Some Direct Estimates for Linear Combination of Linear Positive Convolution Operators

B. Kunwar; V. K. Singh; Anshul Srivastava

doi:doi:10.11648/j.acm.20160505.14

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Some Direct Estimates for Linear Combination of Linear Positive Convolution Operators

B. Kunwar, V. K. Singh, Anshul Srivastava

Published in Applied and Computational Mathematics (Volume 5, Issue 5)

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

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Abstract

In this paper we have estimated some direct results for the even positive convolution integrals on C_2π, Banach space of 2 π - periodic functions. Here, positive kernels are of finite oscillations of degree 2k. 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.

Published in	Applied and Computational Mathematics (Volume 5, Issue 5)
DOI	10.11648/j.acm.20160505.14
Page(s)	207-212
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2016. Published by Science Publishing Group

Keywords

Convolution Operator, Linear Combination, Positive Kernels

References

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[2]	P. C. Curtis, “The degree of approximation by positive convolution operators”, Michigan Math. J., 12(1965), 155-160.
[3]	P. P. Korovkin, “Linear Operator and Approximation Theory”, Hindustan Publishing Corp., Delhi, 1960
[4]	E. L Stark, “An extension of a theorem of P. P. Korovkin to singular integrals with not necessarily positive kernels”, Indag. Math. 34(1972), 227-235.
[5]	G. Bleimann and E. L. Stark, “Kernels of finite oscillations and convolution integrals”, Acta. Math. Acad. Sci. Hungar. 35(1980), 419-429
[6]	P. L. Butzer, “Linear combination of Bernstein polynomials”, Canad. J. Math. 5(1953), 559-567.
[7]	W. Engels, E. L. Stark and L. Vogt, “Optimal Kernels for a general Sampling Theorem”, J. Approx. Theory 50(1987), 69-83.
[8]	O. Shisha and B. Mond, “The degree of approximation to periodic functions by linear positive operators”, J. Approx. Theory, 1(1968), 335-339.
[9]	G. G. Lorentz, “Approximation of functions”, Holt, Rinehart and Winston, New York, 1968.
[10]	A. Zygmund, “Trigonometric Series”, Vol. II, Cambridge university Press, Cambridge, 1959.
[11]	J. Szabados, “On Convolution operators with kernels of finite oscillations”, Acta. Math. Acad. Sci. Hungar. 27(1976), 176-192.
[12]	C. J. Hoff, “Approximations with Kernels of finite oscillations”, Part I, Convergence, J. Approx. Theory 3(1970), 213-218.
[13]	C. J. Hoff, “Approximation with kernels of finite oscillations”, Part II, Degree of Approximation, J. Approx. Theory 12(1974), 127-145.
[14]	R. Bojanic, “A note on the degree of approximation to continuous functions”, Enseign. Math., 15(1969), 43-51.
[15]	R. A. DeVore, “The Approximation of Continuous Functions by Positive Linear Operators”, Springer-Verlag, New York, 1972.
[16]	Gupta Vijay and Ravi P. Agarwal, “convergence estimate in approximation theory”, New York, Springer, 2014.
[17]	P. Maheshwari, S. Garg, “Higher Order Iterates Of Szasz-Mirakyan-Baskakov Operators”, Stud. Univ. Babe_s-Bolyai Math. 59(1) (2014), 69-76.
[18]	Tuncer Acar, Vijay Gupta, Ali Aral, Rate of convergence for generalized Szász operators, Bull. Math. Sci. (2011), 99–113.

