ON INTEGRAL EQUATIONS
Their Solution by Iteration and Analytic Continuation
by
ALAN J. PERLIS
B. S., Carnegie Institute of Technology (1942)
S. M., Massachusetts Institute of Technology (1949)
SUBMITTED IN PARTIAL FULFILLMENT OF THE
REQUIREIENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
(1950)
Signature Redacted Signature of Author...... W...... Department of Mathematics, May 12, 1950 Signature Redacted Certified Thesis Supervisor Signature Redacted
* *9*00 @ 00*0* - 0i0*0 **00 * e*0 Oe e 0. 0 0C 0e* O OC C e OC C 0C 0e C O C * *CCS Chairman, Departmental Committee on Graduate Students 7/ Table of Contents;
Page
Acknowledgment......
Introduction...... 1 I. Iterative proceeaures...... 12
II. Remarks on the Resolvent as an Analytic function..36 52 III. Solution by Analytic Ontinuation...... -___
Acknowledgment
The author wishes to express his sincere gratitude and appreciation to Professor
Philip Franklin for his encouragement, assistance, and guidance in the preparation and development of this thesis. ii
Abstract
This thesis is concerned with the linear integral equation with fixed finite limits, known as the Fredholm
equation of the second kind. The kernel is not assumed to be
symmetric; but, as with all other functions involved, is
assumed continuous in the unit square 0h X3 I A 1. Other than the Fredholm procedure involving infinite
determinants, solutions for large ) have not been exhibited, at least in forms not demanding prior explicit knowledge of the
characteristic values and characteristic functions. From the
standpoint of computing, it is desirable to obtain solutions
requiring only evaluation of the kernels and iterated kernels
of the equation. This thesis exhibits solutions for all
but (i) characteristic values, Ai , of the kernel, and (ii)
those A satisfying simultaneously I o / 1 and
art Ak av -Ai for some i 1, 2,... . Furthermore, the solutions are in the form of power series; or uniform limits
of power series; or as uniformly convergent infinite inte- grals; or as limits of iterative processes.
In Chapter I, are considered certain iterative processes yielding solutions of the integral equation for certain . Firstly, we consider the well-known method of successive approximations which is a simple iterative scheme defined by:
and show that convergence to the solution (always taken to be unifon) obtains if and only if I N/A, 41 where I t| is the iii characteristic value of minimum modules. An upper bound on the error after n steps is obtained.
A variant of this method, useful where numerical cal- culation is to involve operations at a fixed number of points, is obtained where sequences 1n(x)I and Kn(x-y4 converging uniformly to g(x) and K(x,y) are employed. Then the scheme defined by: 1 i k / converges to the solution if
For some values of 1 outside the circle, yields r~izt v . The method of back substitutions an iterative procedure converging to the solution. The iterative algorithm is given by: LAO C X) 4 continuous on
oll", but otherwise arbitrary, functionj where C is a fixed parameter, and the residue at tne (n - 1)th step. This procedure is actually Euler summability applied to the series whose par- tial sums are generated by the method of successive approxi- mations. The necessary and sufficient conditions for con- vergence of 1 VsiL1 to the solution of the equation are (i) I t 44 I' and (ii) Vi where V is the set of all ? satisfying + <
This method in particular yields solutions for all except, of course, any on the circle >- >.d
Since the method of successive approximations defines iv
a power series solution, by successive continuation of this
power series in a chain of circles in the X plane having
only regular points in their interiors, a solution is obtained for those A for which, simultaneously, (i) no I Nj I S- AI and 60$ = K for any i. The solution is obtained by iterating successively by successive approxima-
tions a finite set of equations, eacn a finite number of
times to yield a solution accurate to wit-oin an error whose maximum is explicitly evaluated.
Finally, Newtc's method for finding reciprocals of numbers is applied to tne solution of integral equations.
The algorithm >1 KCit
yielding a solution satisfies
which is the solution of the integral equation. The necessary and sufficient condition for cmnverging to RL'Aj4) is that:
ft& 2 Kb+.t), (t)4, - k $4: K, Et)Jt Kt k,(tS) C<
This method has the advantage of being of the second order or one whose error at the nth step is the square of that at the (n - 1)st step, whereas the previously-mentioned processes were of the first order. Indeed, by combining various "Newton's" methods, trivial processes of arbitrary order may be obtained. These "Newton's" methods are the only "polynomial" iterative forms, which converge to the reciprocal kernel. j
V
In Chapter II, is introduced a formal treatment of the
Fredholm integral operator as an element of a complete normed ring. This is done to bring forth in a clear way the rela- tionsnip between analytic functions defined by power series and the resolvent or reciprocal kernel of the integral equation. This procedure allows the use of the apparatus of complex variables in describing the resolvent. Indeed, the resolvent is exhibited as a meromorphic function of % and capable of representation by the Cauchy integral theorem.
Further, it has a Taylor's series wnose radius of caivergence is I)I and, hence, by the uniqueness theorem for such power series is the Neumann series.
In Chapter III, by using the representation of the resolvent by the Cauchy integral formula and the concept of the Mittag-Leffler star, the solution is obtained for large by using the following sunmability methods:
(i) The Euler method, which yields a solution for interior to the Borel polygon of summability, as does
(ii) The Borel integral method, whidn expresses the solution as an infinite integral in A converging at every interior point of the summability polygon.
(iii) The method of caivergence factors, which yields a solution for all N in every bounded region of the Mittag- Leffler star of the resolvent. These solutions are exhibited as limits of uniformly convergent power series. INTRODUCTION
1.
This thesis concerns itself with the specific problem of providing algorithms for solving the linear Fredholm integral equation of the second kind. The intent is parti- cularly to obtain solutions for large values of a parameter without specifically employing the characteristic functions and values of the Fredholm kernel. Throughout, the approach is from the standpoint of eventual application as numerical methods.
In the not-too-distant future, the advent of rapid large-scale digital machinery will require quite general methods for handling entire classes of functional equations.
At the present stage of coding for these machines, iterative procedures would appear to be the most advantageous for at least two reasons: (1) random errors will not tend to increase without bound in the calculations; and (2) loop or iterative coding, at present, the most widely used, places a premium upon those schemes involving a relatively simple process, or (1) sequences of such processes repeated a large number of times.
This thesis will be restricted to developing algorithms for the solution of the linear Fredholm integral equation of the second kind. These algorithms specifically indicate under what conditions solutions exist, the nature of such solutions, and the truncating error in approximate solutions (the sense of the error to be in the norm of a "uniform topology"). -2-
It is pertinent to mention those methods available at present for effecting the solution of these integral equations.
The Fredholm procedure involves the generation of two entire functions of a parameter either through an iterative procedure or by the evaluation of a succession of deter- (2) minants of ever-increasing degree.
The Hilbert-Schmidt procedure for symmetric kernels involves the representation of the solution as a generalized
Fourier series and requires a previously obtained knowledge (3) of the characteristic values and functions of the kernel.
The Schmidt method is to approximate the kernel by a degenerate kernel with a remainder, the degree of the poly- nomial being so chosen that, in norm, the remainder is less than one. Using the degenerate portion of the kernel, the procedure then is to solve a set of simultaneous equations and then, using the remainder, to generate a solution by (4) successive approximations. The method of successive approximations may be applied for a small enough value of a parameter in the equation.
The above-mentioned methods are most useful in proving existence of solutions, but have not - with the exception of the latter - been found tractable for numerical work. The numerical procedure most extensively employed thus far is to reduce the integral equation to a system of simultaneous linear equations. The problem then is an alge- braic one and is treated by any of the available methods for (5) inverting matrices. -3-
The existence theorems and the nunerical procedure cited above will be discussed in more detail at the end of the introduction.
