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. 00C 0e* OOC C
e OC C
0C 0e C O
C*
*CCS
Chairman, Departmental Committee on Graduate Students
7/
Table of Contents;
Page
Acknowledgment.............
......
.................
1
Introduction. ........................................
I.
Iterative proceeaures.......................
. 12
II. Remarks on the Resolvent as an Analytic function..36
III.
Solution by Analytic Ontinuation.................
52
-___
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,
A
those
art
Ak
Ai
, of the kernel, and (ii)
satisfying simultaneously
av
-Ai
for some i
1,
2,...
I
o / 1 and
Furthermore, the
.
solutions are in the form of power series; or uniform limits
of power series; or as uniformly convergent infinite integrals; or as limits of iterative processes.
yielding solutions of the integral equation for certain
.
In Chapter I, are considered certain iterative processes
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
An upper bound
characteristic value of minimum modules.
on the error after n steps is obtained.
A variant of this method, useful where numerical calculation 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.
k
i
1
/
Then the scheme defined by:
converges to the solution if
For some values of
1
outside the circle,
t
The method of back substitutions yields
.
v
r~iz
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,
the residue at tne (n - 1)th step.
and
This procedure is
actually Euler summability applied to the series whose partial sums are generated by the method of successive approximations. The necessary and sufficient conditions for convergence of
(i)
I t 44
1 VsiL1
to the solution of the equation are
I'
and (ii)
is the set of all
?
Vi
satisfying
where
V
<
+
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
A
for those
and
60$
for which,
=
K
simultaneously,
for any i.
(i) no
I Nj I S-
AI
The solution is
obtained by iterating successively by successive approximations 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)
ft&
2
is that:
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
This is done to bring forth in a clear way the rela-
ring.
tionsnip between analytic functions defined by power series
and the resolvent or reciprocal kernel of the integral
This procedure allows the use of the apparatus of
equation.
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
is
I)I
has a
Taylor's series wnose radius of caivergence
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").
-2It 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 polynomial 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
successive approximations.
(4)
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.
-3The 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.
set of elements forming a ring.
G
represents a
U, V, W represent domains
C denotes contours in the complex
in the complex plane.
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:
('
-41
1)
g(x) is continuous for 04 x
2)
K(x,y) is continuous for 0 o x, y
3)
1
is an arbitrary complex constant.
A
Corresponding to the inhomogeneous eCquation (1), there
is the homogeneous equation:
=
A? (x)
A
(x , y)*(y)
(2)'
dy
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 conditions 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)
the closed interval
If g(x) and K(x,y) are continuous over
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
-5orthogonal 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.
X
Assuming
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
k :
quotient of two entire functions of
00lu
R(x,y ; A )
=
r;
DC x)
(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:
dv+
-
(v+
)
K(x,y), d0
d (xy) .
=
1
dv (x,x) dx
(6.1)
dv(x,y)
=
K(x,y)dv
+
and
K(X,t)d,-l(ty)dt
(6.2)
*
0(x
a characteristic value, means that there exists a
t 0 such that (2) is satisfied for this value of
.
yields the coefficients d,(x,y) and dv of (5).
A
-6do
Since D(o)
=
1 and the characteristic values are
isolated*, there exists a neighborhood about
is a holomorphic function of
00
A
.
A -= 0 in which
Thus:
and hence:
The product of these two uniformly convergent series
(for
near enough to zero) yields the Neumann series
Il
00
Rj7
j6).in)
(6.3)
The Hilbert-Schmidt theory for symmetric kernels
makes extensive use of the following theorems and
definitions:
For a real, symmetric, continuous kernel K(x,y),
there exists at least one characteristic value
1.
.
Theorem:
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
Definition:
(x).
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.
-7to be complete if there exists no contnuous function,
h(x),
such that
t~~~~
51 Ztv)
For th6 case of continuous functions defined on
Co,17
,
0
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-%
Theorem:
K. X
(K
(1)
*
If
(x)
form a complete ortho-normal set of
,
characteristic functions for the real symmetric kernel K(xy)
o
is uniformly convergent on Co,l
and
then:
This is the bilinear formula.
The iterated kernels of K(x,y) are defined by:
Definition:
CC
(Xj
(8))
from which follows:
(i)
If
(
v
is
the set of characteristic values
-8is the set of characteristic values for
f
K(x,y), then
Kxn)(x,y) and the symmetry of K(xy) implies that of KLnl(xy),
00
and
(ii)
M.,
The series
uniformly convergent on
n k 2 is
,
co,13
and equals K(nl(xy).
Then, the Hilbert-Schmidt representation of the solution 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
RLAK
)
4jf A. %
0j)
is then
(10)
i.e., is
and
hK(x) K( )
of the form
(
3
linearly independent.
with the
(
Consider now the case where the kernel is degenerate;
Then (1) becomes
NI
hi,(X) S)dy (11)
Sty)-: 'gL))+ NA
Multiplying by
{(U)
x
,
integrating, and introducing
obvious notation yields
-:
(
h
- (12)
hK
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
09QgNy)
.
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
Dividing
and any y, the osciliation of K(xy)
2
(o,l]
< 43
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
NN
whiich is a continuous function onI
So for X in that part of In where
I
-
I
C
I
N
n
bn(x)
1:
~'CX,)
k (xn,)[
For X in that associated interval where hn(X) is linear
increasing:
N
'Ej
h. (X)
(X,
i~K
C).,)
N
IV
-10-
Since hn(X) 4- hn+ (X)
1
=
1, the above can be made
nil
hn(x) K( )Id-xd I
rI 0 I KCx
)
Hence:
e zS
(13)
and for N large enough, the difference may be made as small
as desired.
