American Journal of Applied Mathematics
Volume 3, Issue 6, December 2015, Pages: 243-249

New Improved Approximation by Linear Combination in Lp Spaces

Srivastava Anshul

Mathematics, Department of Applied Sciences, Northern India Engineering College, Indraprastha University, New Delhi, India

Srivastava Anshul. New Improved Approximation by Linear Combination in Lp 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

Contents

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.

(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

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2. C.P May, Saturation and inverse theorems for combinations of a class of exponential operators, Can. J. Math.,XXVIII, 6 (1976), 1224-1250.
3. D.D. Stancu, Approximation of function by a new class of polynomial operators, Rev. Roum. Math. Pures et Appl., 13(8) (1968), 1173-1194.
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5. R.K.S. Rathore, Linear Combination of Linear Positive Operators and Generating Relations in Special Functions, Ph. D. Thesis, I.I.T. Delhi (India) (1973).
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7. S.Goldberg and A.Meir, Minimum moduli of ordinary differential operators, Proc. London Math. Soc., 23(3) (1971), 1-15.
8. B.Wood,-approximation by linear combination of integral Bernstein type operators, Anal. Nume’r. Theor. Approx., 13(1) (1984), 65-72.
9. A. Zygmund, Trignometrical Series, Dover Publications, Inc., N.Y. (1985).
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11. Gupta Vijay and Ravi P. Agarwal, convergence estimate in approximation theory, New York, Springer, 2014.
12. Gupta Vijay and Neha Malik "Approximation for genuine summation-integral type link operators", Applied Mathematics and computation, 260 (2015), 321-330.

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