The notation to be employed throughout the thesis is as follows:
Lower case letters of the lower alphabet, a, b, etc.,
represent numerical constants. Those of the middle alphabet represent continuous functions of a real
variable; i.e,, f, g, h, etc. Those of the upper
alphabet represent independent variables, both real
and complex.
Upper case letter, H, K, L, M, R, represent both continuous functions of two real variables and the
corresponding integral operators. G represents a set of elements forming a ring. U, V, W represent domains in the complex plane. C denotes contours in the complex
plane.
And, of course, i, J, k, n, n, m, p represent summation indices.
2.
This section deals with some of the properties of the equation under consideration.
The equation is:
&:%P~~(-~ ~ISO G (' and the following assumptions are made: -4-
1) g(x) is continuous for 04 x 1
2) K(x,y) is continuous for 0 o x, y 1
3) A is an arbitrary complex constant.
Corresponding to the inhomogeneous eCquation (1), there is the homogeneous equation:
A? (x) = A (x , y)*(y) dy (2)' It follows at once that every bounded or integrable
f(x) satisfying (1) is a continuous function for 0 & x Ar 1,
and all bounded solutions of (2) are continuous.
Any finite closed interval C a, bJ can be mapped onto
[0,13 ; hence, this restriction imposes no loss in generality. At this point, it might be mentioned that the continuity con- ditions can be replaced by integrability conditions (say, in
the L2 sense) throughout and the results correspondingly strengthened. However, it is felt that continuity condi-
tions suffice for the purpose of txiis thesis, they being more likely to occur in a given numerical problem. Obviously in the kernel, K(xy) continuity in X and integrability in Y
suffice to ensure continuity in f(x), g(x) being continuous. The basic existence theore; is:
Theorem (Fredholm) If g(x) and K(x,y) are continuous over the closed interval CO,11 and if X is such that the homogeneous equation (2) has only the solution ^2 (x)= 0, then there exists a unique continuous solution f(x) defined on C0,1 . If, hovever, the homogeneous equation has m independent solutions, then there exists a solution f(x) to the nonhomogeneous equation, if and only if g(x) is -J
-5-
orthogonal to each of the m solutions of the transposed homogeneous equation
AK(y,x) (y) dy (x) K (3)
This is the basic existence theorem for equations of
this type.
Assuming X is not a characteristic value*, the solution of (1) can be written as
f (x) = g (x) + 0 R(x,y; A ) g (y) dy (4)
R(x,y; X ) the solving kernel is to be found.
The Fredholm procedure yields R(x,y; A ) as the
quotient of two entire functions of k :
00lu R(x,y ; A ) = DC x) r; (5)
The roots of D( X) are the characteristic values of
the kernel K(x,y) and to each of them corresponds at least
one O(x) 0 0 satisfying (2). The couble iterative process defined by:
d (xy) . K(x,y), d0 = 1
dv+ - (v+ ) dv (x,x) dx (6.1)
and
dv(x,y) = K(x,y)dv + K(X,t)d,-l(ty)dt (6.2) yields the coefficients d,(x,y) and dv of (5).
* a characteristic value, means that there exists a 0(x t 0 such that (2) is satisfied for this value of . A
-6-
Since D(o) do = 1 and the characteristic values are isolated*, there exists a neighborhood about A -= 0 in which
is a holomorphic function of A . Thus: 00
and hence:
The product of these two uniformly convergent series (for Il near enough to zero) yields the Neumann series
00 Rj7 j6).in ) (6.3)
The Hilbert-Schmidt theory for symmetric kernels makes extensive use of the following theorems and
definitions:
Theorem: For a real, symmetric, continuous kernel K(x,y),
there exists at least one characteristic value 1. . Furthermore, all such characteristic values are real.
Definition: A symmetric kernel is said to be positive definite if all characteristic values are positive or,
equivalently I I
for every continuous (x). Definition: A set of continuous functions [(x)j are said
* The set of characteristic values has no limit point in the finite complex plane. See p. -7- to be complete if there exists no contnuous function, h(x), such that 51 Ztv) t~~~~ 0
For th6 case of continuous functions defined on Co,17 , every set of independent functions not complete may be made complete by the introduction of new members into the set.
Theorem: To every real symmetric kernel, there belongs a complete set of real characteristic functions 4P%9 satisfying-%
(1) (K K. X
Theorem: If * (x) form a complete ortho-normal set of characteristic functions for the real symmetric kernel K(xy) o and is uniformly convergent on Co,l , then:
This is the bilinear formula.
Definition: The iterated kernels of K(x,y) are defined by:
CC (Xj (8))
from which follows:
(i) If ( v is the set of characteristic values -8-
K(x,y), then f is the set of characteristic values for
Kxn)(x,y) and the symmetry of K(xy) implies that of KLnl(xy),
and 00
(ii) The series M., , n k 2 is uniformly convergent on co,13 and equals K(nl(xy).
Then, the Hilbert-Schmidt representation of the solu- tion of (1) for X is not a characteristic value
and the series is absolutely and uniformly convergent on
ro,l11 . The representation for R(x ) is then
RLAK ) 4jf A. %0j) (10)
Consider now the case where the kernel is degenerate;
i.e., is of the form hK(x) K( ) with the ( and ( 3 linearly independent. Then (1) becomes
NI hi,(X) S)dy (11) Sty)-: 'gL))+ NA x
Multiplying by {(U) , integrating, and introducing obvious notation yields
-: ( h hK- (12)
a set of n linear equations, whose matrix is
Thus for other than a root of the characteristic poly- nomial of the matrix, the equation (12) can be solved for 09-
QgNy) . Then substitution in (11) yields the solution of
(1) for K(x,y) a degenerate kernel.
Suppose now that K(xy) is not of the degenerate form.
However, being uniformly continuous in E OI3 , there
exists an integer N such that on an interior X interval of
length 2 and any y, the osciliation of K(xy) < 43
Dividing (o,l] into N intervals In, let xn be the midpoint
of In, then set
hn(X) is constructed in the following fashion:
oL X 1+
N IV N
NN N IV
whiich is a continuous function onI n So for X in that part of In where bn(x) 1:
I - I C I ~'CX,) k (xn,)[
For X in that associated interval where hn(X) is linear increasing: N 'Ej h. (X) (X, i~K C).,) -10-
Since 1, the above can be made hn(X) 4- hn+ 1 (X) = Hence:
nil r I KCx ) hn(x) K( )Id-xd I e z S (13) I 0 and for N large enough, the difference may be made as small as desired.
So now choose an N such that
h~C~c) 121 , S'I Ix,1) - , say, N) Kn(K) OI .1 7- then put I'CX)I) - (J 4 ) and write (1) as:
+ - CX) = CL7) 6n C-A) C4 (15) 0
+ '*A' H N11) i( ) a
Now for H(x,y), the method of successive approximations for example, (see p. 1-3 ) provides an R( Y); A ) such that
(X)= C) f ? S'0 3 IV)1 (16) where g*(y) is given by the first two terms on the right hand side of (15).
