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
To cite this article:
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
Consider the singular positive convolution integral,
, and (1)
where, a kernel , is a sequence of positive even normalized trigonometric polynomial .
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.,  showed that,
which implies that is identically constant provided,
P. P Korovkin  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 , 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,
with coefficients , the being certain given naturals.
Here, kernel be a sequence of even trigonometric polynomials of degree atmost , which are normalized by,
Thus, and , with and .
Here, Fourier cosine coefficients are defined as usual by,
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,
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,
2. Some Definitions
Definition 2.1.  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,
Some properties of these numbers are,
Definition 2.2.  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,
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,
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.
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.  A kernel is said to have the expansion index i.e, , if for all , there holds an expansion,
Mostly known kernels belong to a class .
3. Auxiliary Results
Lemma 3.1.   Let or and The following assertions are equivalent for a kernel:
the being given as in definition 2.3.
Lemma 3.2.   Let and be different naturals. The unique solution of Vandermonde system of equations,
is given by,
Here, system-determinant is given by,
Let us suppose, , with or , , to be a positive kernel, we set,
We consider linear combination, of even trigonometric polynomials of degree , as,
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,
, 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,
using linear combination
Theorem 4.1. Let be a linear combination for the positive kernel with . Then there holds for
the following expansion:
Proof. A mixed algebraic-trigonometric Taylor’s formula for is,
where the remainder term is given by,
Here, denotes a continuous function independent of and lies between and .
According to Landau,
Now, with the help of (18) and (19), we can see,
Now, we will estimate ,
Using (8), (9) and (13), we get,
So, using lemma 2.1, we see that,
For inequality ,
Taking, , and using (9),
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,
for , we can easily show (23),
Using (23) and (20), we can easily prove,
, where, (24)
Using, and (24), we have the theorem.
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 .