New Sequence Spaces and Function Spaces on Interval [ 0 , 1 ]

We study the sequence spaces and the spaces of functions defined on interval [0, 1] in this paper. By a new summation method of sequences, we find 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 P[0, 1] and C[0, 1]. We prove that these spaces of sequences and functions are Banach spaces.


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, 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 [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 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 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 first segment  1 is composed of the first  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 , Perhaps we would find that ( 1 ) ∩ ( 2 ) ∩ ( 3 ) = 0. 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 [0, 1] in this paper.We have observed recent studies on the sequence spaces, for instants, [6][7][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  = ( 1 ,  2 ,  3 , . . .,   , . ..) 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  () =  1 () +  2 () +  3 () +  4 () . ( 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  ∈ [0,1]; () is called the generation function in the classical queuing theory.Note that the generation function is not a continuous function defined on [0, 1].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 1 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 1 ≤  ≤ , 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 [0, 1], denoted by [0, 1], if the norm is defined as where ‖‖ ∞ = max 0≤≤1 |()|, 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  ∞ [0, 1] defined as The set is in fact a linear space.We define a norm on it by          = sup then it becomes a Banach space.Now, for the function spaces over interval [0, 1], there are the following inclusion relations But the completion of ([0, 1], ‖ ⋅ ‖  ) is not the space ( ∞  [0, 1], ‖ ⋅ ‖  ).Let  ,0 [0, 1] be the completion space of ([0, 1], ‖ ⋅ ‖  ).Clearly, And the integral and differential operations are bounded operators.
Theorem 1.The set  ,0 [0, 1] has the following representation: Moreover, the space C ,0 [0, 1] is isomorphic to the sequence space  0 , and the isomorphism mapping is given by Furthermore,  , [0, 1] is isomorphic to the space ℓ  , and the isomorphism operator is given by Theorem 3. The space  ,∞ [0, 1] has a representation:

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  ≥ 1, in definition of these spaces, the term means that the terms ∑  =1 ‖ + ‖  ∞ are small as  is large enough.

Summation of Absolutely Dominant Operator.
Let {  } be a sequence (real or complex number) and satisfy lim  → ∞   = 0. We define an operator  by where It is called the absolutely dominant queuing operator.
Removing the first  terms, the remainder sequence { + ;  ≥ 0} 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  0 from {  } ∞ =0 , we define the second number  1 , that is, the absolute summation of the first two terms in { +1 }, 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  0 and  1 from {  } ∞ =0 , we define the third positive number  2 , that is, the absolute summation of the first three terms in {{ 2+ }}, that is, Generally, we define positive number   as  (32)

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 {  } = {6, 5, 4, 3, 2, 1}.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 {ŝ  } 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.

Distribution of 𝑠-Sequence.
In this subsection, we shall consider the distribution of the sequence {  ,  ≥ 0}.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  = 2+1, the sequence has even terms 2( + 1); then In this case, the -sequence has a -type distribution shown as follows: If  = 2, the sequence has odd terms (2 + 1); 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  = 2 + 1, the sequence has an even term 2( + 1); then If  = 2, the sequence has odd terms (2 + 1), 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.
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 {  } = {1, 2, 3, 4, 5, 6}, it is easy to see that (  ) = 15/9.
where  denotes the generalized summation and  denotes the -norm in finite-dimensional space.
The proof of Theorem 9 is similar to that of Theorem 8; we omit the details.

Comparison of Spaces.
So far, we have introduced some Banach spaces.The question is whether there is an inclusion relation  , [0, 1] ⊂  , [0, 1] for  > .In general, the answer is negative.
In fact, for any  ∈ N, Clearly, Due to it holds that For any  ∈ N, Define a function as Obviously, Let us consider the following function: Since    (0) = 1 (1 + ) Define a positive number where (, 1) denotes the ℓ 1 sum after the generalized 1summation.Define the function space by Then, we have the following result.