So now choose an N such that
h~C~c)
,
N) Kn(K)
OI
.1 7-
(J
I'CX)I)
-
then put
and write (1)
say,
4
)
, S'I Ix,1)
-
121
as:
+
-
CX) = CL7)
0
6n C-A)
C4
(15)
H N11) i( ) a
+ '*A'
) provides an R( Y); A
for example, (see p. 1-3
(X)=
C)
f
? S'0
3 IV)1
)
Now for H(x,y), the method of successive approximations
such that
(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)
-11and
A,,()~
Multiplication by
-A-~ R+ %*
hp(j)c1j
(-x) and integrating on X over
the equation to the form (13).
(17.2)
[o,1
]reduces
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 "size t of the kernel (a concept
to be defined more vigorously at a later point).
-12I.
Iteration Procedures
In this chapter are exhibited five methods for obtaining
the solution of (1) by iterative procedures.
is assumed that
kernel.
Throughout, it
is not a characteristic value of the
X
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.
larger values of
X
The fourth method extends to much
a solution of the integral equation by
iteration and as applied to integral equations is here presented 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.
-131.
The Method of Successive Apprximations
Select an arbitrary continuous function 4ALk,)
over
to,1J .
Then construct a sequence
defined by:
defined by:
I is the sequence
u #Lo)
Corresponding to the se.uence
VAnLK) i
Uo()
defined
\/(x) -
1An
C(x) - Ut(N).
From the definition of the iterated kernels, the sequence
satisfies:
5'IkCx) vr1C1)&i~
VnC)
(Cho1
(%) Vo1)cb
MLL)1
If the sequence
(1)
then
t40)
Since
converges uniformly to a limit
then follows that
.(,Lx) , it
function
(19*2)
LaKC)
is a continuous function for every n,
is a continuous function, and from
-n-a Unr
)
j(7
+
KCJ)ul
ne
follows
Un('X)
LA(')h lirn
n-fto
The limit function 4LtC)
is a solution.
(ii)
is independent of
UOCx).
Hence the solution s3 obtained is unique since if W(x) were a
as solution
WL)
,
.Cx)
, putting
W,(.)z WCx)
hence equal identically to
A necessary and sufficient condition that
verge uniformly to
tt(%)
is that
i
yields
(&C)
.
solution different from
%1"(01con-
converge uniformly
to zero.
Since the solution is independent of the initial
function
- (4(N
kC-A)
where
put Vo(.It.(y,.)
,
0
characteristic function associated with
value.
is a
a characteristic
satisfies
Cx)
Since
4CY)
<>gyA) :
then
-A,,
KCxl) kCOJ)d
(~VL
V,~41C~)
('tC-)
(20)
From which two necessary cufnditions for convergence follow:
kK)
(i)
for no k since then
V. (-)
=
1 1,
4cCl)
-- -.
and there is no convergence.
(a)
<|
all k, which implies
,
(iib)
Furthermore,
(iib) is sufficient for uniform convergence
since
lim-)
I volt(cyo
COO
L? IO
km
05 )O
( -K
i V'I
tvOwxit4 *A
0 1 Xi
I
""I
KZ6
generated by (18).
Cn2
(x I4I
I"
by the Cauchy-Hadamard theorem, since 1L)11
convergence of the series (Neumann)
I
is the radius of
-15Hence for
IIV. 1 (y.) I =0
,
w.hich proves convergence.
If the kernel has no characteristic val ue other than
the above method converges for all finite
oo
A
Summing (18) yields:
(21
kL
with
LACV-X) -.
),
V1+ =I
]p1.1
(22)
at the nth step is:
Hence, ihe error
Cxm) PPcil
pn;%
O!X
and if
Y"A
OiX91
Tn-a x I K
V! Xjjf I
then
Fri
IIL'X)l
(23)
.4-ft 1
2.
Successive Approximations
of
A Variant on the Method
With the error at each stage in the process of 1., there
will exist a truncation error endowed by the arithmetic ap~proximation to the integral.
This being so, the following theorem
is of interest:
Theorem:
and
Given any sequences of continuous functions fgn(x)3
[ 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-
d
and
converges uniformly to the solution of (1) is
that
7)
on .
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
-.
i A-U
3)00*inw)+
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
and KAC)
%(-A)
S ICA)
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
-
f Al IV,, CA)Il
W2
U -A +y
E
tkk) ita)j
Mi
I
*
Let now
-17Then
where
Vo(A)
Hence the series for
that for
U4(v)
converges uniform1yto_
g ,I4
converges uniformly and so must converge to
(tC.)
the solution
3.
ethod of Back Substitutions
The
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
Un+-i(X%)
=
U.V()L)
[o,C7
,
but otherwise arbitrary.
+ Ck r'n (A
(26)
wfiere
LX)-~ A
Lj)=Uttx)
and
ck
(27)
is a constant to be chosen so that the iteration pro-
cedure converges.
Hence
t5"a-
E: m4
Y,(x)
IYn
1g
)1
is the residue at the nth iteration.
is the error after n iterations.
It follows that
(i)
Every
(ii)
If, and only if,
UICA)
is continuous
Im
En
jk-- a*
, does
U'jyol
converge to a limit function tL(x)
and satisfy equation ().
) will be
Furthermore, the convergence will be uniform and
continuous.
V.t-
(ii)
mations and
sequence
gives the method of successive approxi-
vh o
yields only the trivially convergent
.
(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)
X
and hence, defining
k=o
where
XLI)
is an arbitrary continuous function defined by
(27)
.