Hence:
-X)= q(X) LA,( ) 1(1)4()j (17)
where
-x) -: 5 (-A) + -A 0 (17.1) -11-
and
A,,()~ -A-~ R+ %* hp(j) c1j (17.2)
Multiplication by (-x) and integrating on X over [o,1 ]reduces the equation to the form (13). This is the method devised by
Schmidt to resolve the equation. It demands a combined solu-
tion by the inversion of a matrix and the method of successive
approximations by constructing in a straightforward fashion
the approximate degenerate kernel. It is pertinent to observe at this point that the
difficulty in effecting a solution by the method of successive
approximations depends criticlly on the value of 12'i at least
in proportion to the "normtt or "sizet of the kernel (a concept to be defined more vigorously at a later point). -12-
I. Iteration Procedures
In this chapter are exhibited five methods for obtaining the solution of (1) by iterative procedures. Throughout, it
is assumed that X is not a characteristic value of the
kernel. Otherwise, restrictions on its value are stated where pertinent. Of these methods, the first is well known and yields solutions for small enough . . The second is a rather obvious variant of the first and will have value in those calculations where it is pointless to provide more accuracy in the functions than the algorithm permits in the
solution at any stage of iteration. The third method, long known for inversion of matrices, was only recently and inde- (6) pendently of this work applied to integral equations. It was this method that led to the interpretation and formulation expressed in Chapter III. The fourth method extends to much larger values of X a solution of the integral equation by iteration and as applied to integral equations is here pre- sented for the first time. The fifth method is one of great power and is based on Newtonts method for finding reciprocals (7) of numbers. This has been applied to the inversion matrices and is here applied to integral equations. Herein is intro- (8) duced the concept of the order of an iterative procedure and from Newton's method are derived admittedly trivial variants of any arbitrary order. -13-
1. The Method of Successive Apprximations Select an arbitrary continuous function 4ALk,) defined over to,1J . Then construct a sequence Uo() defined by:
Corresponding to the se.uence u #Lo)I is the sequence
VAnLK) i defined by: \/(x) - 1An C(x) - Ut(N). From the definition of the iterated kernels, the sequence satisfies:
VnC) 5'IkCx) vr1C1)&i~ (Cho1
(%) Vo1)cb (19*2)
If the sequence MLL)1 converges uniformly to a limit function .(,Lx) , it then follows that
(1) Since LaKC) is a continuous function for every n, then t40) is a continuous function, and from
-n-a Unr ) j(7 + ne KCJ)ul follows
(ii) LA(')h lirn Un('X) is a solution. n-fto The limit function 4LtC) is independent of UOCx). Hence the solution s3 obtained is unique since if W(x) were a solution different from .Cx) , putting W,(.)z WCx) yields as solution WL) , hence equal identically to (&C) .
A necessary and sufficient condition that %1"(01con- verge uniformly to tt(%) is that i converge uniformly to zero. Since the solution is independent of the initial function - (4(NkC-A) , put Vo(.It.(y,.) where 4CY) is a characteristic function associated with 0 a characteristic value. Since Cx) satisfies
<>gyA) : -A,, KCxl) kCOJ)d then
('tC-) (20) V,~41C~) (~VL From which two necessary cufnditions for convergence follow: (i) kK) for no k since then V. (-) = 4cCl) 1 1, -- -. and there is no convergence.
(a) <| , all k, which implies (iib) Furthermore, (iib) is sufficient for uniform convergence since km COO lim-) I volt(cyo L? IO
""I I Cn2 (x I4I 05 )O ( -K i V'I
tvOwxit4 *A I" 0 1 Xi I by the Cauchy-Hadamard theorem, since 1L)11 is the radius of convergence of the series (Neumann)
KZ6 generated by (18). -15-
Hence for , IIV. 1(y.) I =0 w.hich proves convergence.
If the kernel has no characteristic val ue other than oo
the above method converges for all finite A
Summing (18) yields: (21
kL
with LACV-X) -. ),
V1+ =I ]p1.1 (22)
Hence, ihe error at the nth step is:
pn;% Cxm) PPcil O!X
and if Tn-a x I K Y"A IIL'X)l V! Xjjf I OiX91 then (23) Fri .4-ft 1
2. A Variant on the Method of Successive Approximations With the error at each stage in the process of 1., there
will exist a truncation error endowed by the arithmetic ap~proxi-
mation to the integral. This being so, the following theorem is of interest:
Theorem: Given any sequences of continuous functions fgn(x)3 and [ Kn(x,y)1 converging, over 0 X ' 1 !I uniformly to g(x) and K(x,y) respectively; then the
sufficient condition that the sequence PACL)1 defined by 4
-16-
and d
converges uniformly to the solution of (1) is that on . 7) for all but, sya, a finite number of n with M satisfying ll M < Ortainly if ( L-IL4 uniformly then tx) is the solution of (1) for
-. 3)00*inw)+ i A-U 1PA(14) dLgrL)dj The interchange of limits being permissible. The above
yields by the hypotheses on 1,a(sjand'KAtC1))Ithe solution of (1). As a matter of fact the particular sequences %(-A) S ICA)
and KAC) kAl1) show that
Now to prove uniform convergence of the Rixt)). By app-
lying the iterative process on the nth and n +1 st step one obtains
Take N large enough so that for n, N, one has
Let now
- W2 U -A +y tkk) ita)j *
f Al IV,, CA)Il E MiI -17-
Then
where
Hence the series for Vo(A) converges uniform1yto_ g ,I4
that for U4(v) converges uniformly and so must converge to the solution (tC.)
3. The ethod of Back Substitutions
This iteration procedure has been used for the inversion
of matrices and has also been treated by Bucknr for the Fredholm integral equations though his method of proof differs
from that given here for convergence, and he makes no evaluation of the truncation error.
A sequence f20c.9 is constructed as follows:
UOCXc) is continuous on [o,C7 , but otherwise arbitrary.
Un+-i(X%) = U.V()L) + Ck r'n (A (26) wfiere
Lj)=Uttx) LX)-~ A t5"a- (27) and ck is a constant to be chosen so that the iteration pro- cedure converges. Y,(x) is the residue at the nth iteration.
Hence E: m4 1gIYn )1 is the error after n iterations. It follows that
(i) Every UICA) is continuous
(ii) If, and only if, Im En , does U'jyol jk-- a* converge to a limit function tL(x) and satisfy equation ().
Furthermore, the convergence will be uniform and ) will be continuous.
(ii) V.t- gives the method of successive approxi-
mations and vh o yields only the trivially convergent
sequence (OC-)= J&(:) . Hence, we eliminate at least these two values.
Consider, then, the sequence 0( r'Cs)) . From (26) and (27), it follows that
.(28)
(,v0(.'j)C0 and hence, defining X
k=o where XLI) is an arbitrary continuous function defined by (27) .
Again, since convergence is to be independent of an initial function, let
where satisfies:
-A Q x4) 4i4A (30)
(29) now yields:
) YI+( .( ) (30) -y)d:L -Jt(
From this, necessary conditions for convergence follow: (i) 1e for any (31.1) -19-
(i) i+4 <-A' (31 2)
And, since ' o is a limit point for the or a singular value if the kernel has no H values, then from (31.2) follows a third condition:
(iii) t <(.) Ordering the D for (30) such that:
then (iii) and (+) I +4 - V ;(
4. The Method _of StepDwise Continuation
Thus far, the values of X , for which solutions by iterations have been obtained, have been quite restricted.
This method removes the restriction somewhat. Indeed, it all but solves the problem of obtaining an iterative form of the solution for any ' not a characteristic value. Later, the restriction imposed here will be lifted completely and the problem will be completely solved.
