Applied and Computational Mathematics
Volume 5, Issue 5, October 2016, Pages: 207-212

Some Direct Estimates for Linear Combination of Linear Positive Convolution Operators

B. Kunwar1, V. K. Singh1, Anshul Srivastava2, *

1Applied Science Department, Institute of Engineering and Technology, Sitapur Road, Uttar Pradesh Technical University, Lucknow, India

2Applied Science Department, Northern India Engineering College, Guru Govind Singh Indraprastha University, New Delhi, India

(A. Srivastava)

*Corresponding author

B. Kunwar, V. K. Singh, Anshul Srivastava. Some Direct Estimates for Linear Combination of Linear Positive Convolution Operators. Applied and Computational Mathematics. Vol. 5, No. 5, 2016, pp. 207-212. doi: 10.11648/j.acm.20160505.14

Received: July 30, 2016; Accepted: August 12, 2016; Published: October 18, 2016

Abstract: In this paper we have estimated some direct results for the even positive convolution integrals on , Banach space of - periodic functions. Here, positive kernels are of finite oscillations of degree . Technique of linear combination is used for improving order of approximation. Property of Central factorial numbers, inverse formulas, mixed algebraic –trigonometric formula is used throughout the paper.

Keywords: Convolution Operator, Linear Combination, Positive Kernels

Contents

1. Introduction

Consider the singular positive convolution integral,

, and (1)

where, a kernel , is a sequence of positive even normalized trigonometric polynomial [1].

For non-negative trigonometric polynomials  of degree atmost and

Here,  being the Banach space of  – periodic functions  continuous on real axis  with usual sup norm,

Clearly,  is a bounded linear operator from  into itself, i.e.,

, we use the notation,

Philip C. Curtis Jr., [2] showed that,

which implies that  is identically constant provided,

Where,

P. P Korovkin [3] states that there exists an arbitrary often differentiable function, , such that,

Above result led to the fact that convolution integrals associated with these type of kernels has a better rate of convergence than .

Using extension of Korovkin Theorem [4], if we multiply our positive kernel η by a trigonometric polynomial, then approximation rate would be , where, is a kernel of finite oscillation of degree . Here,  has  sign changes on  for each .

In this paper, we will consider linear combination of positive kernels thus of convolution integrals for improving rate of approximation. Earlier, several authors [5-9] has worked on the special cases. Here, we will introduce rather general method for obtaining better rate of approximation.

If  is a positive kernel, we shall consider linear combinations, , given by,

(2)

with coefficients , the  being certain given naturals.

Here, kernel  be a sequence of even trigonometric polynomials of degree atmost , which are normalized by,

(3)

i.e.,

(4)

Thus,  and , with  and .

Here, Fourier cosine coefficients  are defined as usual by,

(5)

Here, Fourier cosine coefficients are referred to as convergence factors.

The Lebesgue constants are given by,

In order (1.1) defines an approximation process on , i.e.,

it is necessary and sufficient for the kernel  to satisfy,

(6)

This is due to the well-known theorem of Banach and Steinhaus.

In view of Bohman-korovkin theorem, for positive kernel,

i.e., (1.6) reduces to,

(7)

2. Some Definitions

Definition 2.1. [9] Let for ,

Here,  denote the central factorial polynomial of degree

The central factorial numbers of first kind  is uniquely determined coefficients of the polynomials,

Similarly, central factorial numbers of second kind  is uniquely determined coefficients of the polynomials,

where .

Some properties of these numbers are,

i)

ii)

iii)

iv)

v)

Definition 2.2. [10] Let  be a kernel for, ,

is called trigonometric moment of order 2.

We can also write,

either for or

The algebraic moment of order 2, is defined by,

(8)

Here, trigonometric as well as algebraic moments of odd order vanish, since kernel is positive.

For, , 0,

one deduces for positive kernels immediately the estimate,

(9)

By the well-known inverse formulas,

and the property (v) of the central factorial numbers, the trigonometric moments can be expressed in terms of the convergence factors and vice-versa.

In fact,

,

We can reduce our study of the asymptotic behaviour of the trigonometric moments to the asymptotic expansion of the difference  in the negative power of

In order to derive approximation theorems, we have to replace (6)(ii) by an asymptotic expansion of

Definition 2.3. [11] A kernel  is said to have the expansion index  i.e, , if for all , there holds an expansion,

(10)

for

Mostly known kernels belong to a class .

3. Auxiliary Results

Lemma 3.1. [12] [13] Let  or  and  The following assertions are equivalent for a kernel:

i) ,

ii)

the  being given as in definition 2.3.

Lemma 3.2. [14] [15] Let  and  be different naturals. The unique solution of Vandermonde system of equations,

, where,

is given by,

where

Here, system-determinant  is given by,

Also,

(11)

Let us suppose, , with  or , , to be a positive kernel, we set,

(12)

We consider linear combination,  of even trigonometric polynomials of degree , as,

(13)

Lemma 3.3. For linear combination χ convergence factors associated with positive kernel  admits the expansion,

Proof. Using (13) and lemma 2.1, we have,

(say)

Here,

, for, ,

Collecting all but the first non-vanishing term  into the  term, we have the lemma.

Lemma 3.4. The trigonometric moments for the, , admits the expansion,

Proof. Using definition 2.2,

Now, by lemma 3.3,

Again using definition 2.3, we have,

Using property (v) of central factorial numbers, we have the lemma.

4. Direct Results

Kernels defined by linear combination satisfy (6), so, the corresponding convolution integral defines an approximation process on .

Here, we will try to improve order for,

(14)

where  with

using linear combination

Theorem 4.1. Let  be a linear combination for the positive kernel  with . Then there holds for

the following expansion:

(15)

Proof. A mixed algebraic-trigonometric Taylor’s formula for  is,

(16)

where the remainder term is given by,

(17)

Here,  denotes a continuous function independent of  and  lies between and .

(18)

Here,

According to Landau,

So,

(19)

Now, with the help of (18) and (19), we can see,

(20)

Now, we will estimate ,

(say)

Using (8), (9) and (13), we get,

So, using lemma 2.1, we see that,

(21)

Now,

For inequality ,

Taking, , and using (9),

This implies,

(22)

Now using (19), (21), (22), we have,

As, we have the theorem.

Theorem 4.2. [16-18] Let χ be the linear combination of a positive kernel  with  as in,

. Then there holds on estimate:

Proof. For  and , we have,

There exists  with  and so,

Iteratively, we get,

(23)

for , we can easily show (23),

Using (23) and (20), we can easily prove,

, where,  (24)

Using,  and (24), we have the theorem.

5. Conclusion

By taking linear combination of suitable positive kernels,

We have raised the approximation order of  on .

The trigonometric moments of  upto order  grow in a linear manner, whereas, the moments of linear combination  upto order  behave asymptotically all like .

References

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