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APA Style

B. Kunwar, V. K. Singh, Anshul Srivastava. (2016). Some Direct Estimates for Linear Combination of Linear Positive Convolution Operators. Applied and Computational Mathematics, 5(5), 207-212. https://doi.org/10.11648/j.acm.20160505.14

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ACS Style

B. Kunwar; V. K. Singh; Anshul Srivastava. Some Direct Estimates for Linear Combination of Linear Positive Convolution Operators. Appl. Comput. Math. 2016, 5(5), 207-212. doi: 10.11648/j.acm.20160505.14

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AMA Style

B. Kunwar, V. K. Singh, Anshul Srivastava. Some Direct Estimates for Linear Combination of Linear Positive Convolution Operators. Appl Comput Math. 2016;5(5):207-212. doi: 10.11648/j.acm.20160505.14

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@article{10.11648/j.acm.20160505.14,
  author = {B. Kunwar and V. K. Singh and Anshul Srivastava},
  title = {Some Direct Estimates for Linear Combination of Linear Positive Convolution Operators},
  journal = {Applied and Computational Mathematics},
  volume = {5},
  number = {5},
  pages = {207-212},
  doi = {10.11648/j.acm.20160505.14},
  url = {https://doi.org/10.11648/j.acm.20160505.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20160505.14},
  abstract = {In this paper we have estimated some direct results for the even positive convolution integrals on C_2π, Banach space of 2 π - periodic functions. Here, positive kernels are of finite oscillations of degree 2k. 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.},
 year = {2016}
}

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TY  - JOUR
T1  - Some Direct Estimates for Linear Combination of Linear Positive Convolution Operators
AU  - B. Kunwar
AU  - V. K. Singh
AU  - Anshul Srivastava
Y1  - 2016/10/18
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JO  - Applied and Computational Mathematics
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PB  - Science Publishing Group
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AB  - In this paper we have estimated some direct results for the even positive convolution integrals on C_2π, Banach space of 2 π - periodic functions. Here, positive kernels are of finite oscillations of degree 2k. 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.
VL  - 5
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ER  -

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Author Information

B. Kunwar

Applied Science Department, Institute of Engineering and Technology, Sitapur Road, Uttar Pradesh Technical University, Lucknow, India
V. K. Singh

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

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

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APA Style

B. Kunwar, V. K. Singh, Anshul Srivastava. (2016). Some Direct Estimates for Linear Combination of Linear Positive Convolution Operators. Applied and Computational Mathematics, 5(5), 207-212. https://doi.org/10.11648/j.acm.20160505.14

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ACS Style

B. Kunwar; V. K. Singh; Anshul Srivastava. Some Direct Estimates for Linear Combination of Linear Positive Convolution Operators. Appl. Comput. Math. 2016, 5(5), 207-212. doi: 10.11648/j.acm.20160505.14

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AMA Style

B. Kunwar, V. K. Singh, Anshul Srivastava. Some Direct Estimates for Linear Combination of Linear Positive Convolution Operators. Appl Comput Math. 2016;5(5):207-212. doi: 10.11648/j.acm.20160505.14

Copy | Download

@article{10.11648/j.acm.20160505.14,
  author = {B. Kunwar and V. K. Singh and Anshul Srivastava},
  title = {Some Direct Estimates for Linear Combination of Linear Positive Convolution Operators},
  journal = {Applied and Computational Mathematics},
  volume = {5},
  number = {5},
  pages = {207-212},
  doi = {10.11648/j.acm.20160505.14},
  url = {https://doi.org/10.11648/j.acm.20160505.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20160505.14},
  abstract = {In this paper we have estimated some direct results for the even positive convolution integrals on C_2π, Banach space of 2 π - periodic functions. Here, positive kernels are of finite oscillations of degree 2k. 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.},
 year = {2016}
}

Copy | Download

TY  - JOUR
T1  - Some Direct Estimates for Linear Combination of Linear Positive Convolution Operators
AU  - B. Kunwar
AU  - V. K. Singh
AU  - Anshul Srivastava
Y1  - 2016/10/18
PY  - 2016
N1  - https://doi.org/10.11648/j.acm.20160505.14
DO  - 10.11648/j.acm.20160505.14
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 207
EP  - 212
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20160505.14
AB  - In this paper we have estimated some direct results for the even positive convolution integrals on C_2π, Banach space of 2 π - periodic functions. Here, positive kernels are of finite oscillations of degree 2k. 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.
VL  - 5
IS  - 5
ER  -

Copy | Download