Indeed, we prove the following:
Theoreml If for a given value of ' in equation (1), no characteristic value of (1), 2.j satisfying IX < I>1 lies on the line 0= AVI- * in the complex ' plane; then, for an E7a . there exists an integer , and a set of integers n1 , n2 ,.-'-'''np such that the iteration by the -20-
method of successive approximations of a set of p Fredholm
equations, the first n, times, the second n 2 times, ... and the pth np times, yields a function Lp(70) such that
M3x,) Firstly, observe that if there exists an Rtx'cyA.) such that
f(IX) d (34)
and
)C-x t) R(1, jj X) d (35)
are satisfied, then (34) is the solution of (1). For,
multiplying (1) by RCxtj -A) and integrating on t over c o,1] yields:
0(t) R(xtN)at= (-t) R xstX) d t (36)
Sf (t*)( ' RCK,'vp) dt) (y)d . + 0
Inverting the integration:
(36.*1)
;) R(x t )
if (34) holds. By (35) the left hand side is S Lt) KCX&)*ct Hence, if (35) is satisfied, an f(x) given by (34) satisfies (1). W-
-21-
Suppose that R Ex) t j -A) is a function satisfying
(Ay) = (.* ) -t P (37)
and f(x) is a solution of (1) for 'A -::.4 . Then
-R(x ') RcxY I j /A) = (- - .)-gx p)R~-, iJti)dt (38)
This result will be proved later. We assume it now. The method of solution is as folows:
From the E are calculated the integers p, n,...n A set of kernels are calculated and the R u'I j j AI)to following iterative scheme is employed: 6" O'R A (x, ; j z I,
R(xI tj6 2k) P, (39)
The kernels ?, L~ are defined by
JR) . RCx 41;0) = (cj 'i) (40) by (38) and j)~) is defined by:
SNWRC-I;Q 1 ID
13(xi t) C t D (41) for l j jDh The function L(A,() defined by
will then be such that
1Wn) I tI4() - ) < 0 ik Al is satisfied.
To prove this, we need the following lemmas which are proved
later (p. 1Z ):
Lemma 1 The resolvent R(X)i ' ) commutes with R X; for j- 1, 2,...... wherever they both exist. .emma 2 If R(x ) has as resoivent
then the expansion of the kth iterate of R(x, j ) takes the form 00 R xL13 i .) k! Z~ LRKCnK" Ji Qq) P 1%lP1
and all iterates exist in the same domains as do their respective resolvents.
Lemma 3 If (1) has a solution given by (34) for a particular value of A, ', it has solutions for all X in a suf- ficiently small neighborhood (domain) of A'A. We now prove the theorem:
Since there exists a solution in a neighborhood of
AO , the Neumann expansion yields:
Rx ) R ,,o) ( 'A) '
------We have incorporated the term of ( )"in the sums. -23-
a uniformly ccnvergent series in A , for a suitably chosen integer p. Each term is dominated by:
rri~~~~~KIt 1 nL;J*('3 I~
where C is the maximum value attained by RC) '1 ' in the domain consisting of the intersecti on of the circles of radius I containing the origin, 2 and the straight line connecting them. We need estimates of the error induced in R (Li , jj i ) by the finite number of iterations employed in approximating the R C,Y 1,2,...... ,j. We usefor the relation: p 00
Consider the above equation for j a 1. Then each term on the right is dominated by:
C1e, Ch ~wpikp10! an s '
Now pick p, I%, and k such that for tihSN4y,
That this can be done we show later. -24-
Hence: CK 7 rI ci( ja.+r )1 RW ~,~ . 93j0') N
where C I! Vi' Accordingly:
'R (X, ' + E L(O ; ) W p r"L~ k
where C- vii. !E n I I Elk I I- I lk 211, ) Combining: +A, t) = L two&#I& R tx,%;o PRE') (.
"mo
where I (r -n)! rij I el ~ -Rri (I00 I k+4II
I* p C K R'%O ~ j
)IK hr! Vw"o I h where p is so chosen such that II. h~ I 3. Hence C 4. z st~f C" I k~ I~I~ Since n2 4 n . Now again at the pth stage:
CktA&3 )~~g )~(?I R (x, fly: R ( X) %1 T1 ? 'p
+ ~ -25- I rI NIL I I
and s o: l2gi I I COIL * UMM% ( *)1'IP r- 'RkC Lf~ )= *t"i DP! p C'at*) a a a 6 a a * fI~:o
4; nio +ta t
where Ci oel J~. I IVIV1 I and I I + ki I FIC I 1 k I a We now use these inequalities. At the first iteration:
*) i' 001x R,, N, I; ~j~o) IJ I
At the second iteration:
+ R"ps'j) R r I(X14; )rtI + aIEI.
Since the iterates are approximate, the error induced is wIgobtained from: R"ICX,'f; I E ,Sp * 4), )! dmom nj! 0 ) pp L
IvmF + ( So the 'total error is less than:
Similarly, after (p - 1) steps:
r'~. I. ;
2 C and and then L(A) and f(x) differ at most by: where the primes are introduced because the resolvents were calculated with the already absorbed in the resolvent. We have now to show that, with p such that we can find a sequence nj j:1, 2, ...,p and a k such -27- that the inequalities are satisfied. The basic inequality is and if true for k is certainly true for k < k . We pick a descending sequence based on nl. For if n1 p2P and k a =2p2p - 2, the inequality is satisfied for p > 10. And then if the values are assotgned as follows: and the inequality is maintained. Admittedly, these inequalities are poor for numerical analysis; however, they can be improved. One means of doing so is by improving the basic inequality for determining the n . Another is by improving the iterative method. This we now proceed to do by introducing Newton's method which allows improvement in both lessening the number of steps in the continuation process and in decreasing the error for a given number of iterations. 5. Newton's Method Suppose we consider the equation F(y) , 0 which we transform to y m f(y). We seek a y m Y > F(Y) = 0 and Y = f(Y). Suppose an iterative scheme constructs a sequence of yn) satisfying: -28- and let We seek obviously a criterion by which F 1 , in some norm, converges to zero. But we also seek information about the rate by which -ape IS a function of n. Now form a Taylor expansion about Y yields &+,-~ ;(Y) + ;'(Y) En + VIC Y) 7 and te,= (YOr. + I ti Y) FA - - assuming the derivatives exist. Now if the iterative scheme converges for large enough n, providing ft(Y) 0. Such an iterative process whose error at the nth stage is proportional to the error at the n - 1st (8) stage is called a process of the first order. Suppose, however, that D(Y) = 0. Then: S,,, for n large enough, and the process convergent. Hence Y)~ ( I Y) )1 ( pI()F Such a process where the error at the nth stage is roughly the square of that at the (n - 1 )th is a second order process. -29- th Hence, a k order process is one in which the error at the nth step is the kth power of that at the (n - 1 )th step. Hartree proves that having an nth order process and a (n - 1 )th order process for any equation implies the non- uniqueness of the (n - 1)th order process. Conversely, given two (n - 1 )th order processes whose nth derivatives, fn(y), are different, then there exists an nth order iterative process for solving the equation. Rather than introduce the concept of Taylor's expansion in operator equations and defining differentiation over certain combinations of operators, we use the definition of order of these processes as given by the definition. Now for F" to converge to zero in norm, it is necessary and sufficient that the first non-zero term of the expansion be less than one in norm. Thus far, the processes considered are all of the first order and are relatively inefficient on that basis. However, these first order processes suffice to provide solutions for almost all which are not characteristic values of (1). ------* The A values for which a solution has not been exhibited are those which originate in the L plane such that a characteristic value is on this ray between, X and the origin; i.e., for which a "A exists and satisfyiqs (i) a & We call these the exceptional set. -30- We now exhibit iterative procedures of arbitrary order which converge in such regions as to allow us to apply the methods of 14. to obtain solution for large i and yet use higher order processes to attain this desired result. Tnis, then, effectively solves the iterative problem for the Fredholm integral equation of the second kind, for we exhibit iterative procedures og arbitrary order solving the equation for all ;L except those lying in the exceptional set. We consider Newton's formula for obtaining the reciprocal of a real number X : By choosing yo properly, this method converges I yn to . Multiplying by x and subtracting the above equation from 1 yields: Now, letting the above equation shows Newton's metnod to be of the second order. Hence, pick yo so that xy0 satisfies II f- and convergence is attained. Consider now the equations and -31- Substitutirg (34) into (1) gives an identity and in this sense the two equations are reciprocal of one another. Thus we desire to obtain an RnL"8 ) "reciprocal" to K(x,y) by Newton's method. Hence, we generate a sequence Ra6x, %ijA) by the algorithm: R (x) A)= 7- KCa)l> and where, whenever multiplication is to take place between any (X % ) and an 2aLN' 1;) for any n, or vt th K(x,y)integra- tion over the second variable of the first factor and the first of the second is always implied; for example: RvA) ,)-A KC-x,1 Sf'*A ( A>t ) K(t~j) dt(cs With this convention in mind and for the sake of simplicity od notation, we replace (CI) and (.at) by: RO I- and R R E- Rk (v-)h C-) 3 respectively. The following may be said of commutativity: (i) If Ii'm InaU 'R,- RI=0 it V OD SIII then every Rn (o, j) commutes with K(xy) and with in the sense of (63). -32- Firstly, Ro (X, 4 -A)>> commutes with K(xy) and commutes with R(.