I ntrnat. J. Math. & Math. Sci.
285
(1978)285-296
Vol.
A REPRESENTATION THEOREM FOR OPERATORS
ON A SPACE OF INTERVAL FUNCTIONS
J. A. CHATFIELD
Department of Mathematics
Southwest Texas State University
San Marcos, Texas 78666 U.S.A.
(Received May 4, 1978)
ABSTRACT.
numbers.
Suppose N is a Banach space of norm
All integrals used are of the
main theorem
ETheorem 3
I"
and R is the set of real
subdlvlslon-refinement type.
The
gives a representation of TH where H is a function
from RxR to N such that H(p
+,
p+),
H(p,
p+),
H(p-, p-), and H(p-,p) each exist
for each p and T is a bounded linear operator on the space of all such functlons H.
TH
In particular we show that
+
+
+
+
(1)/bfHd
(xi_l,
xi_ I)-H (xi_ I Xi_l) 8 (xi_ I)
a
i=l
+
,,
where each of
i=l
(xi,x )-.Cxi,xi)
and @ depend only on T,
Is of bounded variation, @ and
@ are 0 except at a countable number of points,
depending on H
and
H(p,p+)-H(p+,p+)#0
{xl}
or
fH
is a function from R to N
i=l denotes the points p in
EH(p-,p)-H(p-,p-) #0.
a b.
for which
We also define an interior
J. A. CHATFIELD
286
interval function integral and give a relationship between it and the standard
interval function integral.
i.
INTRODUCTION.
Let N be a Banach space of norm
I"
and R the set of real numbers.
The
purpose of this paper is to exhibit a representation of TH where H is a function
H(p+,p+),
from RxR to N such that
H(p, p+), and H(p-,p-), and H(p-,p)each exist
for each p and T is a bounded linear operator on the space of all such functions H.
Functions H
for. which each of the four preceding limits exist have been used
extensively in the study of both sum integration and multiplicative integration,
(for example see
2J).
In particular we show that
+
+
(1)fbfHd + i=l H(Xi-l’ x+. l)-H (xi-i
Xi-l)
rH
a
l-
+
where each of
,,
i=l
H(xi,xi)-H(x,x[)] 0 (Xi_l,Xi)
@ depend only on T, e is of bounded variation, 8 and @ are 0
fH is
[a b for
a function from R to N depending on
except at a countable number of points,
and
{x i OO
i= I
(xi-I)
denotes the points p in
[H(p-,p )-H(p-,p- #
0.
which
H (p
p+) -H (p+ p+) #
0 or
We also define an interior interval function integral
and give a relationship between it and the standard interval function integral.
2.
DEFINITIONS.
If H is a function from RxR to N, then
H(p+,p+)
lim
+ H(x,y)
and similar
x,y+p
meanings are given to
H(p,p+), H(p-,p-),
and H(p-,p).
The set of all functions
for which each of the preceding four limits exist will be denoted by OL
0.
If
H is a function from RxR to N then H is said to be (i) of bounded variation on
the interval
[a,b
subdivision D of
and (2) bounded on
[a,b2
such that if
[a,b]
D’
if there exists a number M and a
n
{xi}i= 0
is a refinement of D then
H,
287
REPRESENTATION THEOREM FOR OPERATORS
n
F,
i=l
(I)
IH(Xi_l,Xi)
Further, H is said to be integrable on
[a,b]
b
A
<
n
a refinement of
Y,
D, then
i=l
i)_
H(Xi_l,X
< M, respectively.
if there is a number A such that
[a
> 0 there is a subdivision D of
for each
IH(Xi_l,Xi)
< M and (2) if 0<i<__n, then
{xi }n
by bH
a
D’
such that if
and A is denoted
is
when
D’
such an A exists.
In our development we will also find a slight modification
of the preceding definition useful.
xi_ I <
r
i
< s
<
i
H(Xi_l,X i)
a,bJ.