Again, since convergence is to be independent of an
initial function, let
satisfies:
where
-A Q
(30)
x4) 4i4A
(29) now yields:
-y)d:L
-Jt(
) YI+(
.(
)
(.'j)C0
(,v0
(30)
From this, necessary conditions for convergence follow:
(i)
1e
for any
(31.1)
-19-
o
And, since '
(31 2)
<-A'
i+4
(i)
or a singular
is a limit point for the
values, then from (31.2) follows
value if the kernel has no H
a third condition:
<(.)
t
(iii)
for (30) such that:
D
Ordering the
and
then (iii)
I
(+)
+4 -
;(<I
V
(31.4)
The proof of this we leave
are sufficient for convergence.
until a later section (p.
'.
4.
The Method _of StepDwise Continuation
Thus far, the values of
X
,
for which solutions by
iterations have been obtained, have been quite restricted.
Indeed, it all
This method removes the restriction somewhat.
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
characteristic value of (1), 2.j
lies on the line
then, for an
integers
0=
AVI-
*
'
in equation (1), no
'
in the complex
E7a . there exists an integer
n1 , n 2 ,.-'-'''np
IX < I>1
satisfying
,
plane;
and a set of
such that the iteration by the
-20method 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
d
f(IX)
(34)
and
)C-x t) R(1, jj X) d
are satisfied, then (34) is the solution of (1).
multiplying (1) by
c
o,1]
(-t)
RCK,'vp) dt) (y)d
and integrating on t over
-A)
R xstX) d t
+
.
(36)
0
Inverting the integration:
(36.*1)
;)
R(x t
)
'
For,
yields:
0(t) R(xtN)at=
Sf (t*)(
RCxtj
(35)
if (34) holds.
By (35) the left hand side is
Hence, if (35)
S Lt) KCX&)*ct
is satisfied, an f(x) given by (34) satisfies (1).
W-
-21Suppose that R Ex) t j
is a function satisfying
-A)
(37)
(Ay) = (.* ) -t P
and f(x) is a solution of (1) for
.
'A -::.4
Then
')
-R(x
RcxY I j /A) =
(- -
p)R~-, iJti)dt
.)-gx
(38)
We assume it now.
This result will be proved later.
The method of solution is as folows:
From the E
are calculated the integers p, n,...n
set of kernels R u'I
A
are calculated and the
j j AI)
to
following iterative scheme is employed:
O'R A (x,
;
j
6"
z
I,
(xI tj6
2k)
P,
R
The kernels ?,
by (38) and
are defined by
L~
RCx 41;0)
=
(cj 'i)
(40)
.
JR)
(39)
j)~)
SNWRC-I;Q
D
is defined by:
1
ID
Ct
13(xi t)
for
l j jDh
(41)
The function
defined by
L(A,()
will then be such that
I tI4()
1Wn)
0 ik Al
) <
-
is satisfied.
To prove this, we need the following lemmas which are proved
later (p. 1Z
j-
for
.emma 2
) commutes with R X;
The resolvent R(X)i '
1, 2,...... wherever they both exist.
R(x
If
)
has as resoivent
then the expansion of the kth iterate of
R(x,
j
)
Lemma 1
):
takes the form
00
i
R xL13
.)
Ji Qq)
LRKC
k! Z~ nK"
1%lP1
P
and all iterates exist in the same domains as do their
respective resolvents.
Lemma 3
If (1) has a solution given by (34)
value of A, ',
it
has solutions for all
ficiently small neighborhood (domain) of
X
for a particular
in a
suf-
A'A.
We now prove the theorem:
Since there exists a solution in a neighborhood of
AO
,
)
R
,,o) ( 'A)
'
Rx
the Neumann expansion yields:
-----------------------------------------------We have incorporated the term of
(
)"in
the sums.
-23a uniformly ccnvergent series in
chosen integer p.
It
rri~~~~~K
A
,
for a suitably
Each term is dominated by:
1 nL;J*('3 I~
where C is the maximum value attained by RC) '1 '
domain consisting of the intersecti on of the
radius I
2
containing the origin,
line connecting them.
induced in
R
(Li, jj i
circles of
and the straight
We need estimates of the error
)
by the finite number of
iterations employed in approximating
1,2,.......,j. We use
for
the relation:
in the
the
RC,Y
p
00
Consider the above equation for j a 1.
Then each term on
the right is dominated by:
C1e, Ch
~wpik
p10!
Now pick p,
I%,
an
s
'
and k such that for
That this can be done we show later.
tihSN4y,
-24Hence:
CK 7
where
ja.+r )1
93j0')
.
rI ci(
RW ~,~
N
C
I!
Vi'
Accordingly:
'R
)W
;
where
(X,
'
p
+ E
k
r"L~
C-
I Elk I
I lk
I-
vii. !E n I
211,
)
L(O
Combining:
t)
PRE')
=
L
two&#I&
+A,
R tx,%;o
(.
"mo
where
(r -n)!
I
~
I el
-Rri
I
(I00
rij k+4II
C
p
I*
~ jK
R'%O
)IK
hr!
Vw"o I
h
where p is so chosen such that
I I. h~
3.
I
Hence
z
C
Since n2 4
n
st~f C"
I
k~ I~I~
.
4.
Now again at the pth stage:
RT1 ?
(x,
fly:
CktA&3
R
(
)~~g
X) %1
'p
+
~
)~(?I
-25-
I
and s o:
COIL
*
'RkC
I
NIL
Lf~
)=
l2gi
*t"i
I
UMM%
DP!
fI~:o
(p
*)1'IP
r- C'
I
a
at*)
a 6
a
4;
nio
+ta
where
I
o
el
IVIV
1
I
t
Ci
J~.
and
I
FIC
+ 1k
I
We now use these inequalities.