~14 -)*. x) ; -A) Assume R . ' commutes with K(x,y) and RCx I ) . Then: R"R+l 0 -A K) RI 1(a k) -( a. Ct-x) R, ' (I- ) C-AK) R The same proof holds for R. Hence, the assertion is true by induction. Consider now the equation: 1 -(0 -A k" Rn,I= I - (1->iK)Rvj1 l-NR' = ( I - ( - A 1) 1?'")i (44.1) The equations Lf.4-() and 4-%) represent the error after (n +1) iterations. Now, consider the sequence of functions 9 (multiplication here implying an integration). Multiplying both sides of 64.2 on the right by g(y), the equation Ci-Z becomes: * The proof of this is delayed until the second part of this thesis. See p. S-L -33- If the right hand side, as n --4 c converges to zero in the sense: rn f Ili M a [- (l- NK) R~j It I)' 0 n-MjV e"tt N then aC) converges to the solution of the equation (1) and is given by: RC'IE'4A)3441 Since the limit will be uniform. Now, remembering that I-?t is R (o) and letting pA p and then by expanding out (6q) taking absolute values if: 06 WS1 and Max O t 91451(n W'K11k-. ) s convergence occurs, ai d convergence of the R s to R is uniform. Hence, we have exhibited a solution occurring for at least the values of A given by t6l) . Furthermore, the process is a second order process. This method may be applied advantageously to the procedure outlined in 14. This is so for two reasons. Along the ray, the domains of convergence given by (0j) have a maximum distance. This means that the number of jumps, p, can be effectively reduced. Furthermore, the error at each jump will be attained with much fewer iterations precisely because of the second order nature of the process. It is a trivial matter to generate, using Newton's method, procedures of arbitrary order having the same region of convergence. For consider the process given by: where (I ; order, since However, this is merely Newton's method taken n steps at a time. for k a 2n; for k a 2n+'l, it is such a process plus a first order one. Indeed, these "Newton" methods occupy a central position in such iterative procedures. For, every process of the form where ( is a polynomial in y, that converges to I is a Newton t s method of order equivalent to the degree of yg(y). Furthermore, for such polynomials, no power of y may be omitted because of the role the successive derivatives play in determining the order. For solving integral equations, we are limited to such processes as these since we have no, as yet, infinitesimal or divisional operations defined on the operators involved. 6. Summary of Chapter I Hence to conclude this chapter, we summarize by pointing out that the generation of procedures of arbitrary order and over regions which include all N but those lying in the exceptional set has solved the problem of finding iterative procedures for generating solutions of the Fredholm integral equation. With the exception of the method of 5., these algorithms bear a definite relation to the problem of obtaining the continuation of an analytic function of a complex variable and of summing divergent series. This relationsnip will be developed in Chapter II, where the problem will achieve its ultimate solution in terms of the above-mentioned concepts. -36- II a. In this chapter, we systematize the results of the preceding one and dvelop that theory enabling us to construct a solution of the integral equation (1) by analytic con- tinuation and summability methods. The relationship of the previously-described methods with the summability theory of divergent series is more than accidental, as will now be shown. This chapter is devoted to detailing a general theory of "analytic" operators; the next to a general summability theory and the relation of those metAods previously discussed with the results of this chapter. We begin by introducint the concept of a complete (to) normed ring. A set of elements I is called a normed ring if the following conditions are satisfied: (i) If Ki and K 6 , then Ki + Kj e and K 1 K e q1. (ii) If Z is a complex number, then 1< E G implies 71< C- and zK-:- K-z. (iii) qj is an Abelian group with respect to addition, the zero element of G7 being the identity of the group. (iv) Multiplication is associative and distributive: i.e., kL M e C1 implies (kL ) M ::K ( LM and RL + KM; (tAI)(<- LK +MK and for 2., V1 7 (the set of complex numbers): (-Z K=-ZK+WK=1( -+) .A -37- and (-w) K = 7. (W K) and Z(K+L) = Z k + zL ; 1- k= k-i= k (v) To every K t Gi corresponds a non-negative real number I KI called the norm of K satisfying: AKI >o K=i and the inequalities ( K) L. E G1 , % V 7. ); IK+LI 11<1 + ILI ; IKLJ t IKI ILI ; Izk- izi lKI A distance is defined on 6 . For K and LE , the distance between them is I K - L I and it satisfies: 70IKK-LI1L-HlkL L CO X -a L and the triangle inequality: IK-LI +IL-MI > 1K-I. Convergence is always in norm; and a sequence of elements t6 I K - K Ir . converges to an element K if and only if a We write this as It ie K -m K. The normed ring is said to be complete if the satisfying of the Cauchy convergence criterion, for an arbitrary F-7 and an n, n' > N( t ),q [ Kn- k- | < E implies the existence of a the K r G, towards i*ch sequence Kn converges. We now consider a complete normed ring (3 in which the elements q Eallsetof.theGZvariableacomplexarefunctions complex numbers. If KL) I Gi and if for all 1 i U C 1 a K(2) is differentiable in Z, then K(Z) is said to be analytic in U ; i.e., lim i-. -- . js=kcz) )0 and dC') c- >o 7 '9 - for 14z1 < . A function K (") 9 is analytic at Z0 if it is analytic in some open set containing 20 . It follows from the definition of the normed ring and of analyticity that the sun and products of two analytic functions is analytic in that region U C Z which is the intersection of the regions of analyticity of each function. Further, the product of a natural complex valued analytic function f(z) with an analytic function 1) E is an analytic function i j. 2. We now define the integral of K(%) e G over some region U C . Let 0) tG be a continuous function of x r U . Let Zo and Z be any two points in U connected by a rectifiable Jordan curve, C, (a "path") in U . Then if A-ZVI= 7,m- zSS- and ,, is an arbitrary point on A Z, which lies in the path, the integral defined by R(z) cdt I i M Z -)0 :i -. 39- exists independent of the division points 2 n 3 provided that b n i n I A -M 6 and is in C1 IV %^t0 The proof is exactly that outlined in Knopp*; i.e., it is possible to select two subdivisions It1 [A,} Z'of the) path ( 70 ,. F) such that for an arbitrary E , there exists a &iI)andan IV(F) and M1( ) such that for "31 X A Z i r" XI I it follows that I M N Every such sum is in C and since G is complete, the above defines a limit in called the integral. This integral has the properties: , and (iii) M L where M (7) and L is the length of the path. XIC Cauchy's integral theorem follows: If K(z) is analytic in a simply connected region U C I for which C is an arbitrary closed path entirely within U , then * Knopp, "Theory of Functions", Vol. I, Dover 1945, pp. 34 - 37. -1 -40- The proof is precisely that of KnopP*. The formation of approximating polygonal arcs leads to sequences of integrals converging in norm to SE C . The theorems 1 and 2, pp. 56 and 57* follow at once. Furthermore, it follows that if K(z) is continuous in U , 0 arbitrary and - 2 U that H('.) S Cw)clw for any path in U is in G1 and, furthermore, that H (M) is an analytic function of Z for a C U and its derivation is K(C). Cauchy's integral representation is vital to the results of the following chapter* Briefly, if K CZ) t 1 is analytic for 2 C U , then e z ku Cz) - holds for every simple, closed, positively oriented path, C, and every Z in its interior if C and its interior lie within U Proof: cI. = aGgG 1 it E c & .