(IH)/bHGa and termed the
is a function from R to N,
Also, if each of f and
exists means there is a number
> 0 then there is a subdivision D of
A such that if
{xi }ni-0
n
is a refinement of D and for
f(ti)(xi)-(Xi_l)]-
Y.
i=l
H(ri,si)G(Xi_l,Xi),
interior integral of H with
then the interior integral of f with respect to
D’
is replaced by
in the approximating sum of the preceding definition then
xi,
the number Ais denoted by
respect to G on
If
A
<
[a,b]
0<i<n,_ xi_ I <
and A is denoted by
t
i
such that if
< x
i’
(1)fbfd.
a
D’
is a function from R to
If
N,
(p+)
lim
x+p /
(x), (p-)
(x), and
lim
x+p-
d denotes the function H from RxR to N such that for x<y, H(x,y)
If each of H,
on
[a,b]
then
[a b]
if
is a function from RxR to
> 0
means if
}ni=0 of [a,b
IH(rs)-Hn(r,s)
1{x
D
HI, H2,
i
.
x__l+/-
< r < s
and
respect to D,
all such
M’s.
xi_ l
_<
lHl ID
H uniformly
r < s
_< x.1
for some
0<i<n,_
If H is a function from RxR to N, then H is bounded on
means there is a number M and a subdivision D
0<i_<
Hn
there is a positive integer N and a subdivision
such that if n > N and
<
N, then lim
n-o
(y)-(x).
_<
x i, then
IH(r,s)
< M.
{x
i
}i=0
of
[a b] such
The norm of H on
[a,b
that
with
is then defined as the greatest lower bound of the set of
J. A. CHATFIELD
288
T is a linear operator on OL 0 means T is a transformation from OL 0 to N such
that if each of H I and H are in OL 0 then
2
r[klH I + k2H2] kITH I + k2rH 2
for
kl,
k
in R.
2
ITHI MIIHII D
T is bounded on
[a,b]
means there is a number M such that
for some subdivision D of
a,b.
For convenience we adopt the following conventions for a function from RxR
to N and
m
to N for some subdivision D
(i)
(2)
(3)
(4)
H(a-,a)
H(a-,a-)
{xi }ni=0
of
H(b,b +)
[a,b]:
H(b+,b +)
0,
H(Xi_l,X i) Hi, 0<i<_n,
(xi)-(Xi_l) Ai,
n
Z H(Xi_l,Xi)
ZDH i.
i=l
D
3.
THEOREMS.
We will begin by establishing a relationship between
/bHda
a
and
(IH)fbHd
a
which will require the following lemmas.
LEMMA i.
on
[ab3
then
If H is in OL 0 and a
bHd
a
is a function from R to N of bounded variation
exists.
This lemma is a special case of THEOREM 2 of
LEMMA 2.
Suppose H is in OL
sets such that p is in S
p is in S
and S
2
2
1
if and only if p is in
is a finite set.
if and only if
LEMMA 3.
is an interval,
> 0, and S and S are
1
2
a,b] and IH(p,p+)-H(p+,p +) I
IH(p-,p)-H(p-,p-)I e. Then, each
if nd only if p is in
2,
We note that it follows
a,b]
0, a,b]
2].
[a,b]
lemma page
and
and
of S
1
498].
from LEMMA 2 that if S is the set such that p is in S
H(p,p+)-H(p+,p+) #
0 or H(p-,p)-H(p-,p-) # 0 then S is countable.
If H is in OL 0 and a is a function from R to N of bounded variation on
then (i) if p is in
a,b]
each of
(p+)
is a sequence of numbers such that if p is in
and (p-) exists and (2) if
[a,b]
and
H(p,p+)-H(p+,p +) #
{x
0
i i=l
REPRESENTATION THEOREM FOR OPERATORS
# 0, then there
+
+
or H(p-,p)-H(p-,p-)
Y.
(i)
i=l
is an n such that
(x )?
[H(x i,xi)-H(x i,x+)][(x)i
i
P=Xn,
289
then
exists
(2)
It follows from the bounded variation of
INDICATION OF PROOF.
p in
[a,@
(p+)
each of
Since H is in OL
on
[a,b]
0,
and (p-) exists.
it follows from the covering theorem that H is bounded
and that there is a number M I such that for each positive integer i,
+ +
IH(xi,x)-H(xi,xi)
< M
I,
and, furthermore, for n a positive integer and 0 < i < n, let x
n
Z
s(x
i=l
that for
+
i)-(xpi )]
< i.
n
Y.
i’l
> x
Pi
i
such that
Hence,
+
+
H (xi, xil-H
(xi, x)7 (x)_(xi)]
_< MI [ Yn (x)- (XP )] +
I (XP
i=l
< M
where D is a subdivision of
in D for each 0 < i < n.
