<(,t-r) /r a<(r-trJ/ II / 0 ( k-/J / e L0 e" (<r) d <r a 7 • Then ~k~~ :::;..z:/·_z; } =f1t-r)le-1 ;f.-<{,i-rr)/(<rJdcrclrl= [ r ;i(;l-r)k o<Ji-r) f(r.l d r -1J/ o J /<J k e r / . 0 1 after the inevitable integration by parts. ?71 ""kt-I Hence the theorem is true for and by induction for any rn ~ / • Other important properties of integrators can be inferred directly from the definition by methods like those used above, in which the intimate connection between integrators and derivators does not appear conspicuously. It will be more convenient, however, to obtain these re- sults after the derivator-integrator relationship bas been fairly well developed. r 16 4. Revers:k9.,n of Derivators The derivator fl::: C<. 4 1 ..&2-<t. Al-< 77 and the integrator X= f ~' Itfz. --- ~..., are coindicial if 15, / ;Sz. /- --/S>t of -G, J o(.l. 7 - For any -<Srr = -<,, =-I 7 then A and f eF, ~L;:.f =~~,(JI ~--x: r ==: ~ since - - if moreover j of.. fl: .lo ./'(,t) dt] : T l;a. • f is a permutation C<.. X • Hence Ioe is a right reverse of More generally, suppose A and X are reciprocal. Then AXf = L9<><::, A1..cz. ---..&.(~ Ic!_'h? IcX.n,_,·- -I-<-,-/ ::./, (5) in applying the preceding result ~ times. If~· ::: X of A and the notation Hence ][ is a right r everse for the integrator reciprocal to A is justified (see Sec. 1) and will be used hereafter .• From (~ 5) , for any f, f£ F The converse is false for s~~ / , is equivocal in :f :::: If ~ ;-- implies fl f :: ? . 't, since the correspondence But if the linear condition T t! F..,...,*" is sub- , the result is univocal in f and equivalent to /- • fl T = 'j joined to , f =A -1>-T for any . / 't ~ F . Many properties of A * may be inferred more readily from this equivalence than from the definition in the preceding section. For any deri~ator THEORE1Vi 8 and f E. F..,..., *" and any if and only if · /- ::::: /I integrator reciprocal to Proof: rl *l Je F ~ F .6).( f.; is continuou.s, ..<'.9-.,c f:::: 'J and / 0 f-1 ;l .= 't , where A~ is the fl 1 is normal. =O is equivalent to f V\\e remark is equivalent to r = I..<:. i ~ E ~ fro] =I,,(_ 1 ~ 7o [e-<.J]. Hence J fl • Without loss of generality suppose that, since ;'5 / • = I.c '/ =.:&..( 1f { , 17 be the indices of fi =- ~ 4~- --Alh-i.' Now suppose 4, ( ~<><~ - - - Then ,,&lo<)'>, f) ~"L4?..&..(4 - - - ~~ f /o : :; c • f) ::: Lt, 'j /SJo<b-1 . 1 -== - - - 1f-- ~ ~»7 / • and by Theorem . J Hence by the preceding remark, ..6).t, (-&of - - 3 3 , ~-r:~ A<>l 4 Again by Theorem .67.c 3 "'9.,(q_ and as before and /I/ = / ' so that 11- ,.,&-< n, f =- f .Loe;>.. Io<, J, : : o, ,<l)of'I?? / • / = .[o<..,..., ..t.o{,_,, --1 Repeating this argume~t, we finally obtain Conversely, if the latter is true then • A- f ==• .&./ .&/ --- ......,""' A"J,, L o<.,,.., 4-o{.,..., / ..._, ""'t. "' -1 - - - r oe, .L == / ..c• r£ Vle can now show that the indices of an integrator are unique except for order: Egua1 int§_grators have equal modules and the same set THEOREM 9 of indices. Proof: Suppose that the integrat ors X ~ A I..<, T-< ~ - - ~ Z-< n-, Let A~ 13 are equal. be the derivators reciprocal to )C 'r' • on F /~ F Let ./t • ,q j{.. = ~j(r-= cept for order. and B and ,/.3 11 ;:::: x ???~ (rn, "'n) • hence the domain of By Theorem . 1,.. II modules of fl 'T , (R ~ /3 ) X r X = , and since Then AX == J3 'r' .::: and 13 and 13 == o for any By the preceding theorem, x r: includes x- • ;t are cogredient, and by Theorem 2 the are equal and their indices are the same ex- The conclusion now follows, since the modules of /9 A.-' / ~ _, and their indices are o<, , /52. / - - - , /3.,, • • are / be a normal derivator of degree I ::: A *'l ::0:: - 7?? ~ L ...., L.,,::z.. • - - - For / t Lo(n-, 1 · F .1 let -< l. ; - - - " o(~ 18 By Theorem 6, • T = { /],../...,(, i- Az. .L~l.. T f-- An, L.e<",_,_, } I) where /1, = (<><, ot,_ - --o<.,..,J-'.1 ,q, = {-<z. ..<3 -- _ ..<'>-> o<tF' - - -/ A,., = (..<n-, -<r ..(i. - - -./m-, (6) The individual terms on the right in this equation be- • long to r--; since i l. r fk for k // • belong to is continuous, but in general they do not Yet the entire right member belongs < F-;o-n / since by Theorem 8 1 /1 r = J and F n-. ~ to .f ~ F,,,,., -Jf • It will be instructive to verify this by means of Lemma 1. on F_, ,,&f- = [ o(, /-}I Io(, + { 1-l, -f- ;c:lz_ I- Now by Lemma .& I- -f .(, /. / - - - r- An-, } - - 'I- ,,(n, /.1 ">'>-? - r:. n, } i l. and { ~ 2. R, ~I }!} "i.j- =- 7- the second expression in braces vanishes, hence F, i A z.. I.( z. -(- { o!, A I f- ..{,_ 7.. 4. L-<L -1- ~.... /-} l.. f -1- - - - 7- -<.,,., ~ /-l~ Z:ofl?-i} i f- .o(?->? /-J.,,.,., } - - - I . Again by the lemma, the second expression in braces vanishes, so that ~7-J t Fr • Continuing iu this way, we obtain (7) k { ..<., k A,--'-<, -1- .<, k /'12. I-<1... -1- - - _ r- <'(,.,., ,Kl.,_,., .L. ~,,_, for k = o,. , _, - - -, )?-7 _, /SJ If>,/- = / and (.BJ .(),,,,., / ~ { cl., -f-- 7-n /], l °"// L «, -;- ...<,2. ,....., /-1 -z.. 4~~.... "J'n-1 / I r71 / f- Ol z. Jn-y f- r-/ ;z. -f - - - - ~ } 't r- o(.,.,.,7>? /?,_,_, L.<_,,,, if - - r ,,,(J?? where this time the second expression in braces From ( 7) we verify that 771-/ /-j ? ? ; j J) = / • for le:::: q I/ - --/ m-1 19 rn fl f == ( a~ 2: k 0 -k ni since o<., / =<. . ./ . are the zeros of {'1e ct .,..,., -11:.. / in c. ~ k m If the deriva tor Tr.: 19 == ..<'.9.,...N is applied to the matri."t: ~ of the derivator fac- tors annihilate the integrator, i.e. reduces it to [ t:? o<k+r I: remaining factors nullifies zero. Thus -/. ~~ =-· [ <Z ~,:t:-] n.. are in the null domain of and 1 A J I and one of the i.e. reduces it to Jl p n.,, ; n.. -z. ./ ---,It_.,,,,,_, and it is easy to see that they are linearly independent with respect to S : 1F 1 -/- <T,.,,,_I 10(, j_.,(~ - -- Lo<.,,..,_, [ (?-<..,,,,. :i With 0-0 / v; J - - - I 0-:;n _ / 5 ~ then 4, -t.., / o = V, J . J hence ~ :::- o J implies ~ == = er,,,., _ -::= ~ • u; ::: _ 1 Moreover, it is shown be- low (Theorem 11) that the A-.:., span the null domain of A form a basis with .S for that domain. ,,<9/ °"' is .eguivalent...:!!.Q - - - -f- 0.,.,... -I and hence f/ = 0. -- o I J f ::: r-1 1'- 'J -f- lu 1 -/.__,.,,,.._, -1 I tf9.t., 4 . _ - - - ..<'.9.,1,...., where fu : : :. v; ~ -1 ; : 0 lo : : U ~ rr; fu, -r- • From the remarks in the preceding paragraph, ?'>? - /1 ~ -: : : o I 20 are respectively o;;_., <J, / -- / q;;.,,_1 • This and Theorem 3 imply that the simultaneous equations /1 ;I= j 1 f lo = <To / "'°°" :f/o = rr, ~,,.,_ 1 l/o =er;._, are equivalent to f-1(/-£) =o 1 - • J(. -_, 1 , is repeated as o6l'..1 odJ..->.. ___ 4 rn-1 f-1 =~- 1 fo i§ eauivalent to 1 f .::: u;, { O -f- er, £I f- - - Heplacing tj THEOP.E?~ 12 ? The important case 7 == o , corresponding to the whence the theorem. ;8 f I -u. O<, lo - 1 .J - - ~ o&l..c1. - - - and ;/--A.