R,, N, I;
*)
i' 001x
I
I
kiI
a
At the first iteration:
~j~o)
IJ I
At the second iteration:
R"ps'j)
+
)rtI
R r I(X14;
+
aIEI.
Since the iterates are approximate, the error induced is
wIg
obtained from:
Ivm
F
(
pp
E ,Sp *
4), )!
mom
d
0
)
I
nj! L
+
R"ICX,'f;
a a
*
I
rI
So the 'total error is less than:
Similarly, after (p - 1) steps:
r'~.
I. ;
2C
<l
and
Hence write:
and if
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
-27that the inequalities are satisfied.
The basic inequality
is
and if true for k
is certainly true for k < k .
descending sequence based on nl.
For if n1
We pick a
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).
Y = f(Y).
of
yn)
We seek a y m Y
> F(Y)
=
0 and
Suppose an iterative scheme constructs a sequence
satisfying:
-28and let
We seek obviously a criterion by which
in some norm,
,
But we also seek information about the
converges to zero.
rate by which
F1
IS
-ape
Now form
a function of n.
a Taylor expansion about Y yields
&+,-~ ;(Y)
+ VIC Y)
+ ;'(Y) En
7
and
+ I ti Y)
-
FA
-
(YOr.
te,=
Now if the iterative scheme
assuming the derivatives exist.
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) =
S,,,
Then:
for n large enough, and
the process convergent.
Y)~
Such
0.
Hence
( I Y) )1 ( pI()F
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.
-29Hence, a k
th
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.
two (n -
1
Conversely, given
)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.
for
F"
Now
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.
-30We 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
to
X :
By choosing yo properly, this method converges I
yn
.
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
and convergence is attained.
Consider now the equations
and
II
f-
-31Substitutirg (34) into (1) gives an identity and in this
sense the two equations are reciprocal of one another.
)
RnL"8
Thus we desire to obtain an
"reciprocal" to
K(x,y) by Newton's method.
%ijA) by
Hence, we generate a sequence Ra6x,
the
algorithm:
R (x)
7-
A)=
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:
( A>t
RvA) ,)-A KC-x,1 Sf'*A
) K(t~j) dt(cs
With this convention in mind and for the sake of simplicity
and (.at)
od notation, we replace (CI)
RO
by:
I-
and
E-
R
R
Rk (v-)h C-) 3
respectively.
The following may be said of commutativity:
(i) If
then every
InaU
Ii'm
it V
'R,-
RI=0
OD SIII
Rn (o,
j)
commutes with K(xy) and with
in the sense of (63).
-32Firstly,
commutes with
Assume
x) ; -A)
RCx I )
Ro
(X, 4
R(.~14
commutes with K(xy) and
-)*.
commutes with K(x,y) and
'
R .
.
-A)>>
Then:
R"R+l 0 -A K)
'
RI 1 (a k) -( a. Ct-x) R,
)
(I-
The same proof holds for R.
C-AK) 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
The equations Lf.4-()
(n +1)
9
iterations.
and
4-%)
represent the error after
Now, consider the sequence of functions
(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
(44.1)
-33-
If the right hand side, as n --4 c
converges to zero in
the sense:
rn
then
Ili M a
tN
n-MjV e"t
f
(l- NK) R~j
[-
It
I)' 0
converges to the solution of the equation
aC)
(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
Ot
91451(n W'K11k-.
)s
s to R is
convergence occurs, ai d convergence of the R
uniform.
Hence, we have exhibited a solution occurring for at
least the values of
A
given by t6l) .
process is a second order process.
Furthermore, the
This method may be applied
advantageously to the procedure outlined in
14.
This is so
Along the ray, the domains of convergence
for two reasons.
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
For consider the process given by:
of convergence.
(I
where
;
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
Indeed, these "Newton" methods occupy a
a first order one.
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.
-36II
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 conThe relationship of the
tinuation and summability methods.
previously-described methods with the summability theory of
divergent series is more than accidental, as will now be
This chapter is devoted to detailing a general theory
shown.
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)
A set of elements
normed ring.
I
is called a normed ring
if the following conditions are satisfied:
If Ki and K
(i)
K
If
and
Z
is a complex number, then
qj
is an Abelian group with respect to addition,
the zero element of
(iv)
1< E G implies
zK-:- K-z.
and
(iii)
i.e.,
e
e q1
(ii)
71< C-
Kj
.
K1
, then Ki +
6
G7
being the identity of the group.
Multiplication is associative and distributive:
M
kL
e C1
implies
(kL ) M ::K ( LM
RL + KM; (tAI)(<-
and
and for
2.,
V1
7
LK +MK
(the set of complex numbers):
(-Z K=-ZK+WK=1( -+)
.A
-37and
(-w) K
=
7. (W K)
1- k= k-i= k
;
Z(K+L) = Z k + zL
and
corresponds a non-negative real number I KI
(v) To every K t Gi
called the norm of K satisfying:
AKI >o
K=i
and the inequalities ( K) L. E G1 , % V
IK+LI
);
IKLJ t IKI ILI ;
11<1 + ILI ;
IK
Izk- izi lKI
6 . For K and LE
A distance is defined on
distance between them is
7.
, the
L I and it satisfies:
-
L
CO
X -a L
and the triangle inequality:
+IL-MI
IK-LI
>
1K-I.
t6
I K - K Ir
.
Convergence is always in norm; and a sequence of elements
a
converges to an element K if and only if
We write this as
K -m K.