I
(i) + M
[a b]
i=l
i
i
)-(xi)
D l(xi)-e(Xi-l)l’
I
containing x
i
and x
as consecutive points
Pi
is of bounded variation there is a number
Hence, since
M such that
n
Y
i=l
+
+
[H(xi,xi)-H(x
i
x)] [(x)- (xi)]
< M.
Therefore,
i=l
[H(xi,xi)-H(xi,x [a(x )-(xi)
exists.
In a similar manner it may be
shown that
i=l
[H(xi,xi)-H(xi,xi) [(xi)-(xi)]
THEOREM i.
variation on
If H is in OL 0 and
[a,b]
then
(IH)fbHda
exists.
is a function from R to N of bounded
exists.
J. A. CHATFIELD
290
+
If g > 0 then it follows from LEMMA 2 that each of the sets A and
PROOF
Ag
to which p belongs if and only if
or
IH(p-,p)-H(p-,p-)l
> E
P is in
[a,b
IH(p,p+)-H(p+ P+)I
and
respectively, is a finite set.
Let A+
g
{c
m
2
{d i}i__l,
and A+ and A- denote the
A
p is in
[a,b]
H(p,p+)-tt(p+,p+)0
and
each of A+ and
or
Since a is of bounded variation on
di,
and
0 < i < m
fi-<
2
<
si
}I
i I_
sets to which p belongs if and only if
H(p-,p)-I-I(p-,p-)0, respectively.
is a countable set then let A+
A-
> E
-fa,b,
Since
{yi }’i=i"
+ A-
then for each c i, 0 < i < m and
1
there is an e. > c. and an f. > d. such that if e. > r. > c. and
i
I
I
then
di’
i
+
l(ci)-a(ri)
i
g
<
la(d)-(si)
and
v1ml
i
i
<
g
16m 2
From LEMMA 3, it follows that there is a positive integer N such that if n > N, then
(i)
i=l
[H(Yi,y)-H(Yi,
yl
i=l
(y
8
and
i=l
E
i=l
Note that for some
[H(yi,yi)-H(yi,yi [(yi)- (yi
Yi s, H(y,yi)-H(y,y
bHd
a
Since, from LE i,
D I of
a,
(3)
D’
and
of
i
(4)
fbHda
a
(5)
if 0 < i
Further,
2
such that if D
snce
a,b
or
<
g.
(yi,yi)-H(Yl,Yl
may be zero
exists, then there is a number M and a subdivision
xi }ni=0
is a refinement of
DI,
then
<M,
Z
D’ HiA i
!
n, then
H is in OL
such that f
O,
D’
<
4’
+
+
+
[H(Xi_l,X_l)-H(Xi_l,Xi_l)
< M and
using the covering theorem we may obtain a subd[vsion
n
{x} 0
is a refinement of
D2,
0 <
_<
n, and
291
REPRESENTATION THEOREM FOR OPERATORS
xi_ 1 <
r < s < x i, then
x)[
(6)
[H(r s)-H(x
(7)
[H(r s)-H(xi_ I
(8)
IH(xi_ I X:_l)-H<xi_ I xi)
and (9)
Let
ment of
for each
<
xi_
64M
I)]<
]H(x’xi)-H(Xi-l’Xi)l
D
DI+D2+A:+A+ i {e
D, and for each 0 < i
xi, 0 <
i
_<
6M
<
64
64M
2
+
_<
g
<
_<
Hence, for xi, 0 < i
in
follows from (6)-(9) that
n, in
D’
IN (xi-i
D’-
Wl[a(y:_l)-(Yi_l)]
[Z
+ Y.
A+
x
i.
(xi-i
i-i
Ym }
Y2’
+
xi_ I
Wl[a(Y:_l)-a(Yi_l)
D
+
+
+
o
Z CH(x.
D’.