~ Fm* ; by Theorem 8 this conjunction is equivalent to / -4 == /1 null domain of R -/ For any f- {)_.,..., -/ h h"> - / • by fl f in Theorem 10 yields r .?' F,..,, /::: /}~/7/-r-A. (9) where oo./ o., .) - -- v..,,,,,_, r..fil.:e the in~l values of r,1 4 f / - - -, ..<O.,i:, "'2.r. . · - - 4 ., . , _/ ' ~ A. -=:; u;, t.b + <:r,-fu, 1- - - - +- cz;...._, A. »->-1 • Thus associated with each derivator /-l of degree J-n is~, the resolution ( 9) of f ~ ~ into its projection its projection t.. /I -JI.4 / on ;::::;.,,, .;..' and on ;Vfl- • By Theorems 3 and 11, the intersection of F,,., !P and /./"' contains only o ~ F. Hence the resolution (9) is unique, in the sense that if /-= d' + t..., = J-.. +A.~ where the ;J's are in Fn-, 11- 1 '3' - 1 • = /......_ -"', = o , since J' - ;r. ~ G and the X .s in /VA , then a.nd *- 'A-.. - t..., ~ ~. There is another very important way of regarding (9): It determines the matrix /~~ in terms of the following data: (1) the matri..X AJt, 4,_ ' - - ~. . . f f ~ and (2) the elements ~°"' ...£'.9.J,_ - - - ptf}_,,,,.._, f' lo I J of ..61.r, T /o S . In many practical problems (differential equations of physics and engineering in which (9) solves the problem. f is the 1unknmm ') these data are given and 21 ~.. Commutation For any two elements !::; <iJ of L , the operator r is the commutator of /-? Q o p Q , ~p . r of r The domain in 0£ r be the domains in F is ~ I • 4 2-• is in general a proper subset ~ of Ll , · ~ 2 (see Sec. 1) if ~ = 4, P Let ~ , ,, 4 2 , = Ll z_ $ of The null domain o P am Q commute This iscerta.inly the case if G are derivat ors of degree rn,, n ( 4 =L.:l , = LI z. ::= Fn-rf >J) or in- tegrators of order m n (A-:::.~ 1 ==-A 2.- ==.r) l!l of degree Q ?n >O and is the11 F n If P is a derivator an integrat.or of oroer /'! ~r then 4, = F arrl ~ 2. = F>'>'I ~ Ll / , so that P of T> .::: P Q - Q P F~ = and Q do not connnute. The do.T11ain rn-, , but the null dome.in of r is not it is F?'>) *- if ???~ n (Theorem 13) but isharder to de... so simple: scribe if 7n > n (Theo1"em 14). A connnutation condition is a condition C ( ./ J such that C {I'} implies equation r T = ~ r /-== o • A. commutation rule is an , Where in tre domain of for all / on :/ < r r is an ~Jtpr:ession in ¢ . / , valid In this section we shall gi·ve several commutation conditions and commutation rules .for where /f , .8 are derivators of de- --n respectively• gree P7 , THEOREM 13 ( 413 + Proof: -If 7n ~ e *,.c;) / n , then the null domain of T' is F,.,_, --x- - = 0 Since the domain of i f and only if /-: /. E / r-.,;; . r :ts ~ , suppose / E r-, o since operator mul·lii- plication is distributive, and associative, derivators commute and {3 13 I'- == I (we shall usually omit such formal details )a Hence by . rf = o Theorems 11 and .3, this is equivalent to 7 is equivalent to / ( c Fn nlf/J c r;_:Jf A 1..J t c r,,., X- /' since ?.> A k < >i-»t ./) », 1 /( < k for ~ Hence e AP.> =o 1 , · • k-. - , J J , am E F,.. -Jf Jr9~r./ffo = , since the degree of the derivator ~~A is then (by Theorem 8 the range of /3 -lf is F.,. "*' if * I r ~ }f ). Clearly 0 ,{) '>i. J,, Jf if am only if I& € 1-R ¥Pf - 0 If.Id t4- • • There exist unique derj.vators Qk. of degree k arrl 1C'k of degree }, -hr f.K < ~ such that JP A:: /)le 13 I .e* , as may be seen by ap- A ( GJ plying the division algorl thm t,o the abstract polynomials cogredient with A t3 /.3(e} o 1 Then , ~h-h.f.~13*ft ~ ifVkf/o 1-1\ic!J*ffo , and the la.st term vanishes s :ince the is less than the order of ~ -7t (the degree of 13 ) e degree of f!1c Hence l'f =- o if and only Qkfl =o and this is equivalent to since r ,If)/ ~1- f /0 ~ k for ,t := 01 11 f ~-:: IJ!la = '· • = A>J..-tf/o =- '" J),,., - t , 0 is a linear combination of the in.i.tial values of k lr9 /? in which the coefficient of j) f' is :/:- 0 • THEOREM 14 If ~ > J.i , ~ (A/3 )(-_ 131fA) f =o if am only ,J,4' are certain linear vomP...,_1' I!...,_ i,.,,, ... !...,, -1 11'\ J • • • ' f 0 , ' J nations of k = 0) I, Proof: I I /!,, .. o, f i,, ->r _ / h, - I . As in the preceding proof, r1 =0 i f and only if • There exist unique derivators Q of degree ),., -)., and R of degree <,, such that A ~ Q 13 f R. Hence for k_ -: 0, I J • " 1 n - f lt9 ~ J3*fl =: ~ ~ f /0 f lJ ~ /3 *flo : : O 0 is a linear eq ua.tion in coefficient of f f, , ,· // P),y _ >11- if. f >., _ Jt f I_ 0, 1 s =F o • in "?hich the B!£19;pe J?J :":. e:::..s fgz• tke;;e 23 1 = K'~ Rn-1 - - - I<'~ --- R, / Let (2) either (,l ) for every le(~-'.£. k =I_,. z/ and 12k = ~ ~~ or (i. ...i.) for every .le/ (3) for any k/ Pk is an integrator of order derivator of order ffk (/~11..-1-.fk-1 J • THEOREM_.1.2 1n case J1e ~ <\k~ ~ r ~ rt\ where (1) 1 • ~At. pk 1 l'k and Q1e i\~-:::: A...,._(P,-j'rJ I.et - ;--exists and is a -r- ("!' ... -/._) -t- f'o =Jo = o (To avoid exceptions, let (~ ) : Rk ~ ~ F _ _ -1- - )• if for e.J!]l.I.Y k_ ./ If t his condition is satisfied, then 'j is unaltered when ~k.. J'k ~k are transposed if defined recursively~ ?"7 =f where the /-<-j In case ("-.,<.) : _,,.<... >o .Qr ~ 0 _,.<. < ,i 7 I == ,,£. or - )• i exists r- if for every fi? ./ j' ~ ~ r- i6-k. E k If th.i.s condition is satisfied, then are transposed if recursivel;y b_x > and /Ai= /'J-i .,._ </;-i - J;-1 • (For any int eger ,£. o according.fill ~ %k Po ::: .? ~ f • k is unaltered when Q-<u /~ , where the yk 7-J are defined "11 == < 171 _1 -t- 7-J -1 -,1;-1 >· ,4.No • r=- we need only as- Since the domain of the integrators is certain that the operands of all the derivators are within the domains ·t.hereof. Suppose case (_,t.. ) • The operand of if and J ' <: A.1 =- r\ If since the range of ;I clearly Q, / is Qk. is Tk ~ Rk-r Rk-z.. / exists ~ r;o, * <:: /)z., t l-17.,_ .,.. (\ .... - 7.. =r" "3 etc 0' and finally t Fi\.,,,.,._, Assume that the conditions and certainly and Q-z.. P , Q, f Fi\.,-y, 'Repeating the argu- • ment, if k = / . J'k ~ ?\. k /z. = A.. Q ~ P Q, T '}- exists are satisfied. then If c r-;-rµ/-$1 R Q, f We Call but if' ' QI r only assert and ~ r- , L and then P,Q, L r, =- P, Q, / in either eventUa.lity, Continuing the argument, we see that for / .t:. r-,,r'r </<-1-p > - F, L r- -;If ~ -"K- F * _.µ:~ and by Theorem l3 commutation of Le:!!, '77J /-l == derivator of degree n-, . the " a 7n-k ,,6Jk 7?'7 Then for any , ~ L3 any derivator of degree /" .!" ~ Ak ::- ~,e ak-.( .t1_ ~ ::: ,,,(S ~ /)o Proof: Then :Ek k ~ and will not • The proof for case (ii) is similar. '.l.'HEOREM 16 ~ • ~ ~ E • Hence for any 77-/ , then alter r ~If-. Let • I : : : (A r n - r ,,(J) n I = 0 7? , ·whe~ A ) ;r 77 ~ /77 • and hence by Theorem 11 (with there equal to one) ,, I 25 For k. < • , since the degree of _&Ill/I is <.>7 o r ,..,_.,.,,., +-k 'U moreover, .J.. NO\V and ..& j -n __ J<::l*°h __ t?'V (Al 71'1-A?. - , For k .:;:;: S '~ - - -,, -:>n-1 .J ?7->-n f-k. ..1.. .; .1())1* o J<)~-k ~ n I . f3,...L By Theorem 8 , / r belong to r?