It ie
The normed ring is said to be complete if the satisfying of the
Cauchy convergence criterion, for an arbitrary
n, n' >
K r G,
N( t ),q [ Kn-
towards
k-
| <
E
i*ch the sequence
and an
F-7
implies the existence of a
Kn
We now consider a complete normed ring
converges.
(3
in which the elements
q
complex numbers.
If
KL) I Gi
and if for all
differentiable in Z, then
1 i U C1
a
K(Z) is said to be analytic in
K(2) is
U
; i.e.,
dC')
for
14z1 <
A function
.
js=kcz)
)0
c- >o
7
'9
.
and
--
-
lim i-.
9
K (")
is analytic at Z 0 if it is analytic in some open
.
set containing 2 0
It follows from the definition of the normed ring and of
analyticity that the sun and products of two analytic functions is
which is the intersection of the
U C Z
analytic in that region
regions of analyticity of each function.
Further, the product of a
natural complex valued analytic function f(z) with an analytic function
1) E
i
is an analytic function
j.
We now define the integral of
Let
Z
0)
tG
U
G
over some region U C
be a continuous function of x r U
be any two points in
C, (a "path") in
K(%) e
.
U
Then if
and
A Z,
Let Zo and
connected by a rectifiable Jordan curve,
A-ZVI= 7,m- zSSon
.
.
2.
,,
is an arbitrary point
which lies in the path, the integral defined by
R(z) cdt
I iM
-)0
Z
:i
-. 39-
2
exists independent of the division points
n 3
provided that
b n i n IA
-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
,
and M1( )
there exists a
&iI)and
an IV(F)
such that for
"31 X
AZ i
<Fs
and
I
r" XI
I
it follows that
N
M
Every such sum is in
C
and since
called the integral.
is complete, the above defines a
This integral has the properties:
,
limit in
G
ML
and
(iii)
where
(7)
and
XIC
Cauchy's integral theorem follows:
M
L
is the length of the path.
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
-40The proof is precisely that of KnopP*.
The formation of approximating
polygonal arcs leads to sequences of integrals converging in norm to
SEC
.
The theorems 1 and 2, pp. 56 and 57* follow at once.
Furthermore, it follows that if K(z) is continuous in
arbitrary and
-
2 U
for any path in
U
is in
analytic function of
Z for
,
0
that
S
H('.)
U
Cw)clw
G1
and, furthermore, that
aCU
H (M)
is an
and its derivation is K(C).
Cauchy's integral representation is vital to the results of the
Briefly, if
K CZ) t
ku
Cz) -
is analytic for 2 C U
1
,
following chapter*
then
z
holds for every simple, closed, positively oriented path, C, and every Z
U
in its interior if C and its interior lie within
cI.
Proof:
it
=
aGgG
E
c
&
e
by the distributive and associative laws of the ring
1
.-
.
Now
By the Cauchy integral theorem, C may be deformed to a small.path, say, a
circle of radius
m-a x
''
about Z, so chosen that
IkCt) - KC6I(
t C
This follows from the analyticity of K(z) for
*
Loc. cit.
2 I U .
Then:
=1
Hence
KCZ)| <E
.L
I
K
', being arbitrary, this proves the theorem.
X V U
that K(z) analytic for
From this it follows
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
is just
1<(z) i
Infinite Series of
I
itz)
If
S
all z for which
((
vergence of the series.
.
element in
where the
. But it
by the Cauchy integral theorem.
Kn
3.
ytic Function in
Z C U C 7
and
)
the set of
converges is called the domain of con-
Of particular interest are power series of the form
are elements of
*
For such series, the domain
of convergence is a circle about Zo as center.
is of importance.
,then
If the series converges, it converges to an
The sequence
The series above is absolutely convergent for
I
and divergent for
I,
[t...'z,1
'
.!f. d)
r) K1
-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.,
such that for any
t U
C
,any
V%+pI
I
n, Or N CG))
then
NL)
3
V '>
i
U
converges uniformly in
.
if given
Every power series converges uniformly in any circle concentric to
the circle of convergence and of radius strictly less than the radius of
convergence.
i
For
G'
and analytic for 7 C U ,we
get the
theorems:
KC)
K
uniformly convergent in some
the sum H(z) is analytic in U
for every
Q
C U
implies
Integration and summation may be interchanged
C
ci
Z1
(iii)
.
(ii)
and is in
U
.
(i)
Cz)
may be differentiated any number of times and
each such differentiation yields a uniformly convergent series and an
HL)
derivative of
'which is precisely the corresponding
IC
.
analytic function
With the circle of convergence as
series
for
K Z
t E U1 .
in
G
for
tZ U'
,
U
it follows that every power
is an analyrtic function in
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
= M! K
H(b
and
W t! a i
k+1
C being the
circle of radius I fI
yields the Cauchy inequality
W H LX
*
ir T
V'"+'
t ttC
I W(W))
MAX
The uniqueness theorem for analytic functions in a domain
now falls out.
H(w)zI
H(W)
C
if W lies on a circle
cU
and Z is interior to
on the right being uniformly convergent
j
V
For, let H(z) be analytic in
I9 G.
any
W- .
C
t. W
, the series
along C since
Integration on C may be carried out term
by term and
W(
W
4E
am%
L.tL
~
W"*m1
Go
Hgww
Hence
I
Y. k(z)
0
is analytic in Z f V
G1.
and in
Hence every analytic function
is representable by only one power series in a given domain
Equality of two elements of
G
a set having one limit point in
L where
Let
W(CZ)I (
,
U'
.
H LT)=
Cw)
, analytic in a common domain U'
VJ
,
,
over
implies equality everywhere in U'.
