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Journal of Function Spaces and Applications
Volume 2013 (2013), Article ID 601490, 10 pages
New Sequence Spaces and Function Spaces on Interval
1Université de Lyon, LAGEP, Bâtiment CPE, Université Lyon 1, 43 boulevard du 11 Novembre 1918, 69622 Villeurbanne, France
2Department of Mathematics, Tianjin University, Tianjin 300072, China
Received 27 April 2013; Accepted 17 August 2013
Academic Editor: Ji Gao
Copyright © 2013 Cheng-Zhong Xu and Gen-Qi Xu. This 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 defined on interval in this paper. By a new summation method of sequences, we find out some new sequence spaces that are interpolating into spaces between and and function spaces that are interpolating into the spaces between the polynomial space and . We prove that these spaces of sequences and functions are Banach spaces.
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 , brain imaging data, optic spectral analysis, text retrieval, financial data, and climate data. These data have common features: (1) there are at most finite 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 different 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 classification 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  for the semiparameter estimation,  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 classification 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 , where denotes the norm in space. In most cases, there is at least one such that . We denote by the feasible set. Can we say that is close to some ? To see disadvantage, we divide sequence into three segments ; the first segment is composed of the first elements, the second segment is made of the next elements, and the third is composed of the others. Similarly, we also divide into corresponding three parts . Now, we reconsider
Perhaps we would find that . Can one say that is a new class? From the above example, we see that we need a new definition of the norm to fit application.
Motivated by these questions, we revisit the sequence spaces and function spaces defined on in this paper. We have observed recent studies on the sequence spaces, for instants, [6–8] for different requirements. Here, the sequence spaces we work on are different from the existing spaces, this is because the spaces are aimed to solve our problem. Now, let us introduce our idea and the resulted sequence space and function spaces.
Let be a DNA sequence. Obviously, there are at most finite many nonzero terms and . To shorten the representation, we can embed into a polynomial. In this way, we can write as
For a different DNA sequence, we have different polynomial . Obviously, it is a simpler reserve form.
How do we extract original sequence from the polynomial? By the classical mathematics, we know that so we recover the sequence.
To extend this form into a sequence of infinite many nonzero terms, we usually take ; is called the generation function in the classical queuing theory. Note that the generation function is not a continuous function defined on . So, the differential operation is not fitting to such a function, although the formal differential is always feasible. To find out a feasible form of , let us consider the operations of integral and derivative. We denote by the integral operation. Operating for the constant leads to
Generally, we have
For any polynomial of -order, , it can be written as
Next, let us consider the differential operation . Taking deferential for function leads to
In general, for any , it holds that
Obviously, using , we get once again the coefficients in (6):
Therefore, the coefficient sequence is given by
Clearly, we should take functions as the basis functions.
Moreover, we note that in the polynomial space over , denoted by , if the norm is defined as where , then it is a normed linear space. In this space, the integral and differential operations are bounded linear operators. To extend to an infinite sequence, we should choose such a function space in which the integral and differential operations are bounded linear operators. What is such a function space?
Consider a subset of defined as
The set is in fact a linear space. We define a norm on it by then it becomes a Banach space. Now, for the function spaces over interval , there are the following inclusion relations But the completion of is not the space . Let be the completion space of . Clearly, And the integral and differential operations are bounded operators.
For space , we have the following representation theorem.
Theorem 1. The set has the following representation:
Moreover, the space is isomorphic to the sequence space , and the isomorphism mapping is given by
Motivated by above result, for , we define a new norm on the space by
The completion of space under this norm is denoted by . Similarly, we have the following result.
Theorem 2. The space has a representation:
Furthermore, is isomorphic to the space , and the isomorphism operator is given by
Theorem 3. The space has a representation:
Moreover, is isomorphic to , and the isomorphism operator is given by
By now, we have gotten a series spaces in which both the differential operator and integral operator are bounded linear operators. Obviously, these sets have the following inclusion relations: We observe that for , in definition of these spaces, the term means that the terms are small as is large enough.
To insert a new space between and , let us consider the sequence formed by , the norm is maximum of the function sequence under space . To define a new normed space, in Section 2, we shall study new summation approach for a sequence. Based on such a new summation approach, we shall define some new sequence spaces. In Section 3, we shall define some function spaces and discuss their completeness. Furthermore, we compare these spaces and give some inclusion relations.
2. New Summation Method
2.1. Summation of Absolutely Dominant Operator
Let be a sequence (real or complex number) and satisfy . We define an operator by where . It is called the absolutely dominant queuing operator.
Removing the first terms, the remainder sequence has a new queuing:
Using the queuing operator, for a sequence of zero limit (this is only used to ensure that we can take the maximal value for such a sequence), we define a positive number by
By removing the first term from , we define the second number , that is, the absolute summation of the first two terms in , that is, where the number in subscription of summation is the number of the terms denotes the absolutely dominant queuing operator.
After removing the first two terms and from , we define the third positive number , that is, the absolute summation of the first three terms in , that is,
Generally, we define positive number as
According to this rule, we get a new nonnegative sequence
Example 4. Let . Then, we have
2.2. Relationship between Summation and Order of Sequence
To explain the thing we concerned about, let us see an example.
Example 5. Let scalar group be . Then, we have
Comparing this with Example 4, we see that the summation of a sequence has a relationship with its order.
From Examples 4 and 5, we see that the sequences and generated by a new summation method have relation of order of a sequence. In the sequel, we mainly discuss the infinite sequence. If a sequence has only finite many terms, we shall complement zero after the last term so that it becomes an infinite sequence.