>? *" o By Theorems 13 and 5, L3 * _[ n / ' 71~ Hence applying B == ,LY n1f- /I 8 ;>\'- ro (//) /f-/ • , Wl.• th q ~ ll'7o ~ - r 11 ( r7/.3*-6*/l)r = e *~.z- (IZJ .::: QB"7f_r'h [!] where the commutation of Theorem 13, since I THEOREM 17 n [ 1] ~ fl as = I 17 / 77 ?'t1 77 Al 111-f /1 -f - - - 7- rt-,,,.,_, [ /J n is justified by and B ~ [!] belong to ~ L] -t-- rT 1 11V <;J 6* [t] 8: Q and of Q / .r Theorem follows on applying Proof: "7-~; .r.J == /l B ~ ~ n * T (13) .L .v . -::=- 77 - ,,.,..., The To(! 2) 0 in the preceding theo1r.em, for any 7 c r 777 -/- = .rl jf--/1 / r(/lo .15J1'>1T-/}/,<9ni-'+ ----I- /-1,.,.,_ 1 ,6) /l:Jf- [I J. Application of 8 /!~ to (10) or (11) yields (13) after commutations am reductions justified by Theorem 13 and Or simply take 8 = /l in Thoorem 16. /? /7 -Jt< = I o ~) / 26 6• Derigrators • For aey derivator // of degree 777 and integrator X or order n , the operator /1 X is a derigrator of degree 711, order 77 and rank n-7"7 • If Y and l3 are the reciprocals of A and X , i.ea ~ =/I 8 = 7'X J then /-J. X = "'" T1 X == /I B 1f- If /-/X ==A/ X / , where /1 1 then ~ X / R : : : *X A / 77 r , 7(- and • is of degree >?J, , and X, of order , after left multi plication by ~x "'"X, , or X /11 ~ = X / ,q * , after right multiplication by ,•;4 ':A/* , and by Theorems 1 or 9, 7'17-T n/ -- n?, -r "7 i.e. -n- -rn , :::: n,,, - .>->? / • '\ll[e have, in effect proved the following: Theorem 18 if /-l B, = /l, 13 Theorem 19 • * an:l A, e,,, are equal i f and only '""' Equal derigrators have the sane rank. Derigrators /113 /-l /3 ~ For aey derigrators .J /J 13 *"-t- ,q 1 13, k = ( fl Br !: /-7, l3) ( l3 ~) is ~o , ~ (rJ87f-}(/-J,~ *) = (/113,) {88,,,) * I ! If the rank of /I .e 7f =:. • /-1 ( 1.31 13,, ;;tc) 8 iF- *o The second part follows from Theorem 130 A derigrator rank of /11 f3/ -:::: (19/1,)(L3e,,)i1r-. The first part follows from Proof: k /I B /1 13 *- is dextral or sinistral according as the * is ~ o or ~ O (thus derivators of rank zero are dextral and sinistral). If /-1 B -)(-- is de:::i:tral, then by 'lhoorem 13 ( B/l ~) (fl t3 *) == / 71 ,q.1.3+ has an unrestricted left reverse 1f(176' ) ioe• is therefore univocal (see Sece 1). 13 -K- = 8/9 "*" am In particular, the integrator is a dextral deri.grator of degree zero with unrestricted left reverse If -x. ( 13 *) = 8 rl t3 o * is sinistral, then (/f t3 ~) ( .6' /-il * ) = / , 27 , ieeo //- 8 n- has an unrestricted Ii ght reverse ( /-1- 8 ?'<) *- = 6 .4 ~ but /-1- t3 "*' is equivocal: by Theo rem 11, the null domain of /7 B *" contains (in fact is spanned by) the non-zero matrices -/!.. n , l{J? 1- 1 / - - - .I -It.,., -1 • In particular, the derivator /l is a sinistral derigrator of order zero ' t h unrestricted right reverse /-1 *o Any element The derigrator concept may be extended as follows. of L will be called a deri.grator i f it equals a derivator-integrator product /-11.3' -1- , hereafter called a primitive derigrator. The rarit of a derigrator is that of any equs.l primitive derigrator (by Theorem 18, this is unambiguous)o By Theorem 19, the class )( of derigrators is closed under ad.di tion a.n:i si btracti on, and the product of two elements of K belongs to I< i.f the rank of the second factor is ~o o Derigrators are dextral or a itlstral according as equal primitive derigrators are dextral or si. nistralo Theorem 20 The class Kd. c I< of all dextral derigrators is a commutative ring containing the unit operator / f<.:;1._ 11 Every element of is univocal. be primitive derigrators equal to Proof: P/ Q ~ K.:;1._ . By Tha:irem 19, o am P Q "" ( ,q A, ) ( 8 (3, ) * o PfQ = ( 178,, T rl, 13)( 8 4 ) 1f- The order of ( 13 8 1 ) ~ (ioeo degree of [3 8, ) is the sum of the orders of B*- , /.3, "1f- ; for ea.ch of /J t:'1 , rt-, 13 , R /1,, , the degree of the rroduct is the sum of the degrees of the factors; the degree of rl /3; :t:: 4~ 8 is not great.... P .:f: <iJ am er than the degrees of the t ermso Hence the ranks of P~ and P~ a.re colll!llutative, and F a.re ~ O • Clearly P+ 9 28 is univoea.1 s:ince A !!J * is. Let F' be an element of .Kd ? .: A/$* deriva.tors 'Wi. th no common indices such that A by that of 13 (if A, , 13, oonditions then by Theorem 18, A 13, ::: A, 13 indices of A and gether, hence A J By Theorem 2 , the o over the product of the modules of of the modules of ,,4,, 13 )o Let A, 13 , A, ard 13 to- 13,, 131 are coindicial and so a.re // 1 • These con- also satisfy the together must be those of 13 be am the quotient~ di tions uniquely determine the indices of ,A , 13 of the module of Let A, 13 of pos.l tive ra.nk!J e More- must equal the product o( 1 1 o( 1.. , 1 • • 1 of_.,.. be the distinct indiCeS Of /3 ~l with ffiUltiplieitieS m,} ).,.,.(' J 111 J ),., ........ Ther e exist unique complex numbers Theorem 21 f;.,lt. (j=I,.<,"·,~; k. =o,,l,·~~>-.,) sueh that y,, ~k (l </J P ::: ,:EI J · L..1...~ f.ik Iot.J J Proof; Let A (el , 13 (eJ be abstract polynomials eogredient -1 ),., • O w.i. th A 1 13 • Unique complex nU!llbers fi It. exist ( 'partial frac- tions' algorithm) sueh that ...,,... \.., - I l. -Y- . (e-oS.e A(e) =- µ L-( · l:k~ fd·k ( e -o<J·) '()ITT lf.Ri=J / Hence t -I- A~ fi ~ )),,'1' • I ti n,.-/ k -r ;J), ~ (t !'ik ~~·) lft=1J· ~ 1¥ Equation I 'Jf 13 == / ('f-e7 follows on applying this expression for A to r; I "l.e -r- ht :;f , Any sinistral derigrator (or dextral derigrator of rank zero) may be resolved uniquely into a derivator aid an elem::nt of J(J l'f. positive rank, to vb ieh (j.i,) IDE.7 be applied. of a 29 For any of ?E ,Y , the matrix F and of its reciprocal *P o F' £1] is t he indicial matrix If the rank of P is positive, P PP [1] is the weighting matrix or density of the matrix *" P • In particular of i t s reciprocal A-Jt[I] and am /J A -Jt [ 11 are the indicial matrix m d tre density (i:f the degree of A is positive) A and the :b1tegrator A * o of both the derivator a.t With the notion of ttle preceding theorem, for at\Y Theorem 22 derigrator P of post tive rank aid any (!sJ r'F =-{f 0 f.[f· z·-,f. ok: r c- F J,(n-.j-1-k) ,J r; (t - ke ~· (t·7J( f(-rJir]. k. r This is an immediate consequence of Theorems 7 and RI Theorem 23 . If the rank of is given by --r ~·-, ,J 0 I' is positive, the density t le ~· [2.· Z 1c ~·,<'In. -1-lcJ -y-r e G= (/ 'J • J tj G = t>F'f:1] . "' This foll ows immediately from the preceding theorem. Thus equation (J ~) may be written as t rt= Lf G(t-~)f(7)Jr]. (11J 0 Theorem 24 With ·the notation of Theorem IC. , if A is a derivator of posi ti-cr~.