New Improved Approximation by Linear Combination in L_{p} Spaces
Srivastava Anshul
Mathematics, Department of Applied Sciences, Northern India Engineering College, Indraprastha University, New Delhi, India
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To cite this article:
Srivastava Anshul. New Improved Approximation by Linear Combination in L_{p} Spaces.American Journal of Applied Mathematics.Vol.3, No. 6, 2015, pp. 243-249. doi: 10.11648/j.ajam.20150306.11
Abstract: In this paper we extend our studies for Modified Lupas operators introduced by Sahai and Prasad. We introduce and develop some direct results for Stancu type generalization of above operators using linear approximation method. Fubini’s theorem is used extensively to prove our main theorem. The anticipated improvement is made through technique of linear combination is well corroborated by the results in the paper. Here, modification of operators through Stancu generalization plays an important role to obtain better approximation results.
Keywords: Stancu Type Generalization, Linear Combination, Order of Approximation
1. Introduction
Modified Lupas Operators introduced by Sahai and Prasad [1] is
(1.1)
Where and .
Stancu type generalization [2] [3] of above operator gives,
(1.2)
Where
Here
And
Howsoever, smooth [4] the function may be, the order of approximation by these operators is at its best at order 0. To improve order of approximation, May [2] and Rathore [5] proposed technique of linear combination of these linear positive operators.
Let be arbitrary but fixed distinct positive integers. Then linear combination of , is defined by,
(1.3)
Where is Vandermonde determinant obtained by replacing the operator column of above determinant by entries 1.
Simplification of (1.3) leads to,
(1.4)
Where and
Let and , Also let denote integral part of
Let Then for sufficiently small η >0, the steklov mean of mth order corresponding to is defined as,
(1.5)
We will use the following results:
a) has derivatives upto order m, and exists a.e and belongs to .
b)
c)
d)
e)
Where are certain constants independent of and η.
Here, represents absolute continuous function on . Set of all functions of bounded variation on is represented by The semi-norm is defined by total variation of f on .For , 1the Hardy-Little Wood majorant of f is defined as,
2. Some Auxiliary Results
Lemma1. [1] For the function, the mth order moment be defined on [0,) as,
(2.1)
i.e.,
or,
Here, is a polynomial of degree m.
Also, is a rational function in n.
Consequently for for (2.2)
Using, Holder’s inequality, we get,
for each and for fixed (2.3)
Here is an integral part of Also, for any given number and , there is an integer such that,
for all (2.4)
Lemma 2. For and n sufficiently large there hold,
Where is certain polynomial in of degree
Proof. From Lemma 1, for sufficiently large , we can write,
= + + ………..+ +………..
Where are certain polynomials in of degree atmost
Therefore, is given by,
= for each fixed
3. Direct Results
Theorem 1. Let If has derivatives on with and then all, sufficiently large,
Where is constant independent of and
Proof. In view of [6], let and ,
(3.1)
Where is characterstic function of and,
For all and .
Operating on this equality (3.1) by and splitting right hand side into three terms, say, , we have,
In view of lemma 2 and [7], let be Hardy Little wood majorant [8] of on .
Now by Holder’s inequality and (2.3),
Using Fubini’s Theorem and [9],
Hence,
So,
For, , such that .
Thus,
= (say)
Using Holder’s inequality and (2.3),
Now, reapplying Fubini’s Theorem and [10],
Using equation (2.3) and [7] [11],
Therefore,
The result follows.
Theorem 2. Let . If has derivatives on with and , then for all sufficiently large
Where D is a constant independent of and
Proof. For our assumption of , and for almost all and for all , we have,
Where is characterstic function of .
for almost all and .
Thus,
+
+
= (say)
Applying lemma 2 and [7] [12],
Further,
For each there exists a non-negative integer such that
Then, we have,
Let be characterstic function of the interval , where are non-negative integers. Hence, we get,
+
Using lemma 1and Fubini’s theorem in the next step to obtain,
=
Hence,
Here, depends on
For all, ∈ [0,∞)\[] and all ∈ , we can choose a δ > 0 such that |t-u| ≥ δ,
=(say)
For sufficiently large there exists constants and such that,
for all and .
By Fubini’s Theorem,
= (say)
Now, using lemma 1,
And
Hence,
Further using (2.3) and [7],
The above estimates of and leads to,
By combining estimates of and , we get the required result.
4. Conclusion
Using technique of linear approximation method, we get direct theorem for linear combination of Stancu generalized modified Lupas operators. Similarly, we can also get local inverse and saturation results.
Acknowledgement
Author is thankful to Dr. B. Kunwar and reviewers for their valuable suggestions and great help throughout the paper.
References