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2 The space of bounded variation and its properties
Definition 2.1.
Let
f
be a real valued function defined on
[ , ]
a b
. Given a partition
1
1
{[ , ]}
n
i
i
i
P x x
of
[ , ]
a b
, let
1
1
( , ,[ , ])
| ( ) ( ) |
n
i
i
i
V f P a b
f x f x
¦
.
The variation of
f
defined by
( ,[ , ]) sup ( , ,[ , ]),
P
V f a b
V f P a b
where supremum is taken over all partition
P
. We say that
[ , ]
f BV a b
if
( ,[ , ])
V f a b
f
.
Lemma 2.2.
If a real valued function
f
is a monotically increasing function on
[ , ]
a b
, then for any partition
1
1
{[ , ]}
n
i
i
i
P x x
of
[ , ]
a b
, we have that
[ , ]
f BV a b
.
Theorem 2.3.
If
,
[ , ]
f g BV a b
and
,
D E
are real numbers, then
[ , ]
f
g BV a b
D
E
and
(
,[ , ]) |
| ( ,[ , ]) |
| ( ,[ , ]).
V f
g a b
V f a b
V g a b
D
E
D
E
d
Theorem 2.4.
If
[ , ]
f BV a b
, then the function and
( )
( ,[ , ])
V x V f a x
and
( ,[ , ]) ( )
V f a x f x
are both
increasing functions on
[ , ]
a b
.
Proof
. Let
1
2
a x x b
d d d
and assume that
( )
( ,[ , ])
V x V f a x
. Since
1
2
[ , ] [ , ]
a x a x
, we have
1
1
2
2
( )
( ,[ , ])
( ,[ , ])
( )
V x V f a x V f a x V x
d
.
Thus
2
2
1
1
2
2
1
1
[ ( ) ( )] [ ( ) ( )]
( ,[ , ]) ( ) ( ,[ , ]) ( )
V x f x V x f x V f a x f x V f a x f x
2
1
2
1
( ,[ , ]) ( ,[ , ]) [ ( ) ( )]
V f a x V f a x
f x f x
1 2
2
1
( ,[ , ]) [ ( ) ( )]
V f x x
f x f x
1 2
2
1
( ,[ , ]) | ( ) ( ) |
V f x x
f x f x
t
.
So the inequality
2
2
1
1
[ ( ) ( )] [ ( ) ( )]
V x f x
V x f x
t
follows from
1 2
2
1
1
1 2
{ , }
| ( ) ( ) |
| ( ) ( ) |
( ,[ , ])
i
i
x x
f x f x
f x f x
V f x x
d
¦
.
Theorem 2.5.
The function
[ , ]
f BV a b
if and only if it is the difference of two increasing functions.
Proof
.
( )
o
Assume that
[ , ]
f BV a b
, then
( )
( ,[ , ]) [ ( ,[ , ]) ( )]
f x V f a x V f a x f x
.
( )
m
Assume that
f
the difference of two increasing functions, thus we have that
f g h
,
where
,
g h
are increasing functions.
By Lemma 2.2 and Theorem 2.3, we have that
,
[ , ]
g h BV a b
and
[ , ]
g h BV a b
. Hence
[ , ]
f BV a b
.