~~ree l,, , then for agy Proof: f E FJ.. and. The direct part of 1he theorem follows immedia:te:cy from The orem 17 and equation (17) vi th suppose (I 4') o From (!') f' =A If o For the converse,, , hence applying A to 30 (!"if) Af == 1 d f AQG. ::: :J f GAG where ~ = Ao ~},,-ti A,~ >.t-'l f ' " f A 'nt-1 o But the k o(. t] elementary exponomials [t e cd in (!') belong to tm null d cma.in of A o vV"ith A G == O Hence 3 =o , i ·t is clear from {;8') and (JG) that every element of the null domain of A is a linear combination with coefficients in S of the elementary exp:momials. Since these are linearly indepen- S , they form abasis W. th S dent with resp:ict to Theorem 2.5 A~ If m .t fk qlit-k iJ er; , II, .1 , , (Jct) ~ v".i., • / • , Af:::q) 0 arrl elements S 1 the system of equations f./01 ==<i';:; / f)f!.,,,rr ... jji,,·'f)/= r>-,,-i ) lo f /J I is equivale~~ J ) t , ~-i f ~A"~ t QA" [ 1] ~ L)G<t--rJ3<1'J 11']+ ~olJ 1 ... + 6,,,_,)a, {}o) A.: where • is a derivat or of f €' !;., , d E F .E_ositive degr~ ),, , then for ayW' ~ for c) i1c. = ~£ £'.?K-.1' a; f?o l- le = 0, I1 1 " , 1,., - I J ~ ~ ~' Q ~ht-k ahel G = A9A -jf[I] k ok 0 Proof': Assume o /;qJ "' v spectively equal to J A I J.t to (z oJ (Ro) • · are re- ~ ht - I (which ar~ defined as in Theorem 16), hence Conversezy-, assume in the .proof' of Theorem 24. Applying . • • fo 1 f,, ". " />-,.,, _1 tic, ~k .,. ~)nrl Theorem 24. n- ~ v0 I v I I Then Then We must verify (~o) A I: follows by d follows,, as ..(:) ~ = Vk_ yields >,. AI"'f': A z":4Jt:J f A l~A"t:1J-= AA"f d f AA"G(Ii..l"J :; J~ + aI,.,[iJ where the commutations of QJ A Theorem 13. and NOV"T applying J),e_ we obtain Q, I.,,,_, are justified by 31 Evaluating at o , as aofk i- Cf,~ •1 f ,,, fa~ fa= bk= tlo CJk +ct, rrft., f '" tale <ro for 1~ k=o,1,.,,,~-1, /k=V"fe· The Method of Integrator Transforms For .i, k == I, 2 1 be a derivator of degree and w~1c/i n"'l'/'1 ~ j €" F , c/;1t h-7, , die = h,~ "( ~·k , !Jc c ~ /le J Consider the equations o A11F, <t1J , ' ' f A1z fz_ -f ,,, -I Ao, fn. = J' , Az1f, fA22f, f .,, 7..4.i.,,fh,-; d.,_, , . . A»,,f, +A,,, lf:. f · .. 1 AJ,, >t t,,., = g~, • For an:i • an integrator of order N~ Let (Q be 32 am hence (Z3) Conversely, i f (.23) holds then by an evident extension of the proof of Theorem 25, (RI) and (Z'Z) follO\'To (21) is equivalant to G?2) and Thus the single system (p3J o AJle. be the cofaetor of in the determinant (defined in the usual way as a rum of products) o Then f.R3J an:l hence ()</} provided that the order /Y of Q islarge enough to enS'J.re that Q~. am Q ~a are within the do-mains of the derivat ors applied to them (see Theorem 15). Moreover, by Theorem 13, i f N islarge enough, (/vi~ M - /vi /Vl'Jf) ¢ =o appearing in (z 4) , 'Vb ere ¢ is arr-J of the <Q transforms , and hence .,,, ~ ~ ),., <.RsJ Q ~· -:: 2_..J M-f_J.. M Q ~· q . t L . 1c. M;f'.J· Cl i "- M a r' 7 , I I ,)J the commutations in the right member being justified, by Theorem 13, if N is sui'fici.ent.ly largee Conversely, (2 ~) follov1s from (..?s-) on applying A.1.l on ...c and reducing the Ii ght menber by , summing f""""·.4.JZ,· /VI 11· :::: d.A/ f'1 33 Jf and /\1 M -= I , - provided that /\/ is sufficiently large. - JtLet f == ~ f. for any f € F , and introduce the equiva- lence relation ra_ , meaning equal for all () with sufficiently large N • Then (2.3) becomes n., (26) 4 AJ'k ~ <Q ),. ~J· -f ?-1c {)J.k [I] I the .§].bsidiary systems for (21) and (22); (25) becomes "' f?,J' /'rr ~ ~~· f fJ,k ~ l· Q ~J· /vf-<.i ()JI<. /Yf'Jf [ i] (27) and the systems (27) and (26) are each equivalent to the double system (21), (22)o Since ~· f ~ , a necessary condition for the existence of matrices f./ satisfying (27) (and hence (21), (22) is that the right member belongs to F1v j' , and a sufficient condition for this is, by Theorem 13, that the ranks of ~ (Q.-== ( L · M .. (), )/vi ~ Jk ..J'J Jf.:.. of (27) may be written as :A I - ; ~ flt· - ( L J. ~ d·) frr I t and . be positive, for then the right member h. ) ~J. /YT';Ad . /vljf(]Jq ' + LJ· k ~·J· Q 'L /vt* [I JJ E ~ . L, ( J't. ~ f ) "f [ ~ I Indeed, the condition that the /'1~· , ~· be de~tral derigrators is sufficient for the existence of unique roots ~· of (27) given by p ~ ~ ~ ~ ( 28) f _. -=: /vf" /vf "f f '). ' I /'1.. /\ 'L /vl [ I] .A , j -1 J CJ ..AJ L>(Jt: L· 7J/1.. which follows from (27) upon application of (3.. .. In general, if the ~· , !t· are not all dextral, then the l' condition that the right member of {27) belong to necessary condition on the /Ir becomes a g;· in order that roots of (27) exist., 34 ~[e have tacitly assumed that M =I o • If this is not the case, then (24) becomes a necessary condition on the 'Ji for the existence of solutions of (21), (22); if it is satisfied, some of the equations in (21), (22) become dependent on the others and the system (21) becomes, in effect, a rectangular rather than a square array. This more general case can be treated by the methods used above. In the preceding section it was shown that the system f>o ' J1Jl/,=f ··· £n.-l/= Aro;/:: dq ) P/:: Io o fa IJ.,-1 '1 -' is equivalent to the •equation 1 (29) where Af1>J/ a j + (Aol2._fA,iJ>..-if ,,. IA-,,,_,/fJ)[7)) i -= a'J(..f. for a:rzy f €" ;: and 7fi means 'equal for any t) of sufficiently high degree 1 • 'rhe binding variable ~ can be eliminated from in the same way that t was eliminated from [f c'2J]$ , where ¢r a.) Vfo introduce •matrices r Q on kJ to r .f'ri) c Q in (29) in ftt) t: Jf6) • is a function of , just as we introduced the matrices [ frt-JJ-1. , where is on A: to S • After appropriate definitions of equality, on, etc. , the class [ FJ of all matrices [ f ( G)] Q is an abelian group with operators t" Ad an,.d,:.C::iquation (29) may be v1ritten Ari)) ["{/Y] = [tJ'lfd J + <Ao iJ~A, lJi.-'1 ,,, +A,.._,~) [Q j(riJ] as ad di ti (the subscript Q having finally been dropped) • ii.e~ of operators on [ F] • The class K:; is a 35 This is a very- satisfactory procedure in many respects and we shall present it in detail elsewhere. We wish here to examine another way of eliminating the binding variable Q • l<e:A. has been extended to a field Suppose that the ring K J.... and the class F correspondingly extended to a complex linear - class F • exist in (.30) Then Q :*- -/ , the inverse of t he :reverse of Q , will KJ , and applying it to (26) yields Q*-' A ( .1.9) Q.,.r = :J -1- Q+'-r( Ao~m + --- +-A,,,,,_,-<!)) Q* II]. As stuning that Q * _, ~ Q .. K- cJ- if the order of ~ is sufficiently E Of-== large, Q* _, ,tj k (q-f- _, .,,& Q -If. )le 1 and distributing Q* -/ forward and Q *" backward in (.30) yields {,31) /-/ ( f') / where f1 = = <j- -f- (/Jo f <!;> ,__/.& 0 * "' ~ -f- Ar f-' >??-If - - - t- fl 1??