- by the distributive and associative laws of the ring . Now By the Cauchy integral theorem, C may be deformed to a small.path, say, a circle of radius '' about Z, so chosen that m-a x IkCt) - KC6I( t C This follows from the analyticity of K(z) for 2 I U . Then: * Loc. cit. =1 Hence .!f. d) KCZ)| ', being arbitrary, this proves the theorem. From this it follows that K(z) analytic for X V U implies that K(z) has derivations of arbitrary order which are analytic for - I U , and belong to ; in addition they are given by: " ticz) KC S) This is proved by showing that the element of G1 defined by is analytic and differentiable any number of times for 7 V U . But it is just 1<(z) i by the Cauchy integral theorem. 3. Infinite Series of ytic Function in If itz) I and Z C U C 7 ,then the set of all z for which S (( ) converges is called the domain of con- vergence of the series. If the series converges, it converges to an element in . Of particular interest are power series of the form where the Kn are elements of * For such series, the domain of convergence is a circle about Zo as center. The sequence r) K1 is of importance. The series above is absolutely convergent for I and divergent for I, [t...'z,1 ' -J I42j The definition of uniform convergence in Z I U for C is the same as that for ordinary functions of a complex variable; i.e., if given V '> 3 NL) such that for any t U C ,any n, Or N CG)) I V%+pI then i converges uniformly in U . Every power series converges uniformly in any circle concentric to the circle of convergence and of radius strictly less than the radius of convergence. For i G' and analytic for 7 C U ,we get the theorems: (i) KC)K uniformly convergent in some U C U implies the sum H(z) is analytic in U and is in Q . (ii) Integration and summation may be interchanged for every C ci . (iii) Z1 Cz) may be differentiated any number of times and each such differentiation yields a uniformly convergent series and an analytic function IC 'which is precisely the corresponding derivative of HL) . With the circle of convergence as U , it follows that every power series K Z in G for tZ U' is an analyrtic function in for t E U1 . The derived series have the same radius of convergence as does the given series and H C.( "! kG j -43- The Cauchy integral formula yields H(b = M! K and W t! k+1 a i C being the circle of radius I fI yields the Cauchy inequality W H LX MAX I W(W)) t ttC V'"+' ir T * The uniqueness theorem for analytic functions in a domain now falls out. For, let H(z) be analytic in V H(W) H(w)zI if W lies on a circle C c U and Z is interior to C , the series on the right being uniformly convergent W - . t. W along C since j any 9I G. Integration on C may be carried out term by term and Go Hgww W( W 4E ~ W"*m 1 L.tL am% Hence I Cw) Y. k(z) H LT)= 0 is analytic in Z f V and in G1. Hence every analytic function is representable by only one power series in a given domain U' . Equality of two elements of G , analytic in a common domain U' , over a set having one limit point in VJ , implies equality everywhere in U'. L where Let W(CZ)I ( , analytic, and single vilued for 7, V U is a concentric annular ring about some Zo as center; i.e., K(z) is 4 analytic for Y, <| -'.I < N . If now is an annular ring about Zo concentric to U , from the Cauchy integral formula follows the uniformly convergent "Laurent expansion": Kz)= K (- V e where Furthermore, the expansion is unique and l <1C,) + where 1%0%CI) is analytic everywhere inside I-' I " and is analytic outside I %--. oI -Yi . Of importance is the concept of analytic continuation of a function q) ianalytic in a region U C ' , Let U1 C ? be any other region such that U r% Ut It . If then () 1-:L%) V r on any set of points, having a limit point, in V11VI and HCZ) t is analytic in U, , then HN14) is the analytic continuation of k09) into U and there is only one such continuation. They are partial representations of the same function in * Of particular interest is continuation by power series. Every function WCZ) f G defined by a power series has at least one singularity on its circle of convergence. We shall be concerned only'with single valued functions, in G , of a complex variable Z * Hence, to uniformize these functions, we introduce appropriate cuts in the complex plane. The cut plane then yields a set of . values for which an analytic extension of W(L-) can be obtained d~~ 'which is single-valued and analytic. The totality of all such extensions defines the analytic functions, represented in some V by f(WC), in the entire cut plane. The construction of the cut plane will be dealt with later. h. Let be the set of functions K(x,y) continuous in 0 2 , E I. The addition of two such functions and multiplication by complex numbers is defined in the ordinary way. Multiplication between the elements of K(.Ki and K 'Citg) is defined by: and . The norm of (L /' 1) is defined by: may. I kE1I Then is a complete normed ring. The completeness follows from the uniform convergence of sequences of continuous functions to a con- tinuous function. There is no element in having the property of being an identity under multiplication. However, the ring may be extended to include such an element with the additional hypothesis that I z E + k I = z 1 + IK for 7. E and K G . We call this the ring Cy* z1- E Consider now the collection of elements f(x) which are con- tinuous on X i I and for which a norm is defined as mO% f and a metric as ay,- jf,(& SL)j . Such a set forms a complete normed linear space, B. A linear transformation of such a space, B, onto itself is called completely continuous -when it takes all bounded sets of B into compact sets in B. For K tw we define -I a linear transformation of B onto itself. This transformation is completely continuous. For, consider the sequence i K(with I p(.9) Cl for every n. Since K tC1 it follows that for every 7o there exists a 9(c.) such that I- ? I < implies I K (x )-K 4 ) Hence: so that (,.,)- ) < for all n-,:O, 1, 2. By ArzelAs theorem, this is sufficient to allow the extraction from the set of jC0 of a subset which converges in norm (or uniformly, the two implying each other in the space B). From this point, we consider the set of linear transformations of B onto itself of the form L -11 1 , where 7k t U C Z , K is the completely continuous Fredholm operator defined above and E is the identity transformation. The set of such transformations form a complete normed ring, C . 