Hindawi Publishing Corporation
Journal of Function Spaces and Applications
Volume 2013, Article ID 601490, 10 pages
http://dx.doi.org/10.1155/2013/601490
Research Article
New Sequence Spaces and Function Spaces on Interval [0, 1]
Cheng-Zhong Xu
1
and Gen-Qi Xu
2
1
Universit´ e de Lyon, LAGEP, Bˆ atiment CPE, Universit´ e Lyon 1, 43 boulevard du 11 Novembre 1918, 69622 Villeurbanne, France
2
Department of Mathematics, Tianjin University, Tianjin 300072, China
Correspondence should be addressed to Gen-Qi Xu; gqxu@tju.edu.cn
Received 27 April 2013; Accepted 17 August 2013
Academic Editor: Ji Gao
Copyright © 2013 C.-Z. Xu and G.-Q. Xu. Tis is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly
cited.
We study the sequence spaces and the spaces of functions defned on interval [0,1] in this paper. By a new summation method of
sequences, we fnd out some new sequence spaces that are interpolating into spaces between ℓ
and ℓ
and function spaces that
are interpolating into the spaces between the polynomial space [0,1] and
∞
[0,1]. We prove that these spaces of sequences and
functions are Banach spaces.
1. Introduction
With development of sciences and technologies, more and
more information are obtaining and need to be reserved
and transmitted in the form of data sequence, such as DNA
sequence, protein structure [1], brain imaging data, optic
spectral analysis, text retrieval, fnancial data, and climate
data. Tese data have common features: (1) there are at
most fnite many nonzero elements in the sequence; (2) their
dimensions have not bounded from above; (3) the sample
size is relatively small. In particular, some elements in the
sequence repeat many times, for instance, there are only four
diferent elements in DNA sequence: , , , and . When
the data have much greater dimension, their record and
reserve also become a serious problem. On the other hand, we
usually use the data to obtain some information, such as the
image reconstruction, sequence comparison in medicine, and
plant classifcation in biology. From application point view,
the basic requirement is that one can draw easily information
from the reservoir; the is to use this data to handle some
things. When the data have lower dimension and the samples
have larger size, the statistics method such as the covariance
matrix can give a good treatment; for instance, see [2] for the
semiparameter estimation, [3] for the sparse data estimation,
and [4, 5] for the threshold sparse sample covariance matrix
method. However, when the data have higher dimensional
and the sample size is smaller, the statistics method shall lead
to great errors. So, we need new methods to treat them.
Let us consider a simple example from a classifcation
problem. Set as a set of some class samples and as
a given data. Is close to someone of or a new class?
A simpler approach is to consider problem inf
∈
‖−‖
,
where denotes the norm in ℓ
space. In most cases, there
is at least one
0
∈ such that ‖−
0
‖
= inf
∈
‖−‖
.
We denote by () the feasible set. Can we say that is close
to some
0
∈ ()? To see disadvantage, we divide sequence
∈ into three segments (
1
,
2
,
3
); the frst segment
1
is
composed of the frst
1
elements, the second segment
2
is
made of the next
2
elements, and the third is composed of the
others. Similarly, we also divide into corresponding three
parts (
1
,
2
,
3
). Now, we reconsider
inf
1
1
−
1
, inf
2
2
−
2
, inf
3
3
−
3
.
(1)
Perhaps we would fnd that (
1
)∩(
2
)∩(
3
)=0. Can
one say that is a new class? From the above example, we see
that we need a new defnition of the norm to ft application.
Motivated by these questions, we revisit the sequence
spaces and function spaces defned on [0,1] in this paper.
We have observed recent studies on the sequence spaces, for
instants, [6–8] for diferent requirements. Here, the sequence
spaces we work on are diferent from the existing spaces, this