-1 tf) [I J J . Thus the dependence of the equation on is lodged entirely in the operator q r. We shall show that ~ and r' can be extended in such a way that the foregoing heuristic argwre11t is justified . The range '"' F of any non-zero element X .E K,,J.. of positive ~ rank :i.s a proper subset of F X '} = T has no root F /-- ~ P • for some / < F , the equation We now construct an extension i= of such t hat thi s equation has a unique root : <! F for every .r~ r . We wish this e~'<:tension ;= to be minimal, i.e. to contain only such elements '/! extraneous to =r r- as are necessary- to satisfy the . Thus F should be the range of the quotient equation X( : r ~ F and 0 t XE Kd.. , with )(: = f for should be an abelian group with operators in " Moreover, F KeJ.. .. We begin therefore 36 with the following postulates :for F o <j, 1f' E' F >' f; ;j € ;= J, X, t/J' In ;f;/ .' With 1) f =- 1t' 2) Every quotient is a reflexive, symmetric, transiti'folie relation. xf (X ;io) belongs to F , and every element of F equals such a quotient. 3 ) ~ ± Y' / )( 4) -"' = 'r' belong to F implies X x rj =- x'f 5) f o f =Vf , and ( = )') implies )(/= P X and .X-/- o implies j-= tf o (x'r'J<! = x(Vt) 7) K( f ±-p) -= -X. f f X: y; 8) ~ .£ ::- f ( xt o) x 6) From these postulates, we infer that, with \ Y :f O (and hence X Y ¥ o since X, V univocal): q ..,, ( x /!) :::- -x ( v ..a:) e fl w v Y t =-X d xp : :- tr e:orv , x -x ~ 9) io) "'II ~ r f ± ~ >= yr x I J t x ( v ~ J -= x~ _!__ + ~ whence x - y =- yf .f X' XY xy d Y(x ~)-= -;;( Y f;J :::: V ~ ll) rf1xd whence 'X *V = These inferences suggest the following definitions: I.et F be the class of quotients and for any element r : :- (\" d and (l'i,,<,) "X ! , ~ of F ·1et f =- f be equivalent to (t.) , ((. .l .) {() -+ f-~ ==- Yf'Xi-yX qd} l( f for f c F and o :.f /I EfJ f- ~ . , .. ') I( l l -v- = f' ./. X y With these definitions 1) -7) are satisfied, but instead of 8) we / ave gJ' I.et Xx::- ( ()(-:/o)· F(be the range in F of .1.I for t F r " For any 1 37 I / - F 2 I ,/ E ~, ,,..__, .T if and only if 7 I write / _f I is an isomorphism (this follows from ( i )1 below) between F,. , F f , thereby malting and F such that 8) holds for / /- , i l) , ( i v) and 12) , so we may without contradiction identify ( if q l ( =/T • Then / '---v T F / -= F and F an extension of rc. r- (and hence from (l i iJ for / c. F : a :;! x KJ... , then X -$- = ;< ~ :::: ~ =- x) = +,) . -X XI).-' -y- <j/ .E: X T Thus the quotient class t= satisfies postviates 1) - 8), and it is easy to see that any other class F-rf- satisfying these postu.. l ates must be isomorphic with r- (for ¢€ 0 </> "\.) ¢ fo be equivalent to ¢ The relation cj ..-....J =.: = r: ¢~<! ¢ ;f- for some re F ¢*is an isomorphism between F , F let ' 0 ~x ~ K.J.. . :){- ). Hence the postulates are consistent and categorical (but not independent). In addition to 9), 10), 11) we shall need 12) IF X::::; 0 J and (x± YJ~ 13) for <f ~ ,P E F and X, Y ~ K,J._ • These follow readily from { .i..} - ( i "it) and 1) - 8). The foregoing is summarized in THEOREM 26 ~_g:y.oti~nt, clas.§. F :><. in tha~inmtitative ri~ Ka • F is an abelimi. group with OJ2e.r.ato:t;:§ is a sv.b .~pup of F • For aux =ljJ if and only if ¢ ~ ~ Un Ilf2;rticular J(t/> :::: :T is eguiyaleµt to ¢ = :. f.Qr r c F ) • <1:> 1 \ / ; .! F and o =/ X ef<J_ ./ x¢ With /- constant in F , the range of X -9- ror x, Y c kJ and Yi- O is of course more extensive than i f only one of X/r'varied. ·we wish now to define g:gerators X (V) r' such that " the transform X y :J _ -4:._ - X y · 38 I.et L be the class of all linear operators on F to F For any X J T' t.. KJ. with ~ /:: O , let ~ be an element of L that for any ¢~ F.J ~ ¢ =- X such :j (= x.$ ; this follows readily from Then with ¢ c ;:= ; ;( 8) and tbi:i) ) • o .1 T'., z ~ K:e_ ; u, v, w c Kj_ and :f:. o (and hence u V ~ o since ~ V univocal): i ~ .::!:. ..,..,v ¢ 14) ( {; !:. .i:..) ¢ == v (by definition of xLJt:> !:. r:__:j: = \/ K .:!::. u 'r' V uv cp, operator sum and difference) :::: ( v X. ±:. U 'r) </::i =- vv V(X<f>) ::!:. UC 'r' ¢) UV by UV..), 12) and the postulates o 15) ( product) ==- c ~) ¢= ~ ( ~ 1>) (by definition of operator 5 ( 'r'.:) ==- rx ~Jr- _ LJ ~ ~ ;:- .. v by ( v) and the postulates. 16) ~ y=~ v v w II= ::: uz xw' X .;£ 0. 17) I.et 'f I:- o . + ¢ = x¢. Ket be the class of quotients yx t- -L for x., 'r' From 14), 15), K,J,. .E Ke1.. and is closed under addition, subtraction, multiplication, and the multiplication is commutative. From 16) 'i<J... is closed under division by non ..zero elements.. Hence K J is a field of operators on ;=:- • -1 Iet /'(,,_ x ~ for i. ~ Kd. of T be the range in Kci )(' ~ i(j , X t KJ. write X '"'J X i f and only if J(' == X' A.. X is an isomophism between l<(,1 r tify ~ and X , thereby making e:x:tension of the ring K.J.. • , f<'J.. o -f For • o Then By 17), we may iden- KcJ..' = !<,)... and the field /<,f an .39 !Jl£L.QUotient c]&§§; cla,s,.l:l K,,t , is a field of line~ 012erators on F to t= which contains t~e ring K.{ • ¢/ Vr E P ~ o F XE K. 1. / x <rf> = ijr- For any if a.fill only if ej :::: X _,1/.t I'( ~the Let /( be the class of derivators and class of integrators (including O , which is not the reverse of a derivator). t<*-t Iet L/ A~r-.. be the class of inverse reverses ,4 if- -/ of elements jf--f (with C> o ). - Suppose !+P-Jf--, 13 9-t:- ~ with 13 =lo . ,4 Iv (i.e. are dextral derigrators) Then X= !J~:: c K:J. 1 13 ~ /1-,1.1~ 5*- - r/(8/j:··z; 89* B* - Q-.. ., .. 1 = /1 B* ~ o , multiplication by ~= yields X == /:;!.-.~ =- (rl <?;))-If -r ( 13 P) 1f 1 and the last expression is still valid if If d>~ F , since o * _, = o . .4 = o , then for some L3Q~ £/\.J with f3 :lo , tP=Ly,. == 6 ~/ BQ <;>-l'- * B (on multiplying on the left with B* and using Theorem 13) =cp r l{If YI .r.1. Hence The_gl.assea F /I , ~ are the range§....of the tran§.forms .,.-1£ / J This result is the main reason for referring to the pro- cess of generating the classes F ./ ~ as iAV:!3.X:§e reversio.n.11. The image of ~ ~ X is the group-theoretic transform >(-I ( ;:/ x) • If RP 7f-- .! K<A -I then p* (fl P *) ::: p "If - J fl 1'-P""'" - ,q*- = R jnderendent of P (the last expression is valid for rl == o ) , i oe • the image of A by P * equals the in'irerse reverse of 11 • Hence for all Q t 1< of su!'ficiently high degree (. L:J. '7 ± a J "*- -/ _ U/ - (11 ± B) Q -1- _ q::>"'° - ~ r q::>* - B C5l 11Q" (p.B}Q~ = (/-lt3)G-t.<4>~ - and (r:l/3) *--'.: 9.;t = /l ;If--, <?*"Q-;'f- '3 ,r- -/ - ,,., ~ -1 __ _+ /7 /,?11'- - / <(...I ..f!.9-!" Bey*" 9*" ~* (u.'3ing Theorem 1.3). -Jf -1 40 Thus we have proved THEOREM 29 is an ~.QD2,hi..[.m. I.et f = ~ *"-'. Then THEOREM 30 degre~ ?rJ , then ,q.c.19J r---' = { ~ t> a..,.,.,_k. ~" J If-_,-== f:t a nt-k ( ~ '9 ;I._, = /-l(r), a,n,d hence ,4 ( .& J * - ,4 (,,iPI . 4-1) == This follows immediately from the preceding theorem. If /1 r ,&J = ..,,, 7?7 F le a~-~ ,(]) k D If! -t-n-1e ~ k 13(.lS>J == 0 with 7?7 ~ n , thes ;7{.tP) 8{.&Jjl- 'r he image 6 (.(pr~-/ Proof.:: /I (~J * -i • t = /7{f>). fj{ I' ! ,4 ( .f5)) 6 (~) 1- } ,4 (,,,<D) I"'" 0 equals Hence by the preceding theorem nrpJ { 1+ ct&J 6(.&J *J = /Jr1>J. ~ THEOREM 32 (.32) 11 C;p) where /Ii= ,4 c& ) as in the :greqeding theorrun, for any / r - /I ( ~) r = (A& 'fo tie {)J-le. at} le f lo /J,-; .,.,.,_, .,_ - - · r l'n-t- FoR ; = C:, I, - . • J l'ln-,-1 f'J Er~ ltJ ?Tl-I• By Theorem 31, equation (.32) follows from (1.3) on applying I.et J =- f' [ I J • Then } 41 and any o; ~ <i;._ ,I - - - ,I <T?