Some Results on the Space of Bounded Second Variation Functions in the Sense of Shiba ()
1. Introduction
The notion of function of bounded variation, or the functions space of bounded variation on
(
) was first introduced by C. Jordan in 1881, [1], when he critically re-examined a faulty proof given by Dirichlet to the famous Fourier’s conjecture on trigonometric series expansion of periodic functions, see [2]. Jordan actually extended the Dirichlet’s criterium (on convergence of the Fourier series of monotone functions) to the class of BV functions. The interest generated by the classical notion of function of bounded variation has led to some generalizations of the concept, mainly, intended to the search of bigger classes of functions whose elements have pointwise convergent Fourier series or to applications in geometric measure theory, calculus of variations and mathematical physics. As in the classical case, these generalizations have found many applications in the study of certain differential and integral equations (see [3]). Consequently, the study of certain notions of generalized bounded variation takes an important direction in the field of mathematical analysis [4]. Two well-known generalizations are the functions of bounded p-variation and the functions of bounded
-variation, due to N. Wiener and L. C. Young respectively. In 1924 Wiener, [5], showed that the Fourier series of function in one variable of finite p-variation converges almost everywhere. In 1937 L. C. Young, [6], developed an integration theory with respect to functions of finite
-varia- tion and showed that the Fourier series of such functions converges everywhere.
In 1972, Waterman [7] introduced the class of bounded variation functions
. In 1980, M. Shiba [8] generalizes this class and introduces the class
. This class
, is the set of all functions
of
-bounded variation on
which definition is as follows:
Definition 1. [9] Given an interval
and a non-decreasing sequence of
positive numbers
such that
diverges and
.
A function
is said to be of
-bounded variation on I (
) if
where
(1)
and the supremum is taking over all partitions
of the interval I.
In [10], the authors introduce the notion of functions of bounded second variation in the sense of Shiba, following the line traced by De la Vallée Poussin [11], in 1908, and M. Shiba [8], in 1980. Inspired by the work done in [7], a sequence of positive real numbers,
, is a
-sequence if it is non-de- creasing and
. Additionally, they use
to denote the set of the partitions
of the interval
, which includes at least three points (
), and then define a function of bounded second variation in the sense of Shiba, as follows:
Definition 2. [10] Let
and
be a
-sequence. The
-th variation of f on
is defined as
where
and the supremun is taken over all the partitions
.
The sum in the definition 2, is called an approximate sum to
.
When
, f has
-th bounded variation on
.
is the space of such functions.
Example 1. Let
defined by
. We will prove that
.
Let
,
a
-sequence and
in
a partition.
Let’s consider
thus, using that
for all
:
Taking supremum
therefore
.
In [10], some properties of
were proved in order to show under what conditions it is guaranteed that an overlay operator acts in that space. However, there are other features that have not been explored and that allow us to improve this space. For this reason we present our following results.
2. Results for Functions of
-Bounded Variation
In [10] it was shown that
is a vector space and that
provides a norm for that space.
Next, we will prove the completeness of the space
.
Theorem 1. Let
a
-sequence, then
is a Banach space.
Proof. Let
a
-sequence and
a Cauchy sequence in
. Then, given
, there exists
such that for
we obtain:
this is,
Thus, for
(2)
and
(3)
On the other hand, using (2) we have, for
:
Therefore,
is a uniform Cauchy sequence on the interval
. Then, there exist a function f defined on
such that
uniformly in
, thus
(4)
Let’s prove that
.
Consider a partition
; that is,
, then using (3)
fixed
and
we have:
(5)
On the other hand, since
is a Cauchy sequence in
, there
exist
such that
, this implies that
, thus
Therefore,
this is,
.
Now
From (4) we get:
then, using this result and (5)
Therefore, the sequence
converges to the function f in the norm
. Thus,
is a Banach space. □
In the next result we show that
is a Banach algebra. It should be noted that this result is not inmediate as it happens in other spaces of functions of generalized bounded variation.
