Trigonometric Approximation of Signals (Functions) Belonging to the *Lip*(*ξ*(*t*),*r*),(r＞1)-Class by (*E*,*q*) (*q*＞0)-Means of the Conjugate Series of Its Fourier Series

Author(s)
Vishnu Narayan Mishra^{*},
Huzoor H. Khan,
Idrees A. Khan,
Kejal Khatri^{*},
Lakshmi N. Mishra

Affiliation(s)

Department of Applied Mathematics & Humanities, Sardar Vallabhbhai National Institute of Technology, Surat, India.

Department of Mathematics, Aligarh Muslim University, Aligarh, India.

Dr. Ram Manohar Lohia Avadh University, Faizabad, India.

Department of Applied Mathematics & Humanities, Sardar Vallabhbhai National Institute of Technology, Surat, India.

Department of Mathematics, Aligarh Muslim University, Aligarh, India.

Dr. Ram Manohar Lohia Avadh University, Faizabad, India.

ABSTRACT

Various investigators such as Khan ([1-4]), Khan and Ram [5], Chandra [6,7], Leindler [8], Mishra *et al*. [9], Mishra [10], Mittal *et al*. [11], Mittal, Rhoades and Mishra [12], Mittal and Mishra [13], Rhoades *et al*. [14] have determined the degree of approximation of 2π-periodic signals (functions) belonging to various classes *Lip**α*, *Lip*(*α*,*r*), *Lip*(*ξ*(*t*),*r*) and *W*(*L _{r}*,

KEYWORDS

Signals; Conjugate Fourier Series; Trigonometric Fourier Approximation; Degree of Approximation;*Lip*(*ξ*(*t*),
*r*)-Class; (*E*,
*q*) Summability

Signals; Conjugate Fourier Series; Trigonometric Fourier Approximation; Degree of Approximation;

Cite this paper

V. Mishra, H. Khan, I. Khan, K. Khatri and L. Mishra, "Trigonometric Approximation of Signals (Functions) Belonging to the*Lip*(*ξ*(*t*),*r*),(r＞1)-Class by (*E*,*q*) (*q*＞0)-Means of the Conjugate Series of Its Fourier Series," *Advances in Pure Mathematics*, Vol. 3 No. 3, 2013, pp. 353-358. doi: 10.4236/apm.2013.33050.

V. Mishra, H. Khan, I. Khan, K. Khatri and L. Mishra, "Trigonometric Approximation of Signals (Functions) Belonging to the

References

[1] H. H. Khan, “On Degree of Approximation to a Functions Belonging to the Class*Lip*(*α*,*p*),” Indian Journal of Pure and Applied Mathematics, Vol. 5, No. 2, 1974, pp. 132-136.

[2] H. H. Khan, “On the Degree of Approximation to a Function by Triangular Matrix of Its Fourier Series I,” Indian Journal of Pure and Applied Mathematics, Vol. 6, No. 8, 1975, pp. 849-855.

[3] H. H. Khan, “On the Degree of Approximation to a Function by Triangular Matrix of Its Conjugate Fourier Series II,” Indian Journal of Pure and Applied Mathematics, Vol. 6, No. 12, 1975, pp. 1473-1478.

[4] H. H. Khan, “A Note on a Theorem Izumi,” Communications De La Faculté Des Sciences Mathématiques Ankara (TURKEY), Vol. 31, 1982, pp. 123-127.

[5] H. H. Khan and G. Ram, “On the Degree of Approximation,” Facta Universitatis Series Mathematics and Informatics (TURKEY), Vol. 18, 2003, pp. 47-57.

[6] P. Chandra, “A Note on the Degree of Approximation of Continuous Functions,” Acta Mathematica Hungarica, Vol. 62, No. 1-2, 1993, pp. 21-23.

[7] P. Chandra, “Trigonometric Approximation of Functions in -Norm,” Journal of Mathematical Analysis and Applications, Vol. 275, No. 1, 2002, pp. 13-26. doi:10.1016/S0022-247X(02)00211-1

[8] L. Leindler, “Trigonometric Approximation in L_{p}-Norm,” Journal of Mathematical Analysis and Applications, Vol. 302, No. 1, 2005, pp. 129-136.
doi:10.1016/j.jmaa.2004.07.049

[9] V. N. Mishra, H. H. Khan and K. Khatri, “Degree of Approximation of Conjugate of Signals (Functions) by Lower Triangular Matrix Operator,” Applied Mathematics, Vol. 2, No. 12, 2011, pp. 1448-1452. doi:10.4236/am.2011.212206

[10] V. N. Mishra, “On the Degree of Approximation of Signals (Functions) Belonging to the Weighted W(L_{p},*ξ*(*t*)),(p≥1) -Class by Almost Matrix Summability Method of Its Conjugate Fourier Series,” International Journal of Applied Mathematics and Mechanics, Vol. 5, No. 7, 2009, pp. 16-27.