[Wi[- [(y_l)-(Yi)[
+
+
D’
3
16
Hence
+
(A+-A+
+ E16M Z
+ 16Mm I
Q-Ag+
<MZ
_.)]Ail
+
Q_A+IWil. [(Yi_l)-(Y
+
xi_ I
for 0 < i
fA:+ (A+-A+)]
+
be a refine-
Choose m > N such that
Q-A
;,
< F,
{yi }, D’={xi }n
nor
x.m
is
32M
+
If Wi=H(Yi_l,Yi_l)-H (+
Yi_l,Yi_l and Q={YI’
+ )-H +
+
,x
Z
i=ZiWi [ (Yi-i)- (Yi-i)
7.
such that neither
+
m
N
there exists a positive integer z < m such
that
Yz X..m
(A++A-) it
+
xi_ 1 < rj_ I < sj <
n,
D’-(A/+A-)
n, in
{f
[(y-I )-(Yi_l)[
+ 16M
D’"
+)
_<
m then
J. A. CHATFIELD
292
m
+
+
+
+
Z H(x ,Xi_l)-H(xi_
(Yi_l)-(Yi_l)Wi[a
l,xi_ I) Ail
+
i=l
D’ "A
Z
3e
16
<
+/--+/-
and in a similar manner it may be shown that
Z
(ii)
i-I
where Z
i[(y)-(yi)J
[H(x,x)-H(x,xi) Ail
D’ "Ay’
<
3E
H(yi,y)-H(y,yi).
i
Using inequalities (I0) and (ii) we are now able to complete the proof of the
In the following manipulations W and Z are as defined for (i0) and (ii)
i
i
theorem.
and
+
+
D’
i
j
a
i=l
Z
<
D’ (Hj-Hi)A i-
<_
D’Z
<
<
<
+
Pi=H(Xi_l,Xi_l)-H(Xi_l,Xi_l
and
+
Wi[(Yi_l)-(Yi_l)
i=l
Z
(Hi-Hi) A i-D, "EA+PiAi-D "EA-QiAi
ilAa
F’!Hj’-H il il DE[Hj-Hi-P
D -D’-(A++A-)
"+
A
32M
Zi[(Y_)-(Yi)]["
-i
i
"M +
32M
--
Qi=H(xi,xi)-H(xi,xi
+
"M +
32M
-M
i
+
m
Zi[(y)-(yi)]l
i=l
Z
+
E
+
E
+ Ei-
+ 3 + 3 + 3
E[Hj-Hi-Qil’IA il
+ 3__g4
D’-A-
E.
Hence, we have a relationship established between
(IH)fbHdaa and /bHdaa
which will be
used in the proof of the principal theorem.
{Hi’i= 0 is a sequence of functions from SxS to N, such that
each i, H is in OL 0, lim
i
Hn=H 0 uniformly on a b, and T is a bounded linear
operator on OL 0 then lim
n-o THn=TH 0.
THEOREM 2.
If
The proof of this theorem is straightforward and we omit it.
THEOREM 3.
Suppose H is in OL 0
T is a bounded linear operator on OL 0
Then
TH
+
+
+
] 8(xi_ I)
[H(Xi_l,Xi_l)-H(Xi_l,Xi_l)
i=l
+ Z
(1)f
fHd
+ Z
[H(xi,xi)-H(xi,xi)] O(Xi_l,Xi)
i=l
for
293
REPRESENTATION THEOREM FOR OPERATORS
, B,
where each of
and @ depend only on T, a is of bounded variation,
are 0 except at a countable number of points,
and {x
pending on H
/
/
H(xi,xi)-H(xi,x
}
denote the points in
i i=l
#0 or
fH
and @
is a function from R to N de-
a,b
for which
#0, i=1,2
H(x,xi)-H(x,x )
B
n.
We first define a sequence of functions converging uniformly to a
PROOF.
0
given function H in OL and then apply THEOREM 2 to establish THEOREM 3.
We
first define functions g and h for each pair of numbers t,x, a < t < b, a < x < b
such that
=I
I’
g(t,x)
if t--x
if
a<
t < x
if
x<
t <b,
and h(t,x)
if
t#x
and using these functions and the operator T define functions
such that
-
, B,
Y, and
(x)=TH(’,x); B(x)=Tg( ",x); y(x)=Tg(x, "); (x,y)=Tg(-,x)g(y,’); and
O(x y) =y (y)
Clearly,
Z
D’
(x y) for x and y in
a,b].
[a,b]
is of bounded variation On
l@(Xi_l,Xi)
Z
D’ i
and we see from
2
lsgn@iTg (- ,Xi_l)g(xl,’)
D’
<
for
D’
MI D’Z sgnig(’,Xi_l)g(xi,’)I
a refinement of a subdivision D of
la,b],
exists and in a similar manner that each of
Z
i=l
Hence,
Y
i=l
l@[Xi_l,Xi)
n
of
[a,b]
it follows that
IB(xi)
and
Z
i=l
Z
(x i)
l(Xi_l,Xi)
exists.
exists.