>J-/ ~ :J /l(..<JJ) ./ = J s , the system ://o-::: tr;, .1 ,<P /Jo= U,./ - - - ~ ~ m-i f } = U- 7 77 - I 0 is equivalent to /-J.{1) ~ (33) . J = E:'~ t'.A.J-le 0 ,4J Proof: ing =? r (/Jo I' I'll::: v:..le r 0 n-i-/ -f- .te. /11 t -,.,.,_<. f- - - - f- /-Jrn-1) tJ 1 o/ (I1 -- I ..I - - - ..I yn - I • By Theorem .31, equation (.3.3) follows from (20) on apply- 11r-;). Theorems 29 - .3.3 provide a rational basis for the methods employed by Heaviside for lumped linear systems, with his mystical operator f' defined as ~ ~ -I and ~ k t.h order 1 , defined as /' le [I J • , the 1 impulse matrix of the Actually, Heaviside used these methods in the case to be considered in the next section, where the elements of F are sectionally cont.inuous matrices, but the theorems of this section remained valid, mutatis mutandis. 9. Section~.J,ly_Qontinuous Matrices. Shift and Jm...Q12erators1 I.et /(;I} be a function of ,,t on L1 C: R 1 the complement L1 c to S , where R. of the domain L'.l of //;t) is scattered, i.eo I the intersection of b.. and any finite interval of R is a finite seto Then / ( r J is defined nearly everywh9re, or for nearly all ;t • ~t d'(-L} be another function of / defined nearly every- rf;i-Jand J(i) are equal near~ evernyhere' or for nearJ:I all ;t : f{;f) "7= j ( ,/-) if the values of / for which / (~.Jr= ? ( ?',) where. form a scattered set. *In this and the fol.lowing sections the theorems are given without proofs. The demonstrations will be given in detail elsewhere. THEORFJ~ 34 f(;f} ~- '} {~ J if and only if for any for all ,,£- ,2Yf...£ic;i.~Jl~l::.Y: near r but :/. r ~ IC1 :rr~J = J(i!) r • /(;1-J is continuous nearly.e:iz:.~:i;:;zwhere if its points of discontinuity form a scattered set. Simil§.rly, ;C(~) is derivable ,llill;!.rly everywhere i f .,& /: f(;I) exists for nearly all 1"' • The jum:g_ onerator , - -- ""' J*'7" f(;/} = ( ""_.,.,.,r ~ i - ~ ~-r ~(£). _) /(~J is sectionally continuous, if it is continuous nearly everywhere and .T~ /{1'-) exists for all r for which • Then values of r .T;, f (IJ :;!: o form a subset of the scattered set of discontinuities of /(;i}o ;/(L) is .§.Qattere!i on a scattered set CJ if ~(;i) is defined everywhere and vanishes on the complement of <r • <T For any scattered §et <r , / [ c ) -t- 2; df 1 ;/("r) 'l'._HEOREM 35 "f" ~d only if .f-(L) i§ ing as ,t = "7" scat·t;ered on o , ,!her§. 4r =/ c>1e o ,g.,gQord- ore ,L- ::i T. let tc°(,,/) be the class of sectionally continuous functions f(.1-J such that /(,/}=o for all sufficiently small ~ , F-n? (;f) < ;:=(,t) the class of 7?1-fold sectionally continuously derivable F-rn*Lt) C :f- { ±-J /-f ,;IJ and F,,...., (:I) the class of "n1 - fold continuously derivable (i.e. /(;I J and its first (-yy.-J) derivatives are continuous everywhere and vanish for all sufficiently small ,L- ) • the class of matrices [:f(,.!.) ] for /r/.J E r::-(./) • be equivalent to /( ,f) JI- L=t F be ~t [ .r::·[;;(J J = [ ;j(A) J J- ( ~) , and define the sum, difference and numerical multiple of elements of F as in Section 1. be the classes of matrices of the elements of r?'Y> (t J, ~ (L) ~respectively. let 43 Let L be the class of all linear operators on subsets of to F and define the elements ..<91 6 c.(. F I.! L Define • Define J,... ~ L Lt:./rrJ £r] by I [fr ,I)]=- as in Section lo a>-' J~ [ .f ( ;t) J = [ Jf. ~(~}]. 1'- Define derivator, integrator, derigrator, the classes K/ K _, I<l _, l\' 5 , K.1., Ko1. J F as in the preceding sections (except that the integrals in integrators are all from - c.o 70 ,i-) • th~ sll1fe opera ~or For MY real number t\ , define11 ~ E L . , where U(.;tJ = Let U-:::: [UC~J] (undefined for 0 ~1e I by Et\ [II,IJ] = r~rr-,u]. according as ;E < J ,C = o ) • THEOREM 36 For anY /E F , an;y- ?\.// E R., ln...J2articular Ji\ ~ U =- d¥ .. J"" ~ ~ = ~;-u / . THEO~__l7 commutes with A - : ~A ,L1 = /+ B"A • ,n k.. .,.,., Let A= IJ~a,,.,,_k p(.Y THEORi"SM 3 8 l.2*: k = s 1, - - - , -rn- 1 .J ;le_i!, o; be a derivator of degree be q_ scatt,ereu~ a funct:i,pn of A .QD R !Q S .§.Qattered .Q!l er-"=- 77? • ¢~ lhfill..!:Qr o any / t Fm and ._ ~ e F , .the ~stem A f ::: ) I ~ J~ ~A. A ¢ ~ , JA .0 f- A ct>~" J --- J eauivalent to (34) /=/! ..,(..'} f- (/lo .6;-rn-f- /J,.6Jn-r-.1 f- i;yhern A; =- t k aJ-k " THBOREM 39 (35) ~¢~E " >-. ~ - f.Qr_ fl1 =.o~ I ,,,I - - - / Yr>-/ • Xillfh A as i:g t~mzeceding th~~!!4 for any / E- !-,,,., 44 . Ok f (~t4)1er JE.i\ where for J--== 1J z1 - - - , ">n-tJ t41 = i;kaJ-le and ({ i is the set of ~Q..ontinu.i. ties of .!:) j r• What shall we mean by L..=; <!/> </J ~ F Iet X 1 'r' ~ K.J.. x <P be such that - F"0 R .1 }-' Then (clearly such F E ;=- are quotients of ele- derigrators exist, since the elements of ments of F by those of !<"- ) • ¢ ? 't' C:.ii._ ~ ¢ = e:-A._ 'r' X ¢ =:- ~ X 'r<!/J • This motivates and justifies the following definition: For any ¢ e F .1 Le:--r EA ¢> ::::: X _, .E;._ X <!/:> where X j.s any ele.[l£m1 of KJ.. X ti ~ F such that reduces to the ear.lieF. meaning bf L-~ </> if .71i e F For anY .IHEOREM 40 with - x E K~ ,\ ..f K:.) • commutes E <'\ x : ~ x <:::: x s1 . Iet f' == ,.()*-I t?'..J Then • 777 If /!{~) = THEOREM 41 ~k ltwr-le ~k of cl.egree 7n , then for any / (.36) ~ This o E F-rn /1 A-rpJf- ,,q rr& ) / = ( , .... 0 1' n... /1 + n , 'I° -,..._, ~;u A r- . .. + ,..,n-._, r ' where the l+j are as in Theorem .39. Let cf .:::rf U . Then .,.... If A(~)=- ~ie a."'n-k ~1e_ ~ 19- pol:vnomial f F-.-... are as ill, Theorem 38, then for ruJY / lmSi d t !=" , the system /l/=7 ./ Ji\ f /ii, A 'f"'AJ J A 4) f = </J A,( - - - ) J ..& A I\ 'n>-1 / == ¢7'1·/ i\ is equivalent to (.37) fl r1-7 JI = 'i -1- <A" r 77?-I -r- /l, where ,]h§. t4j are as in Theorem .38. r Yn- 7.. ~ • • • -r /J n-.-1 ) J' . A 45 These theorems form the basis of a rational 'operational calcuJ.us' completely adequate for the analysis of lu.mped linear systems driven by linear combinations of sectionalJ.y continuous ma.trices e F and •impulse matrices 1 dk ::: ~ k U for R ::: 1, z, - - - Strong Limits of Matri..~ Functions 10. A function ; ; on a class r to F is a matrix function .Qf It on "2' • let [ f,,_] be a s:equence in ;:- ' i.e 0 is a matrix function of >-i on the positive integers. A fn. t. =rt t'tJ} Then , the strong limit g£ J:, ~ }, ~t>0, is an element .,., _,>o(' f-=: [fttJ] E F such that ~ /?. (t) exists uniformly in i on every finite continuity interval of f(t} (i.e. every finite interval not containing a discontinuity of f {c) ). -{!!:;~ is clearly unique i f it exists. J..i-£ is a linear operator on the class of matrix functions of "1 to F • Similar remarks apply to the meaning and properties of k· fx ~_.a x , the strong limit of ~ §§ ;\-> a. , where fx .e including tl. • f X is s,:trongly continuous in X at ~ if £.