Advances in Pure Mathematics, 2012, 2, 45-58
http://dx.doi.org/10.4236/apm.2012.21011 Published Online January 2012 (http://www.SciRP.org/journal/apm)
On Second Riesz -Variation of Normed Space
Valued Maps
Mireya Bracamonte
1
, José Giménez
2
, N. Merentes
3
, J. L. Sánchez
3
1
Departamento de Matemáticas, Universidad Centroccidental Lisandro Alvarado, Barquisimeto, Venezuela
2
Departamento de Matemáticas, Universidad de los Andes, Mérida, Venezuela
3
Departamento de Matemáticas, Universidad Central de Venezuela, Caracas, Venezuela
Email: mireyabracamonte@ucla.edu.ve, jgimenez@ula.ve, nmerucv@gmail.com, jose.sanchez@ciens.ucv.ve
Received September 21, 2011; revised November 11, 2011; accepted November 20, 2011
ABSTRACT
In this article we present a Riesz-type generalization of the concept of second variation of normed space valued func-
tions defined on an interval , ab
[,] ab
f X . We show that a function , where X is a reflexive Banach space, is of
bounded second -variation, in the sense of Riesz, if and only if it can be expressed as the (Bochner) integral of a
function of bounded (first) $\Phi$-variation. We provide also a Riesz lemma type inequality to estimate the total second
Riesz- -variation introduced.
Keywords: Young Function; -Variation; Second -Variation of a Function
,2 p -variation. 1. Introduction
Functions of bounded variation where first introduced in
1881 by Camille Jordan who established the relation be-
tween these functions and the monotonic ones. Thus, the
Dirichlet Criterion for the convergence of the Fourier
series applies to the class of functions of bounded varia-
tion. This, in turn, has motivated the study of solutions of
nonlinear equations that describe concrete physical phe-
nomena in which, often, functions of bounded variation
intervene.
The interest generated by this notion has lead to some
generalizations of the concept, mainly, intended to the
search of a bigger class of functions whose elements
have point wise convergent Fourier series. As in the clas-
sical case, these generalizations have found many appli-
cations in the study of certain differential and integral
equations. Ch. J. de la Vallée Poussin, introduced in 1908
([1]) the notion of second variation of a function. A few
years later, F. Riesz ([2]) proved that a function f is of
bounded second variation on , ab
$1 p
if, and only if, it is
the definite Lebesgue integral of a function F of bounded
variation.
More recently, in 1983, A. M. Russell and C. J. F. Upton
[4] obtained a similar result for functions of bounded sec-
ond variation , in the sense of Wiener. In
1992 the third author introduced the notion of ,2 p
-
variation, in the sense of Riesz ([4]), presenting, also, a
result that generalizes the renowned Riesz lemma for the
class that he called
2
,
p
BV ab , or class of functions of
bounded Riesz
In this article we define the notion of function of
bounded second -variation in the sense of Riesz. We
show that a function F, with values in a reflexive Banach
space, is of bounded second -variation, in the sense of
Riesz, if and only if it is the integral (in the sense of Bo-
chner) of a function of bounded -variation. In addi-
tion, from the main results presented it is deduced an
inequality that generalizes Riesz’s lemma.
2. Vector Value Functions of Bounded
Variation
We begin this section by recalling some known spaces
and results.
We will also assume that all partitions of an interval
, ab considered, contain at least one point , t ab ;
the set of all such partitions will be denoted as
3
The notion of bounded second variation in the sense
De La Vallée Poussin is defined as follows: A function
π , ab .
: , u ab
is of bounded second variation if and only
if
3
2
2
1 2 1
π , 0
; , : sup , ,
m
i i i i
ab i
V u ab ut t utt
where
2 1
1 2
2 1
, :
i i
i i
i i
ut ut
ut t
t t
0 2
π , , , .
m
t t t and
(2.1)
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