5. K.Then if Let . G- - "A VE possesses an inverse in (E ' ) ,irite:w, where TR k Cq is the resolvent of K. The following theorem on completely continuous operators is needed: -47- Either E-?K E G * possesses an inverse E E + -A A or i'- ? K has a solution t G which is not identically zero. (i) 3 converging to K for which the theorem is true. We exhibit them as the polynomials previously con- structed. For they reduce the theorem to one of simultaneous equations. Hence, for a given A , one of three cases may occur: (i) For infinitely many n, there does not exist a unique inverse bounded in norm. (ii) For infinitely many n, there exists a unique inverse bounded in norm. (iii) For infinitely many n, there exists a unique inverse but which is unbounded in norm. Consider (ii): F - A, for infinitely many n has a unique inverse E + . But, let V 4 A&=T and E'+ AA,n = V Then o li Tj T = Em.. X and Vv : For n;large TVj - E I i since I TV. -El < I V*IJ 'T-Tj Hence, t Vi exists, but V , -Tp) V T . Hence, V or V11 -- V . Hence if To, - T and V. , then T V and both are in Consider (i): For infinitely many n 3 A n satisfying for ( V 6: Now, since and b.i A,, = K t 2b sequence fn converges to f and hence - k - for t (3 Consider (iii): This implies the existence of f t3 such that: -.1 with i Let - - Then t A, h I 1 13 , which reduces to case (i). This proves the theorem. The values of A for which (\ does not exist are characteristic values of K and are singular points of A Then, the set of those A for which inverses exist, for all V 13 , of E--AK is called the resolvent set. The compliment in Z is called the spectrum of K and each A in the spectrum is called a characteristic value of K. Hence RA E for all in the resolvent set. Then, a theorem due to Riesz states that for K , the set of A for which Nt does not exist has no limit point in the finite A plane. Hence, C9 C exists for all *t except a denumerable set of V3 having no finite limit point. We then PR.OVE: Theorem: For every I and small enough *A , RA exists, is in C , and is an analytic function of "A Firstly, consider the series: 06 which is a power series in and converges absolutely for Hence, it is an analytic function for all ' IV C E where U is the open set: < Secondly, the series above satisfies: and, hence, may be properly called 6 . Since R x is the uniform limit of elements of ,it too lies in for It U * Riesz: Acta Math. 4il (1918), p. 90. Banach: Operations Tneares. -49- Further, by analytic continuation, . q in any region into which R defined above analytically can be continued. Theorem: Let now R'7 be the resolvent of K C ( . Let U be the set of those A for which R) exists and let ).4' E U . Then This follows at once from the equations and + %06 CG Er E- Further it is seen that (i), U (ii) Fv-um a2.% - :Z(.. C, R with t zo it follows that F - AR2 6 ) +P4.a - 7N R Hence .,= K Thus we get a set of equations E - "AR has resolvent ( y has resolvent . ,etc. t - 6 The solution of these equations permits the analytic continuation of We now come to the important theorem of this section. Theorem: For K f. 1 , .14 is a meromorphic function of . Firstly, the singularities of P. are denumerable and isolated. Secondly, for ?A not in the spectrum of K, R) is a regular function of A . Hence, for 7%. in the spectrum and > )1)* OP>; 0 , the Laurent expansion of exists and is in 2MIWMWMI.Wr -O- Go c Zx V+ H N- *) cc CG1 V A -- 1 t I j " -- with NHA Then rR + 1z 0x where -H ( Nq~ - x * 0 converges folo I h 41'VI' and 00 converges for The equation R S(.X-/AX Rt '- 'k LK* ) implies that the two equations Rx - R -4epj; p-, (L IT) and be true as can be shown by substituting and comparing of terns in like powers of ',4411; from (CS) it follows that Hence So Now H. EG, and hence makes B into some sub-space of B, let it be C. Then = 14.4N-1 indicates N-i makes C onto itself. Since 14-1 1 G*, every bounded part of C is compact. Riesthas * Riesz, F. Loc cit, p. 75, Lemma 5. -Si- shown this implies C is of finite dimension. Hence from ..t i) N.- it follows that, P being a mapping of C onto itself: F" H- Hence, P can be represented as a triangular matrix of finite order h,.V (Aj) such that 4-tj =Co for i 4 . Then: since for any V ' o the series must converge, it follows that the diagonal terms Q. To . So that for i 10 0 Hence - : 4 for 1tV *. Thus is a meromorphic function of A Hence, we have shown tj to be an analytic function 6f , representable as a power series about the origin with a non-zero radius of convergence. Indeed, g x is a meromorphic function of A for all finite . and has an essential singularity at 00 . * The above proof of this well-known theorem is due to Nagumo: Japanese Journal of Mathematics, Vol 1B, No. 1, 1936. III 1. From the resolvent equation: RAUR), jm7)R~(~ which holds where 9A, 2A exist, we show easily that, letting k - h + k exists at least wherever fA exists, for ~ItA: IIM Rho,, RX ji It.'PO S -f 0 and Ii " R h *6 :11r\ . Hence, (. is an analytic h-3)0 function of ;h in some neighborhood about the origin. Thus ZA has a Taylor's series development about the origin: 0 wC"N The uniqueness theae for analytic functions then yields: RD('% = v Likewise, from the Taylor expansion aboit zero: Proof of the lemma on commutativity: Since (E- X C +' t0 IF ( + A0 A K it follows that K R =Ak From the equations: and ~A. 6 it follows that 9.,-b ; i.e., that t. is the resolvent of 2. Further, I-. K commutes with 6-/4,K ; hence their inverses, where they both exist, commute also. From: follows the commutativity of with VI for any X,,. for wnich both resolvents exist. We get ( RN:o Hence 0 2. We now come to the main results of the thesis. We recall that (tA is a meromorphic function of a , analytic in the neighborhood of 'A= , and has a uniformly can- vergent power series of non-zero radius about that point. We now construct th e Mitag-Leffler star for X. From the point A C , we draw a straight line to every , a characteristic value of the kernel and a singularity of RA. Along the continuation of this ray from to 0 , the R\ plane is cut. With the value of A, at a taken as K, this uniformizes vvAA . The entire plane minus the cuts is the star of RI denoted by D. Let now D' be a bounded region interior to D and C a closed simply-connected contour in Dt. Then the Cauchy integral theorem yields for i C and AO -C . In particular, let A lie inside C. 3. Certain theorems and bfinitions in the theory of the summability of divergent series are introduced at this point. If a series Un does not necessarily coverge, we con- sider transformations of the J OAM and/or the partial sums 5,,i T Uts into I Vok and t, . These trans- 0 formations are to have the properties: (i) They define a finite sum to LkZ which exists where the proper sum of Lt" has no value. (ii) Whenever -Z$, - $ , then tin- for any IS41. o c Such a transformation is called a summability method. We consider several such types of transformations. Consider the transformation P, defined by the matrix ( d ): Then the basic theorem on such transformations is: Theorem: For P to transform such that in sn= implies 115 , it is necessary and sufficient that 0 Ala0 (ii) i a (iZi drw z I (C) For the proof, cansult the fine monograph by Szasz. We will need other types of transformations known as series to function transforms. Suppose that we have a sequence of functions ( defined for somne range of X , IZtX$ and l-on fv 6 If 4(w')A% w(1I) converges in some open interval in 0 < 4 and approaches3 r a limit S when then this defines a suimability yielding S - , A - as a generalized sum of a . Then the method is regular (satisfies ii) if and only if for some interval 04 X 4 . This condition is equivalent to A third type of summability is that where the sum is defined as means of entire functions. If with the coefficients non-negative and S NO we define a summfiability method (J) to Un A fourth method is defined by means of moment constants. -56- Let the sequence M be generated by where O(x) is a bounded increasing function of X. Let the integral exist for all n. If we define: formal term by term integration yields: dt X"d)6 We take the integral on the left as defining, where it exists, a sum for We are now in a position to prove that the method of back substitutions yields a sequence of functions convergent to the solution of (1) if the conditions (ii) aid (iii) are satisfied. To do this, we introduce the concept of Ealer summability. Consider the particular matrix ( da.) defined by: -@0W This matrix defines a regular summability method. Now, corresponding to each which is a singularity of 24 , we consider the setof points E defined by: % -J or, in terms of the above: Let V n. V . Condition (iii) comes from the SL particular value A = CA which is a singular point of For any X f V and ay A not in the star of ,it follows that Hence, if it V and then is in the star of A and, hence, a value for which is regular. We point out that V is not empty for every V contains some neighborhood of O . From0 this, it is easy to see that, for a given )4(V , both the boundary and the interior of the circle defined by contain only points of the star. If the boundary of this circle is C, then it is possible to enlarge C to C' so that the same holds true for C'. Hence, for on CI, it follows that Hence, uniformly in ' since the left hand side is a continuous function of Or. Then the series % + uniformly and absolutely for t f C *Henc e ~# 10 00~ - ~ C. A ~rrL tos0 The uniform convergence allows interchange of summation and integration: 'WI (t Pr dT)a k-- k Now depress C1 so that it lies within the circle of convergence of the Neumann series for R and obtain: 60 -n)ti-k lI 17 (% + "t CI : k) 0 " ) ( 0 L let and W00 kw~ A- h R t = which is the expansion of Ry obtained by the back substi- tion method. Hence, we have proved that the Predholm integral equation (1) has the solution (K + 'A Rx", CWIUI X) j(*j)jj *For general C& ,it is defined by - ~~and loTaw The branch we take is that for which log 1 -. 