Each of our approximating functions H
division D
D
n
will be defined in terms of a sub-
determined in the following manner.
J. A. CHATFIELD
294
(IH)fbHda
1
exists and there is a subdivision
xi_ I < ri <
and
H(p+p+)
xi_ I < x <
let J
a
r < s < y <
i
> --or
--n
i i=l
m
H (x y)=
n
Y
i-i
K’={xi }mi= I
such that if
< i where for 0 < i < m
n
IH(x,y)-H(r,s)l
m, then
>
--n
n
m
In {xi}i=0
if p is in
and D
n
K
n
a b]
H(r i
<--in
and
+ Jn + I n
i=l
m
Z
il
[H(x,xi)-H(ri,si)] g(x,xi_l)g(xi,Y)
xi_ I
_< m, xi_ I
0 < i
+
[H(xi_ I Xi_l)-H(ri
si)] [g(x xi)
i=l
[H(x,xi)-H(ri,si)] g(xi,Y)
for each (x,y) such that
<
_<
r.1
It is evident that lim Hn
n-o
x < y
<
s.l
_<
<
xi,
for some 0 < i
_<
m, and for each
x..l
H uniformly on
[a,b]
i
for if e > 0, _--< g,
n
{xi }mi=0’ and xp-1 < x < r < s < y < xp for some 0 < p < m, then
Hn(Xp_l,Xp) H(Xp_l,Xp), Hn(xxp) H(X,Xp), Hn(xp_ l,y) =’H(xp_l,y), and
n
H (x,y)
n
For
be a function from RxR to N determined by
si)[h(x xi)-h(x Xi_l)
m
D- D
a,b]
of
in the following manner:
+ Z
[Xi_l,X
)Ail
i
_<
0 < i
xi,
IH(p-,p)-H(p-,p-)I
n
{x }m
s
that there is a subdivision
each positive integer n, let H
n
of
denote the set such that p is in J
n
IH(p p+)_H(p+,p+)l
D
i
[a,b]
Kn
It follows from the covering theorem and the existence of
Xo.1
H(p,p+)
such that if
Further
<
s.1
(IH)fbHd-ZK’ H(r
of Kn’ then
is a refinement
the limits
and H is in OL 0 then from THEOREM
a,b7
is of bounded variation on
Since
H(r,s).
Since lim H
n-o
n
Hence lim H
n-o n
H uniformly on
H uniformly on
[a,b]
a,b].
applying THEOREM 2
we have
295
REPRESENTATION T}IEOREM FOR OPERATORS
TH
lim TH
n
n
lira 7
n-o
Dn
H(ri,si)KTH(-,xi) TH(-,xI_I)]
where the existenc of each of the infinite sums is assured by LEMMA 3 and the
equality of the last two expressions follows from the definition of Dn.
All that remains to complete the proof of THEOREM 3 is to show that
may be represented by
fH be
(I)bfHd where fH
is a function from R to N.
the function such that for each p in
follows that
(I);51_de
H
a
exists and is
[a,b] fH(p)
(IH);bHde.
a
H(p+p
+),
IlHde
If we let
then it
J. A. CHATFIELD
296
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i.
Goffman, Casper and Pedrlck, George, First Course in Functional Analysis,
Prentice-Hall, Inc., 1965.
2.
Helton, B.W.
3.
Hildebrandt, T.H. Linear Functional Transformations in General Spaces,
Bulletin of A.M.S. 37 (1931) 185-212.
4.
Hildebrandt, T.H. and Schoenberg, l.J. On Linear Functional Operations and
the Moment Problem for a Finite Interval in One or Several Variables,
Annals of Math. 34 (1933) 317-328.
5.
Kalterborn, H.S.
A Product Integral Representation for a Gronwall Inequality,
Bulletin of A.M.S. 23 (3), (1969) 493-500.
Linear Functional Operations on Functions Having
Discontinuities of the First Kind, Bulletin of A.M.S. 40 (1934),
702-708.
Annales de 1 Ecole, Normale Siperleure (3), 31 (1914).
6.
Riesz, F.
7.
Riesz, F. and B. St.-Nagy Lecons d’analyse fonctionelle, 3rd edition,
Akademiai Kiado (1955).
Budapest