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
l <1C,)
Furthermore, the expansion is unique and
1%0%CI)
is analytic everywhere inside
is analytic outside
I
%--.
oI -Yi
I
I-'
"
and
.
where
+
where
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
analytic in U,
of the same function in
*
Every function
They are partial representations
Of particular interest is continuation by
WCZ) f G
defined by a power series has
We shall be
at least one singularity on its circle of convergence.
concerned only'with single valued functions, in
variable Z
*
.
G
,
of a complex
Hence, to uniformize these functions, we introduce
appropriate cuts in the complex plane.
of
is
, then HN14) is the analytic continuation of k09) into U
and there is only one such continuation.
power series.
HCZ) t
and
The cut plane then yields a set
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
the entire cut plane.
by
f(WC),
in
The construction of the cut plane will be dealt
with later.
h.
be the set of functions K(x,y) continuous in 0 2
Let
The addition
of two such functions and multiplication by complex numbers
is defined in the ordinary way.
K(.Ki
and K 'Citg)
and
.
may.
Then
, E I.
Multiplication between the elements of
is defined by:
(L /' 1)
The norm of
is defined by:
I kE1I
is a complete normed ring.
The completeness follows from
the uniform convergence of sequences of continuous functions to a continuous function.
There is no element in
being an identity under multiplication.
having the property of
However, the ring may be
extended to include such an element with the additional hypothesis that
Iz E +k I =
for 7.
E
and
K
G
.
z 1 + IK
We call this the ring
Consider now the collection of elements
tinuous on
mO%
f
Xi I
f(x)
E
Cy* z1-
which are con-
and for which a norm is defined as
ay,- jf,(&
and a metric as
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.
completely continuous.
such that
i
For, consider the sequence
I p(.9)
K(with
K tC 1
This transformation is
it follows that for every
I- ? I <
Cl
for every n. Since
7o
there exists a
9(c.)
implies
I K (x
)-K
4 ) <r
Y I
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
7k t U C Z
where
, K is the
completely continuous Fredholm operator
The set of such
defined above and E is the identity transformation.
C
.
transformations form a complete normed ring,
5.
K.Then
if
Let
G- - "A VE
.
possesses an inverse in
TR k Cq
where
The
needed:
(E
'
)
,irite:
w,
is the resolvent of K.
following theorem on completely continuous operators is
-47-
or
E G*
E-?K
Either
i'- ? K
possesses an inverse
has a solution
t
G
E + -AA
E
which is not identically
zero.
3
(i)
converging to K for which the
theorem is true.
structed.
We exhibit them as the polynomials previously con-
For they reduce the theorem to one of simultaneous equations.
A
Hence, for a given
(i)
, one of three cases may occur:
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):
Tj
li
Hence, t Vi
V,
, then
or
T
(
V
-El
<
I V*IJ
'T-Tj
V
--
V
and both are in
. Hence if
3
An
To, -
satisfying
6:
and
Now, since
converges to f and hence
Consider (iii):
that:
Vv :
. Hence,
V11
Consider (i): For infinitely many n
for
and
-Tp) V T
E'+ AA,n = V
and
since I TV.
exists, but
V.
= Em.. X
i
V
and
T
TVj - E I
For n;large
A&=T
But, let V 4
.
o
for infinitely many n has a unique
A,
E+
inverse
Then
F-
b.i
- k
A,,
-
= Kt
2b sequence fn
for
This implies the existence of
t
(3
f t3
such
T
i
with
Let
t A,
Then
I
h
proves the theorem.
1
13
-
-
-.1
, which reduces to case (i). This
The values of
A
for which
(\
does not
A
exist are characteristic values of K and are singular points of
Then, the set of those
V
13
Z
in
of
,
E--AK
A
for which inverses exist, for all
is called the resolvent set.
The compliment
is called the spectrum of K and each
A
in the spectrum is
RA
E
for all
called a characteristic value of K. Hence
in the resolvent set.
Then, a theorem due to Riesz states that for
set of
A
for which
finite
A
plane.
denumerable set of
Theorem:
is in
V3
For every
C
,
,
the
Nt does not exist has no limit point in the
C9 C exists for all *t
Hence,
K
except a
We then PR.OVE:
having no finite limit point.
and small enough *A
I
RA
,
exists,
and is an analytic function of "A
Firstly, consider the series:
06
which is a power series in
Hence, it is an analytic
the open set:
and converges absolutely for
<
uniform limit of elements of
CE
where U
is
Secondly, the series above satisfies:
and, hence, may be properly called
*
' IV
function for all
6
,it
Riesz: Acta Math. 4il (1918), p. 90.
Banach: Operations Tneares.
.
Since
too lies in
Rx
is the
for It
U
-49. q
Further, by analytic continuation,
in any region into
defined above analytically can be continued.
which R
Let now
Theorem:
A
set of those
be the resolvent of
R'7
exists and let
R)
for which
K C(
).4'
be the
Let U
.
E U
.
Then
This follows at once from the equations
and
E- + %06 CG
Er
is seen that
Further it
U
(i),
Fv-um
(ii)
with
t zo
it
R
:Z(..C,
a2.% follows that
F - AR2 6 ) +P4.a- 7N
.,=
Hence
R
K
Thus we get a set of equations
E - "AR
. ,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
,
Firstly, the singularities of
isolated.
Secondly, for
regular function of
>
is in
)1
)*
A
; 0
OP>
?A
is a meromorphic function of
.14
not in the spectrum of K,
Hence, for
.
,
are denumerable and
P.
7%.
R)
is a
in the spectrum and
the Laurent expansion of
exists and
.