0. -J where A -Adenotes the t erms of the series above, the power of \ in each term being reduced one. The error after n steps is essentially less than (Y lrS.) Os XSI Hence XaX k k& 0 Gw let max I (-'AI raX I~ Then X- S 1 N Z () I Ii-o - lM so that, of course, RI A need lie between 0 and - 1. 5. We consider now another procedure for generating solutions of the integral equation (1) for large A , which converges in the same domain as the above, for cA chosen properly. Firstly, for the given (A , we construct the Borel polygon of summability, C , in the following manner. After drawing a ray from >: to each A, , a pole of ( , we construct normals at ) to these rays. The set of points, ) , which lie on the same side of every normal as does the origin, are interior points of the polygon of summability. We then obtain a representation of A in Q. This method of summability is that of Borel, called (B') by -60- (is ) (11) Szasz and by Hardy. Outside of Q cannot be represented by this method for, the method (BI) has the property : if a power series is (B') summable at A=)& it is a regular function of A at all interior points of the circle having the ray connecting the origin to /4 as its diameter. Hence, if / is outside Q, some tkies within -the circle having the ray 0 as a diameter which contradicts the regularity of P-A inside the circle. To prove convergence inside Q, we consider the moment constant method for which (Y) i-e . Then "= We0 *e t~dtt -A! and we are concerned with the particular series * 0 The sum of the series is defined as: et dt 'h I-) Now let be regular inside and on a closed curve C about I".o ; i.e., in some region interior to the star of S. Let I such that Q lies within C;such that, uniformly, for P16 on C: Then: n C$ O c L .t t t %OA -61- By ( ), the second integral represents an analytic function of , both integrals are bounded and hence the order may be inverted to give: No CO-t - .C e t c l) t Now, contract C to a C' inside the circle of convergence 0* of: R V3 'A ~ Hence, for p4 on CI, the above series and Go' (Ct4_ \/O )' converge uniformly in p. and A respectively. Hence, R),:~2 0t~ ( tK~A CDtV V = e dt and the solution of (1) valid in the Borel polygcn is That XSV (the region of Euler summability) implies ? E Q i.e., Q 0 V, follows from the theorem that every series (IZ) summable (E) (EUler) is summable (B'). Conversely, for any X inside Q, there exists an Q( (or p or q) sucih that -62- V , and hence is (E) summable for that A . For tends to the Borel polygon as q -.)e, p -4 od , That Ct can be so chosen follows from the following argument: if A t Q, then (LA is regular in a circle hav- ing 0 X as diameter. Then a , say % , such that O'h' , as diameter, generates a circle, C', con- centric to OQ and still containing only regu'ar values of RA Now fix ) tQ . Then for t onO', Ri( <) I for all J- on CI; and being continuous takes on a maximum. Hence, for some So for ) , fixed, in Q, take CI as the contour. 6. Applying the same approach, we now exhibit solutions of the integral equation; i.e., representations of SA , valid in every bounded region of the star. Firstly, consider real functions Antfr of the real variable t, the so-called Abelian means and variants of them, satisfying (i) IM it and the series T is given the generalized sum Then if j An(k)} are restricted so that (i) is an entire function of ? for 4 70 and that (ii) if C)0-0 N -63- I then I tb . uniformly in every in every bounded region of the star of "OWA . Then bounded region Q of the ) plane interior to the star "S AatM K A"a uniformly in ;k 0 We may take Q to be star-shaped. Then we can strictly enclose Q in a star-shaped region Q' still interior to the star of . Hence, let C be the boundary of Qt. Then: I I)A but for ' q , )& on C, no lies outside the star of Henc e Lip- b 14, ITt L -16 The limit being uniform, we write: g. 4 t f~ Now contract C within INI and hence, Ib, 'A ~1~o 0 C An(+) K rVil A V1 wt 0 since the uniformly ccnvergent series may be integrated term -64- by term. The result patently holds for every V . Hence, (1) has the solution: +-)0 'AzO Three such sets are given by: (i) (14.) (ii) ,rj -:vi.no g ~: Ijl~~. 0 6 6 (17.) (iii) J...ti.) For proofs that these actually sum uniformly interior in the star of to _LA , consult the above references. 7. Finally, we point out that the method of stepwise continuation introduced in Chapter I is precisely the pro- cedure by which a Taylor's series is continued thrcughcut the star of I . -65- 1. J. von Neumann, "Manual of Coding for Digital Computers," Advanced Institute, Princeton University, 1946. 2. We V. Lovitt, "Linear Integral Equations", Mc-0aw Hill,1924 3. E. Hellinger and 0. Toeplitz, "Integral Meichungen und Meichungen mit unendlichen Unbekannten," Ency. Math. Wiss 2, 1335-1597 (1927). 4. E. Schmidt, "Zur Theorie der Linearen und nicht Linear- en Integralgleichungen," Math. Ann., 63, 64, (1907). 5. P. D. Crout, "An Application of Polynomial Approximation to the Solution of Integral Equations Arising in Physical Problems," Jour. Math. Phys., 19, 34-92, (1943). 6. H. Buckner, "A Special Method of Successive Approximat- ions for the Fredhotm Integral Equation", Duke Math. Journ 15, 197-206 (1948). 7. E. Reich, "The Inversion of Matrices by Iterative Processes", Publication of the Office of Naval Research. 8. G. Hartree, "Some Remarks on Iterative Processes", Proc. Gumb Phil Soc, 49, 229 (1949). 9. E. Borel, "Lecons sur les Series Divergentes,", Gauthier- Villars, 1928, pp190-196. 10. M. Nagumo, "Eine Analytische Untersuchungen in Linearen Metrischen RingenV gap. Journ. of Math., 1j, 61-80(1936). 11. S. Banach, "Operations Lineares", Warsaw, 1932, 12. Q. Hardy, "Divergent Series," Oxford, 1949. -66- 13. 0. Szasz, "Divergent Seriest", Univ. of Cincinatti,1646. 14. K Knopp, "Euler Summabilitat Theorie," Math. Zeits., 15, 226-253 (1922) . Biography Alan Jay Perlis was born in Pittsburgh, Pennsylvania, on April 1, 1922. He was enrolled in the first of a suc- cession of schools, the Colfax Public School of Pittsburgh, in 1927. Then followed, from 1933 to 1939, study at Taylor Allderdice High School. He entered Carnegie Institute of Technology in September, 1939, studied chemistry, and graduated with honor on December 20, 1942. The morning of December 22, 1942, found him in an air cadet school for meteorologists of the U. S. Army Air Forc.e. Immediately upon being commissioned a 2nd lieutenant in the meteorology service, he was dispatched to Air Intelligence School and forthwith sent to Europe where he served eighteen months as an intelligence officer at the operational headquarters of the 9th U. S. A. A. F., and as a squadron weather officer with a reconnaissance squadron. He was honorably discharged in September, 1945, and entered graduate school at the California Institute of Technology. After one-and-one-quarter years of graduate study in chemistry, he transferred to the Massachusetts Institute of Technology, entering in September, 1947. He received his M. S. in mathematics in February, 1949. During the sunmers of 1948 and 1949, he held a position with the digital computer project, Whirlwind, at M. I. T.