~ h exists is a matrix function of -X on an i:nterval of and equals fc 'l'HEOBEM l.J. a.) If • J1 {c) c F {t:) of A, including a. ~ I for all X in an interva:J.. /; (CJ is continue~ in t, X for ~ly all t and for all ~ f' T , then fx =["f;rtJ] ~ strongly continuous in 7\ at a. • d~ fx , the strqng derivati.v<t_Qi: /j in fr at equaJ.s ~C\ ~ _-f . d~ is of course a linear operaCi.. , tor on the class of matrix functions of the real variable ?< · THE:OREM M interval I t X 1 t ;: If d (I:} € r (c) .Qf IC includinf£ ~ ~ for nearly all for all Y .i..n...Jm ~~ !; (t) c and for all tr f" I 1:§. continug.!!2 j.y , ~ cl~ J:/;(t}) 47 l!: o((v) is a complex-valued function o:f )( THEOREM 45 /J: = [ 0 (;/-)] a matrix function of ~9. interva1 I ~ R )c on an including Cl , and the ordinarY derivative ~ YrtJ and the strong derivative&.;,,;_ o((-y) both exist at then ct. , A1~ .('r111 /-y • h_ o(('lt) !; exists equal to 1-- . ~ La A d-v , the strong i~tegral of Tx {;.(. ~ Jr ' is all element ;?c r U.lliformly in / THIWREM 46 in k~ ~ such that £ rl< (,j.) cl_,-y:; eJdsts T (~) on every finite continvity interval of • r-r;f) for all x in the closed inter.., :.M_ .f~ ( t ) ~ ~ (a. 1 .Ir} of R.. and is continuous in J-/ Jc for nearly all f- and for J, S:Jl 1( f (a. / lr J1 then { [ f ic ( t-)] c:1. y. erlsts o It can be shown that equation (17) is valid for sectionally continuous matrices from - C><3 to /- /.::. f f ( j-) ] (and with all integrations ) , i .e • /:- p(~}r = [ L~ Gr~-rJ/rrJdr 1, (38) . Where ft/- and p(~) is a primitive derigrator A (LD) 13(,&) posi-t;ive rank and 6 ::: [ 6 ( J{ J J = .& P(~} U • .;... of With the notion of strong integral introduced above, (38) may be transformed as follows: {.!_!:, G(f-1)/(r),,i r) = [ j~{r) G (L-1Jcf 1]/ -QO since now G ( f) :::: o if r <o (this was not true in the preceding sections, where [I 1 was used instead of =- .... L7(7) [c, {~- -r) ]d r ==-OerJ E,, [Ci (:/-J -==_{./(rJ F, { /S} p(,!!Y)U jd T =-....C,/-(r}E 7 = _{./r7) c:, Pr1 1 d .i. r,, \4/l·/C/e5 cf= V ), Jd T ( f' P(fa) U }d r r u. 4g Thus we have shown (the details will be given elsewhere)1 oG f'{~JF = J~('r) c,,,. ?r?JJ> c/7', (39) -oo which is a resolution of ~(,/P) into a spectrum of •retarded impulses'. It should be carefully noted that the derigrator ;='(,{;)on the left is proper, ioeo an element of N , whereas the 'derigrator' in the in- tegrand is the inverse reverse of ~(,(}) , ioe• the image of ,P(....O) in kj • Nevertheless, since . the rank of .ff/P) is positive ?c_pJ ~ c- F i.e. the integrand, and ' the integral, belongs to r 1Yj)} ~oultf . , If the rank . of / u VJ were e> , the integral ~be interpreted in the 1weak' sense /\. . of the next section. 49 n. Weak_Limit~~t~ix__fu.nctions Let !AJ be a sequence in the complement of F in F , pf,,. are 'improper ' matrices. S.ince IJ.. is not the matrix of a function '3'>. (t} , what meaning cru1 be given to 6.f~ ? i.e. the terms Consider the •images' X.- 1 A Y and Y -'A,.· V of ~ .,,~ o0 ,., _.,..., ,, _ e>Q x, '( { /(/ • Suppose A~),} YI,, are in r and ~u(l -xA,/~~ v1~ where Since the operator ~· is homogeneous with respect both exist. ..,., ---'> oCi to multipliers c ,(/ J v~ xf'!.,, , ,( =- 7,~o<> L· vx.<I~ -= -n~o<' ~- 'Xv~ -=--XL· YAi ""'~ >t~~ o() J -X-I ~ X~- =- Y ~ · Y ~ J --c ~ ~, hence 'h - pC "1-t ~ o<> 'f/i, the image operators cannot produce unequal results. Moreover ..4,... ~" -,., - itself is included) with X = I • OI.' These considerations motivate and justify the following definition: ~~~ ""'1--'>' oC , j{he weak_J,imit of ~ as y -x./ h ~ oQ equals for any I\ f' such that J.i E F and ,, /\'-(A,: x '"' .J,..:. X "1.,_ exists. Since the existential distinction between weak ~-'=' '1-t - "" y-''\.,, and strong limits is often important, we have used the distinct notation k '"'h ~ oC' In terms of -4 it is possible to regard the improper -n-oe elements of /= as limits of sequences of proper elements E F , thereby giving the improper elements an intuitive significance which they This interpretation of the impulse matrices and the other improper elements of F is very helpful in building a 'picture 1 of adequate for the physical applications~ /= 50 The notions of weak continuity and weak derivability are evident modifications of the corresponding •strong' notions and we shall not stop here to give the explicit definitions and state the principal properties. We are especially interested in the notion of 'weak 1 integral': /, £ fx d-x equals~ "X -tJ C\. , .t~k..in~r~ !;, in X !!:Qm a :r& /, , Jx for any A: c ;fj such that X~ E F for a:ny I\ €: ( a., 1') :£ ~ . /, and the strong integral Jx~ ix exists. The weak integral , if it £ fx dx b ~ exists, is independent of X • The formal properties of .6 follow from, and are the same as those of ~ f-x f-x j x- , . t< .f : - [ froJ r §/'er) 1:-:,, rj?' =- [ .f ( t J a) /,J]_, r(t.; a, /,) ::- f { t) _/f Cf < t < b, !gr any I:, E" ~ where :=- f < a. ,;f C O'f' f >h -is undefined for - t::: a. oY I= h The proof of this -~heorem depends upon the following f (t) E F ( c) For allY LEiVllVl.A f}('rJU(t--r)cl'i' t ~ fr.ggi: J t f(7; a.)6) j r -~ It is easy to verify that b _£f/;;~.,. = j}<.,.Jurt-'rJJr ~_[fr.,->Ofa.-TJ.lr-= Hence f e<->f(-rJ U(/, -"r) U( --r) j 7 a_ _ ct:' a. j fr.,.> u(t _,,) cl'f' =f }rrJ V(, _.,.).Ir -J}'(';) t-'rJ.!rt =- Jtr'iJ U(6-7)d'r- J ftrJU(a.-T)d'r t t =- _.[,t( U(b- 'r) - U (a. -7)] J 1" : _.[ f('r; a., J,) J?-, U( '11:" ( -.c> <:13 -r) [ 51 This is comparable with the resolution (39) of the preceding section, but here the integral is weak, whereas that in (39) is strong. 52 12. Appendix Satisfactory rationalizati ons of the 'Heaviside' operational calculus have been developed recently by L. Schwartzl and J . G. Mikusinski2 (the superscri pts refer to the Bibliography), using very different methods. These methods will now be compared with the method of inverse reversion by reinterpreting the fundamental equation (32) in terms of the ideas of, first , Schwartz and then Mikusinski. Let ;=- be as in Section 1, i.e . the linear class of all matrices /:: [ f ( ;//] R. to S • Let <P where fCI:) is everywhere cont:llluous on be the class of all matrices ¢ = [ ¢«!;] where ¥(:!) is on R to C , is 0o - fold derivable everywhere, and vanishes for all sufficiently large I ~I • For any sequence i d-n J cJ , let ~ ¢ be the element ep ~ <J> such that for ~=- 1) z./ - - - _, ,.&; ¢-nc~· J ,,<j;' <Jr~) in 77 77--rc:>o o./ uniformly in ,L:- on every bounded subset of I? • Let F be the 4 to S , i.e. ?