( y has resolvent
has resolvent
2MIWMWMI.Wr
-O-
Go
c
H N- *)
V+
Zx
cc
with
A --
CG
1V
NHA
1
t Ij
" --
Then
+ 1z
rR
0x
where
Nq~ - x 0
converges
*
-H (
folo
I h 41'VI'
and
00
converges
for
S(.X-/AX
Rt '
'k LK*
)
R
-
The equation
implies that the two equations
Rx - R
-4epj;
(L IT)
p-,
and
be true as can be shown by substituting and comparing of terns in like
powers of
from
(CS)
',4411;
it follows that
Hence
So
Now
let it be C.
Since
*
H.
Then
14-1 1 G*,
Riesz, F.
EG,
and hence makes B into some sub-space of B,
=
14.4
N-1
indicates
N-i makes C onto itself.
every bounded part of C is compact.
Loc cit, p. 75, Lemma 5.
Riesthas
-Sishown this implies C is of finite dimension.
it
Hence from
N.-
i)
..t
follows that, P being a mapping of C onto itself:
F" HHence, P can be represented as a triangular matrix of finite order
(Aj)
diagonal terms
-
function of
:
To
Q.
4
i4
for
.
Then:
the series must converge, it
V 'o
since for any
Hence
4-tj =Co
such that
for
0
is a meromorphic
*. Thus
1tV
follows that the
i 10
So that for
.
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.
A
singularity at
*
for
00
all finite
Indeed,
.
gx
is a meromorphic
and has an essential
.
function of
h,.V
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),
which holds where
letting
jm7)R~(~
exist, we show easily that,
2A
9A,
k - h +k
exists at least wherever
fA
exists, for
~ItA: IIM
ji
It.'PO
Ii "
and
h-3)0
R h *6
function of
Thus
ZA
;h
:11r\
.
Rho,, RX
0
-f
S
Hence,
(.
is an analytic
in some neighborhood about the origin.
has a Taylor's series development about the
origin:
w
C"N
0
The uniqueness theae
yields:
RD('% =
for analytic functions then
v
Likewise, from the Taylor expansion aboit zero:
Proof of the lemma on commutativity:
X C +' t0
(E-
Since
IF (
+ A0
AK
it follows that
K
R =Ak
From the equations:
and
~A.
it
follows that
6
9.,-b
; i.e.,
that t.
is
the resolvent of 2.
Further,
commutes with
K
I-.
6-/4,K
;
hence their inverses, where they both exist, commute also.
From:
with
follows the commutativity of
VI
X,,.
for any
for wnich both resolvents exist.
(
We get
RN:o
Hence
0
2.
We now come to the main results of the thesis.
recall that
(tA
is a meromorphic function of
,
analytic
, and has a uniformly can-
'A=
in the neighborhood of
a
We
vergent power series of non-zero radius about that point.
From the point
A C
,
for
X.
we draw a straight line to every
,
We now construct th e Mitag-Leffler star
a characteristic value of the kernel and a singularity of RA.
to
Along the continuation of this ray from
R\
plane is cut.
With the value of
as K, this uniformizes vvAA
is the star of
RI
.
A,
ata
0
, the
taken
The entire plane minus the cuts
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
i C
for
AO -C
and
.
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.
Un
If a series
does not necessarily coverge, we con-
sider transformations of the
5,,i
T
Uts
into
I
Vok
J OAM
and/or the partial sums
and
t,
.
0
These trans-
formations are to have the properties:
(i)
They define a finite sum to
exists where the proper sum of
Lt"
which
has no value.
$,
-Z - $ , then
(ii) Whenever
tin-
LkZ
for any IS41.
oc
Such a transformation is called a summability method.
We consider several such types of transformations.
the transformation P, defined by the matrix ( d
Consider
):
Then the basic theorem on such transformations is:
0
Ala
Theorem:
For P to transform
implies
115
(ii)
,
i
such that
in
sn=
it is necessary and sufficient that
0
a
drw z I
(iZi
(C)
For the proof, cansult the fine monograph by Szasz.
We will need other types of transformations known as
Suppose that we have a sequence
series to function transforms.
of functions
defined for somne range of
(
X
,
I
ZtX$
and
l-on
fv 6
4(w')A%w(1I)
If
4 and approaches
3r
a limit S
0 <
interval in
-
converges in some open
then this defines a suimability yielding S
A
as a generalized sum of
a
.
Then the method is
,
-
when
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.
with the coefficients
If
non-negative and
S
NO
we define a summfiability method (J) to
Un
A fourth method is defined by means of moment constants.
-56Let the sequence
where
M
be generated by
O(x) is a bounded increasing function of X.
integral exist for all n.
Let the
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
-@0W
by:
This matrix defines a regular summability method.
of
24
corresponding to each
, we consider the setof points
%
Now,
which is a singularity
E
defined by:
-J
above:
or, in terms of the
V
Let
particular value
X f V
For any
n.
SL
.
V
A
=
Condition (iii) comes from the
CA
which is a singular point of
and ay
not in the star of
A
,it
follows that
it V
Hence, if
then
A and, hence, a value for
is in the star of
which
every V
and
is regular.
We point out that V is not empty for
contains some neighborhood of
O . From
0 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
Hence,
uniformly in
'
follows that
since the left hand side is a continuous function of Or.
%
+
Then the series
~#
~
-
f C
t
uniformly and absolutely for
10
C.
e
*Henc
00~
A
to
s0
~rrL
The uniform convergence allows interchange of summation and
integration:
(t
'WI
k--
Pr dT)a
k
Now depress C1 so that it
lies within the circle of
convergence of the Neumann series for
60
-n)ti-k
R
and obtain:
lI
CI
17 (% + "t
: 0
k)
" ) ( 0L
and
let
W00
kw~ A-
h
Rt =
which is the expansion of
tion method.