2.3. Distribution of -Sequence
In this subsection, we shall consider the distribution of the sequence . We discuss it according to the different cases.
(1) Let be a positive and increasing sequence, that is,
According to the absolutely dominant summation, we have when , the sequence has even terms ; then
In this case, the -sequence has a -type distribution shown as follows:
If , the sequence has odd terms ; then
In this case, the -sequence has -type distribution as follows:
If is an increasing sequence, then has a symmetrical form; the -data at the medial term is its maximal value.
(2) is a positive and decreasing sequence
According to the absolutely dominant summation, we have
If , the sequence has an even term ; then
If , the sequence has odd terms , then
So, -sequence has distribution as
In this case, the -sequence has a character that the initial original data is the absolute largest; at the first several steps, -sequence arrives at its maximum value; after then, -sequence decreases until it arrives at its minimum value. The final value is minimal.
(3) Let be an arbitrary sequence.
In this case, it is difficult to give a general distribution. Although so, we have the following relation where denotes the value in decreasing queuing, denotes the value in increasing queuing.
From above, we see that -sequence undergoes a great contortion due to different queuing order of a sequence. Denote by the maximum change of -sequence for under different queuing order; then
For the sequence , it is easy to see that .
2.4. Generalized -Summation
From the absolutely dominant summation, we get a new sequence . Now, we define a new -summation for sequence:
Removing the first term , we define the second value by Again, removing the second term , we define the summation of the first three absolute maximum value as , that is,
In general, we define
According to this definition, we get a new nonnegative sequence .
Obviously, when , we recover the above summation sequence . For the sequence , we define a positive number by where denotes the generalized summation and denotes the -norm in finite-dimensional space.
In particular, when as , and hence .
Now, we define the sequence spaces by
We can prove the following result.
Theorem 6. and are Banach spaces.
3. -Type Spaces
3.1. Definition of Spaces
Let . For each , we define positive number:
Example 7. To show the computation method, let us consider the case that and discuss the function .
By computing various derivative functions
We calculate as follows:
According to the definition of , it has the following properties:(1)for any , (2)for each , (3)for any , This can be obtained from the Minkovwski inequality.
These properties show that satisfies the norm axiom, and hence it is a norm on space . But for only is a seminorm (or prenorm), it is continuous in the topology of .
For each , we define the following:
We define the following functions spaces:
Theorem 8. For , the scalar is defined as above. Then, is a norm on , and hence is a normed linear space. Moreover, is a Banach space, which is isomorphic to the sequence space .
Proof. Here, we only prove the second assertion.
Let be a Cauchy sequence, that is, for any , there exists such that
For any , it holds that This shows that also is a Cauchy sequence in . By the completeness of space , there is a such that .
For each , by the continuity of , we can get Thus, we find that This means that the sequence converges to in the sense of norm on . Since is a seminorm, it holds that Therefore, , which implies that is a Banach space.
For each , Clearly, Thus, we have This implies that Therefore, the mapping is a bounded invertible linear operator from to .
Theorem 9. Let be defined as (61). Then, is the completion of space , and
Moreover, space is isomorphic to .
3.2. Comparison of Spaces
So far, we have introduced some Banach spaces. The question is whether there is an inclusion relation for . In general, the answer is negative.
Example 10. Let . Then,
Take . Define a function by
Obviously, , but . Furthermore, we also have .
In fact, for any ,
it holds that
For any , while this shows that , and hence . So, .
Example 11. It holds that .
Define a function as
Notice the identity and so .
Example 12. It holds that .
Let us consider the following function:
Since we have this means that , but .
A new question is whether there is a inclusion relation between and for any ? The following result gives a positive answer.
Theorem 13. For any , the inclusion holds true.
Proof. Set . Then,
and for any ,
From above, we get that Similarly, for ,
Since , it holds that
Let . Set , then Since it holds that
This gives that .
3.3. -Type Space
In the previous discussion, we insert the some spaces between and . The questions is can one insert a Banach space between and ?
For each , whose norm is the corresponding -sequence is
Define a positive number where denotes the sum after the generalized -summation.
Define the function space by
Then, we have the following result.
Theorem 14. Let be defined as before. Then, for any ,(1)the following inequality holds (2) is a norm on . Further, is a Banach space;(3) is a dense subspace.
Proof. For any , we have
This leads to
Obviously, when , it holds that .
Since is a seminorm, the above relation shows that is a norm. In what follows, we shall prove that is a Banach space.
Let be a Cauchy sequence, that is,
Since this implies that also is a Cauchy sequence, so there exists a such that
For any , when is large enough, we have
Using the continuity of with respect to , and taking , we get that
Again taking , we get that So, , and
The completeness of the space follows.
Let be a -order polynomial. Then,
For any , for , there exists a such that
Taking a polynomial and noticing and for , we have
This means that is dense in .
3.4. Some Open Questions
From the previous discussion we see that
Note that the first relation and the relation (see, Example 10). These spaces are based on the new summation approach.
Now, we propose some open questions in mathematics aspect.
Question 1. Does the inclusion hold? yes or no.
Question 2. What is the dual space of ? Is it a reflexive space?
Question 3. Can one find out a Banach space , that is, the least in the following sense: is the completion of under the norm such that, for every Banach space of functions, embeds densely and continuously into ?
The research is supported by the Natural Science Foundation of China (NSFC-61174080). The work has been completed during the visit of Gen-Qi Xu in the University of Lyon, April 2013.
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