\. ti is an additive homogeneous function of c/> on i to S and ~ C\.c/ =o class of linear operators A on -n-C><:J 77 for any null sequence { ¢-r, } in ep • Let L be the class of linear operators on F to F . For ft F , ¢c i , let r fl> -=£/rtJ <j;{;f) d. Then T r/> is a linear function of </> on ~ to S , and it is easy to see that F F c F , is iso:n1orphic with a linear subclass F' of F • Hence upon identifying corresponding elements of F and F ~ If ,,&/ = [ ~ /(~) J c..C , then 53 c:::>o =£FrLJ <j(IJff = /r~)tj(LJ(0 £~r:6J ~ r-"JP, {,&f) ¢ 1 - -/.c:>o/(~)fF{./)U= (.IJ);/)¢-1- /(o) </(o). c More generally, i f LP-.,/ =F; then (40) r-1 .,,,;: /.r,/;)¢{YI}( c) d.I =- ( Jj >'/ f) r/ I- ~ ("l'l~·IJ{o) ~(o) - I (n--..)( 0) ¢ f ( o) f- -/- {-J n-iT (o) f{l5 .,..,_1 (O} _ by (-/'I) ¢ == -/AJ 9 Define Jo f L for ./ c F/ ¢ c ef. Then ( /7 ~ /) ¢ = 17r? r) ¢ -=- - ( f-7 / J ~ r;f ~ / ~ '- ef/ and in gene1•al., ( /' '7 /)f.j =(-Jn / ,6>'>1 ¢ .. Hence i f / c F , r7"'1/)¢ =- r-J'y/ / ~ncj = (-nJ £/r~) ¢c 111 (;/) H. Jn [1] <j == (-) "i" 9' In particular, ( p" J ) ¥ or : : : (- J '» ¢ r ...; ( o) /7 ( ,t.) ,;Lt- =- (-) 7-7-/ ¢ (>?-/./(0) w, rl-1' cf = l7 [ 1 J . J Equation (40) may now be written r.,., >7/ J<f = r~..,;1; ¢ f- { /rn-'ro; r + /r"'-z'ro; rl.1 -- - f- l'roJ f ~I [1 J <P / ·which is equivalent to f.., f = ,6) f- + ( ~-1 f + 4-z f .,_I- - • - +- -fi / 11} [I] /t: (o} • (41) with. ~ =/ >1 Equation (32) now follows, with the notation of Theorem 32. The elements of ;= ar~ essentially distribu.tions in the sense of Schwartzl (except that Schwartz takes /¢-==.f:?r,/J ptl)U for /. t- F , where we have £°". With Schwartz's definition and r as in Section 9, the preceding discussion~ mutatis mutandis, leads to equation (36).) We have had to take some liberties \-ri th Schwai:tz 's notation in order to preserve the form of (32), but on the whole the 54 changes seem to be improvements o For a detailed comparison of inverse reversion with Schwartzian distributions, the exact relationship between F as defined above, i.e. the class of linear operators on F t for /c F as defined in Section 8, i.eo the range of o ::/c. x ~ Kc£ if to S' , and , , must be determinedo This has not yet been doneo For the comparison with Mikusinski, let P- be the class of matrices /= {;C(L-) J where /(,i) is on R + , the non-negative real numbers, to C , is continuous for /- :> o and right-contimious at j- =o o Define sum and difference of elements of F as in Section 1 (but not numerical multiples, for reasons given below). For /; ? ,£ r .I I. ET ~j = [;; ,I-/( T) J (,:/ - r) d r 1. Then (c.f & Mikusinski2) F is a commutative ring without divisors of zero, and hence may be extended to the essentitlly unique quo- = ~ • F has no unit element (there is no fu.nc.., ,t tion tf(i) such that£ cf(r) -/(~-r) d!.'1 = /'c£-J , i.e. Dirac's tient field F 'function' is not a function). is the unit element of the field F o Let ~ = l I J _, , so that cf :=: f ll 1 o Let F 1 <:: F be the class of elements /' [~ ] for Lt- C ,. The correspondence between and 1 I,,,( 1 c F ' is an isomorphism., Hence C c:: F , upon iden- ,( t C tifying .L and r [.t) e Moreover, for any / ~ F .1 ~f == 7 [..t Jr=/' [,(~ /(xJ.U] = [,,.( /f,;&J]_, from which it follows readily that F is now a commutative algebra (linear ring) and F is a linear field (this result makes unnecessary 55 the definition of numerical multiple of elements of r as in Section 1). If .LJ I ~ r , then 1 [t,' j'(~JU-1-T'o and f::: J f = ~ f f- / ~Yl/ E F 0 J:::: f.fo" / 'rL-JcM] + [fo 1 = u<!) / + /o f [ I ] • In general i f , then -; n / = ,,<J. -,, / + ( 4_ 1 / ' + / n-, f' .. + . . • f- /a j -, } [ I ] / ioeo (41) holds, and (32) follows imrnediatelye Thus in this very speciaJ. case, with S -=- C and the domain of f(;f} restricted to e + ' the classes F ' F play the roles of the operator classes L , L in the more general theory .. BIBLIOGRAPHY The literature in the fields of abstract algebra, abstract analysis and operational calculus is enorm.ous. Extensive references to the first two fields are given in Hillel3 and Michall4 below. Gardner and Barnes9 and DoetschlO contain comprehensive bibliographies of the literature qn operationaJ. calculus. The history of this field is reviewed in Coopers. Of the remaining references below, Bana.chl2 and Hassell are standard classics, and to the author's knowledge, Schwartzl and Mikusinski2,3,4 contain the only satisfactory rationalizations of the 'HeaViside 1 calculus thus far published which do not impose unnatural (i.e .. physically irrelevant) conditions on / =:. [f{f)j E r and TJrown7 .. ; brief appraisals follow Heaviside5, Jeffreys6 * Operational CaJ.cuJ.us 1. L. Schwart0, Theorie des Distribl.1tions I, Hermann, Paris, 1950 (Actua.J.ites Scientifiques et Industrielles, Noo 1091). 2. J. G. Mikusinski, 11Sur les fondmnents du calcuJ. operatoire 11 , Studia Mathematica ll (1949) 41-70 .. 3. --------' 11 LVAnneau 11.lgibrique et ses applications dans l'analyse fonctionelle", AnnaJ.es Univ. Mariae CurieSklodowska III (1949) 1-82 *Professor A.. Erdelyi kindly called my attention to this paper. 57 -------' 11 Une nouvelle justification du calcul operatoire", Atti dell 1Acada.mia Nazionale dei Lincttl I (1950) 113-121. 5. o. Heaviside, Electro:magnetic Theory, Longon, 1899. Contains many applications of p-operators and impulse 'functions• to the ordinary and partial differential equations of electromagnetic circuit theory and field theory, guided by remarkable physica.l and mathematical insight and intuition but without adequate definitions and proofs~ 6e H. Jeffreys, Operational Methods in Mathematical Physics, Camb. Univ .. Press, 19.31 (Cambridge Tracts in Mathematics and Mathematical Physics, No. 23). Clearly outlines the relation between cogredient polY'nomials in Ll;- and .£~ (cf.. Section 7 and the proof of Theorem 25 above) but introduces and uses the p-operator with no more rigor and less vigor than Heaviside. 7e B. M. Brown, "Solution of Differential Equ.e.t.ions by Opera~ tional Methods", Ma.t.hematical Gazette XXXI (1947) 145-153. I First 'defines 1 the reciprocal R(('J of a p-p0lynomial -f-; as essentially R(~) (see Section 3 above), ~gives a 'definition' of R.(f) that can only be described as incomprehensiblee Contains vague adu.m.bra.tions of some of ·i:;he theorems in Sections 3 and 6. functionso Does not mention •impulse a 58 8.. J .. L., B. Cooper, "Heaviside and the Operational Calcul:UB", Mathematical Gazette XX:XVI (1952) 5-190 9. M. F. Gardner and Jo Lo Barnes, Transients in Linear Systems I, Wiley & Sons, New York, 1942. lOo G. Doetsch, Theorie und Anwendungen der Laplace Transforw: mation, Dover Publications, New York, 1943 .. Abstract Algebra and Analysis 11., H. Hasse, Hohere Algebra I (2nd ed.) Wo de Gruyter 1 Berlin, 1933 (Sammlung Goschen1 No.. 931) 12. s. Banach, Theorie_iies Operations Lineares, Warszawa,. 1932 .. 130 E. Hille, Functional Ana.lysis~and Semi-Groups, Amer .. Math. Soco, New York, 1948 (Colloquium Publications Volo XX.XI) 140 .A. D0 Michal, "General Differentfal Geometries and Related Topics", Bull, Amer, Math. Socs (1939) 529 - 563.