Ry
obtained by the back substi-
Hence, we have proved that the Predholm integral
equation (1) has the solution
+ 'A
(K
*For
general
C&
,it
The branch we take is
Rx", CWIUI X) j(*j)jj
is defined by
-
~~and
that for which log 1
-.
0.
loTaw
-J
where
of
-Adenotes the t erms of the series above, the power
A
\
in each term being reduced one.
steps is essentially less than
Hence
The error after n
(Y
Os XSI
lrS.)
X
Xa
0
Gw
I
(-'AI
max
let
k
k&
I~
raX
X-
Then
S
I
1 N Z ()
so that, of course,
RI A
-
Ii-o
lM
need lie between 0 and
-
1.
5.
We consider now another procedure for generating
solutions of the integral equation (1) for large
converges in the same domain as the above, for
A
cA
,
which
chosen
properly.
polygon of summability,
C
After drawing a ray from >:
we construct normals at
points,
)
,
which lie
)
(A
,
,
we construct the Borel
in the following manner.
to each
A, , a pole of
to these rays.
(
,
Firstly, for the given
The set of
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 )
Szasz
(11)
Outside of Q
and by Hardy.
cannot be
represented by this method for, the method (BI) has the
property :
A
it is a regular function of
at all interior points of the
circle having the ray connecting the origin to
diameter.
A=)&
if a power series is (B') summable at
Hence, if
is outside Q,
/
-the circle having the ray
the regularity of P-A
0
/4
as its
t
kies within
some
as a diameter which contradicts
inside the circle.
To prove convergence inside Q, we consider the moment
i-e
We0*et~dtt -A!
(Y)
constant method for which
"=
.
Then
and we are concerned with the particular series
*
0
The sum of the series is defined as:
dt
et
'h I-)
Now let
about
S.
I".o
Let
be regular inside and on a closed curve C
; i.e., in some region interior to the star of
I
uniformly, for
P16
such that Q lies within C;such that,
on C:
Then:
n
C$
L
O
.t
c
t
t %OA
-61), the second integral represents an analytic function
By (
, both integrals are bounded and hence the order
of
may be inverted to give:
No
e t c l) t
- .C
CO-t
Now, contract C to a C' inside the circle of convergence
0*
of:
~
Hence, for
p4
'A
R V3
on CI, the above series and
Go'
converge uniformly in
p.
(Ct4_ \/O )'
and
respectively.
A
Hence,
0t~
R),:~2
V
CDtV
=
( tK~A
e dt
and the solution of (1) valid in the Borel polygcn is
That XSV (the region of Euler summability) implies
?EQ
i.e., Q 0 V, follows from the theorem that every series
(IZ)
Conversely, for any
summable (E) (EUler) is summable (B').
X inside Q, there exists an
Q(
(or p or q)
sucih that
-62and hence is
,
(E)
A
summable for that
tends to the Borel polygon as q -.)e,
p
-4
od
For
.
,
V
That Ct can be so chosen follows from the following
as diameter.
all
)
J-
a
Then
%
, say
containing only regu'ar values of RA
centric to OQ and still
Now fix
is regular in a circle hav-
, as diameter, generates a circle, C', con-
O'h'
such that
(LA
,
0 X
ing
A t Q, then
if
argument:
tQ
.
Then for
t
onO',
Ri(
for
<) I
on CI; and being continuous takes on a maximum.
Hence, for some
So for
)
,
Q,
fixed, in
take CI as the contour.
6.
Applying the same approach, we now exhibit solutions of
the integral equation; i.e., representations of SA
in every bounded region of the star.
functions
Antfr
,
valid
Firstly, consider real
of the real variable t, the so-called
Abelian means and variants of them, satisfying
(i)
and the series
Then if
IM
it
T
j An(k)}
is an entire function of
C)0-0N
is given the generalized sum
are restricted so that (i)
?
for
4
70 and that (ii) if
-63-
I
then
uniformly in every
.
I tb
bounded region of the star of
bounded region Q of the
;k
AatM K
Q
to be star-shaped.
enclose Q in a star-shaped region Q'
.
A"a
0
We may take
star of
Then in every
plane interior to the star
)
"S
uniformly in
.
"OWA
Then we can strictly
still interior to the
Hence, let C be the boundary of Qt.
Then:
I
but for
'
I)A
q
,
lies outside the star
)& on C, no
Henc e
of
Lip-
b
ITt L
14,
-16
The limit being uniform, we write:
g.
4
Now contract C within
t
INI
f~
and hence,
'A
Ib,
0
~1~o
C
An(+) K rVil A V1
wt 0
since the uniformly ccnvergent series may be integrated term
-64V
.
The result patently holds for every
by term.
(1) has the solution:
Hence,
+-)0
'AzO
are given by:
Three such sets
(i)
(14.)
,rj -:vi.no g
(ii)
~: Ijl~~.
(17.)
J...ti.)
(iii)
For proofs that these actually sum
to
interior in the star of
0 6 6
_LA
uniformly
,
consult the above
references.
7.
Finally, we point out that the method of stepwise
continuation introduced in Chapter I is precisely the pro-
star of
I
.
cedure by which a Taylor's series is continued thrcughcut the
-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 Linearen Integralgleichungen," Math. Ann., 63, 64, (1907).
5.
P. D. Crout, "An Application of Polynomial Approximation
to the Solution of Integral Equations Arising in Physical
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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.