Theorem 2.
is a Banach algebra.
Proof. Let
a
-sequence. Given
, let’s prove that
.
Consider a partition
; that is,
. Let’s start by studying the expression
by adding the terms
,
, taking
with
and making
, we get
now, by adding the following terms
and grouping these, we get
Thus, using that
and
are bounded (see [10]),
Therefore, taking
now, using that
with
and property of supremum
So,
which implies that
and therefore the space
is an algebra.
In the next results we obtain, an integral representation of the space
. □
Theorem 3. If
and we define
, then
and
Proof. Let
a
-sequence and define
, where
.
Let’s consider
a partition of
. Making
by change of variable in each integral, for example,
we have:
now, using the Jensen inequality to the convex function
:
, with
, and lineality of the integral
Fixed s, with
, let’s notice that
are elements of a partition of
, that is,
Now,
where
is any partition of
.
Thus
So, we obtain
taking supremum over all partitions of
we have:
□
Following the ideas of Giménez, Merentes and Rivas [12], we get the next results.
Lemma 1. Let
a
-sequence, D a dense subset of
and let
be a function such that there is a constant
with
for any finite collection
in D. then
exists for all
, donde
An analogous result holds for
(
), which is similarly defined.
Proof. We will prove that
exists for all
, the case
is treated analogously. We will proceed by contradiction. Suppose that this is not true, that is, suppose that exists
such that
Let
thus
. Then, there exists two sequences
,
in D, such that
and
If
and
are finite, consider
; otherwise we take any
. Thus, there exist
such that
this implies that
which contradicts
. □
Theorem 4. If
then there exists a function
such that
, and
Proof. Let
a
-sequence and
. As
, f is absolutely continuos in
(see Lemma 4 in [10]), which implies that f is strongly diferentiable a.e., with derivative strongly measurable.
Let E be a set of zero Lebesgue measure such that
exists at every point of the set
, then D is dense in
. Given
, choose
ordered points
in D. Now consider
positive numbers:
and
such that
are in D with
Making
(6)
adding and subtracting appropriate terms within the absolute value in each term and using discrete Minkowski inequality:
we have
this implies that
.
Taking the limits in (6), as
and
, and using the above inequality, we get
(7)
If
then we obtain
instead of
in (7). Thus, the derivate
satisfies the conditions of Lemma 1. Now, let us define
, as
By construction,
a.e., let’s prove that
.
Let
be a partition of
. We need to consider several cases.
Case 1. If
then by (7) we get
Case 2. Suppose that there is just one
such that
. Let’s choose
with
. Let’s consider now,
a new partition where all points are in D. So, by the case 1,
taking limit as
, we have
. Thus,
and
therefore
Case 3. If exists j elements in
that are not in D, we proceed as in case 2, j times to get
Case 4. If exists j elements in
that are not in D, we proceed completely analogously as in cases 2 and 3, to we get
In any case, we conclude that
By the result of the Theorem 3,
□
In the following result we give necessary conditions for that the composition of functions belongs to the space
.
Recall that, under certain conditions, the composition of two functions f and g, denoted by
, is defined as
.
Theorem 5. If
is a strictly increasing function and
then
Proof. Let
be a
-sequence,
a partition of
. Then
by adding and subtracting the term:
we can group and get the following:
so, using the inequality
we have that:
where,
y
, are finite, see Lemma 3 in [10].
This implies that
thus,
hence
□
3. Conclusions
In this article, we proved important results in the space of functions of bounded second variation in the sense of Shiba, we study the structure of the space and we obtained that this is a Banach algebra, we also proved an integral representation theorem and we obtained necessary conditions for that the space be closed under composition of functions.
Finally, these results characterize the space of the second bounded variation and are indispensable to initiate other investigations, such as the study of the nonlinear integral equations used for example to describe physical phenomena and dynamic models of chemical reactors, as well as to establish compactness arguments in the study of equations system solutions applied to polymer models.