[11] M. L. Mittal, U. Singh, V. N. Mishra, S. Priti and S. S. Mittal, “Approximation of functions belonging to*Lip*(*ξ*(*t*),*r*),(r＞1)-Class by means of conjugate Fourier series using linear operators,” Indian Journal of Mathematics, Vol. 47, No. 2-3, 2005, pp. 217-229.

[12] M. L. Mittal, B. E. Rhoades and V. N. Mishra, “Approximation of Signals (Functions) Belonging to the Weighted W(L_{p},*ξ*(*t*)),(p≥1) -Class by linear operators,” International Journal of Mathematics and Mathematical Sciences, Vol. 2006, 2006, Article ID: 53538.
doi:10.1155/IJMMS/2006/53538

[13] M. L. Mittal and V. N. Mishra, “Approximation of Signals (Functions) Belonging to the Weighted W(L_{p},*ξ*(*t*)),(p≥1) -Class by Almost Matrix Summability Method of Its Fourier Series,” International Journal of Mathematical Sciences and Engineering Applications, Vol. 2, No. 4, 2008, pp. 285-294.

[14] B. E. Rhoades, K. Ozkoklu and I. Albayrak, “On Degree of Approximation to a Functions Belonging to the Class Lipschitz Class by Hausdroff Means of Its Fourier Series,” Applied Mathematics and Computation, Vol. 217, No. 16, 2011, pp. 6868-6871. doi:10.1016/j.amc.2011.01.034

[15] M. L. Mittal, B. E. Rhoades, V. N. Mishra and U. Singh, “Using Infinite Matrices to Approximate Functions of Class*Lip*(*α*,*p*) Using Trigonometric Polynomials,” Journal of Mathematical Analysis and Applications, Vol. 326, No. 1, 2007, pp. 667-676.
doi:10.1016/j.jmaa.2006.03.053

[16] V. N. Mishra and L. N. Mishra, “Trigonometric Approximation of Signals (Functions) in L_{p}( p≥1)-Norm,” International Journal of Contemporary Mathematical Sciences, Vol. 7, No. 19, 2012, pp. 909-918.

[17] V. N. Mishra, “Some Problems on Approximations of Functions in Banach Spaces,” Ph.D. Thesis, Indian Institute of Technology, Roorkee, 2007.

[18] R. K. Shukla, “Certain Investigations in the theory of Summability and that of Approximation,” Ph.D. Thesis, V.B.S. Purvanchal University, Jaunpur, 2010.

[19] G. H. Hardy, “Divergent Series,” Oxford University Press, Oxford, 1949.

[20] A. Zygmund, “Trigonometric Series, Vol. I,” 2nd Edition, Cambridge University Press, Cambridge, 1959.

[1] H. H. Khan, “On Degree of Approximation to a Functions Belonging to the Class

[2] H. H. Khan, “On the Degree of Approximation to a Function by Triangular Matrix of Its Fourier Series I,” Indian Journal of Pure and Applied Mathematics, Vol. 6, No. 8, 1975, pp. 849-855.

[3] H. H. Khan, “On the Degree of Approximation to a Function by Triangular Matrix of Its Conjugate Fourier Series II,” Indian Journal of Pure and Applied Mathematics, Vol. 6, No. 12, 1975, pp. 1473-1478.

[4] H. H. Khan, “A Note on a Theorem Izumi,” Communications De La Faculté Des Sciences Mathématiques Ankara (TURKEY), Vol. 31, 1982, pp. 123-127.

[5] H. H. Khan and G. Ram, “On the Degree of Approximation,” Facta Universitatis Series Mathematics and Informatics (TURKEY), Vol. 18, 2003, pp. 47-57.

[6] P. Chandra, “A Note on the Degree of Approximation of Continuous Functions,” Acta Mathematica Hungarica, Vol. 62, No. 1-2, 1993, pp. 21-23.

[7] P. Chandra, “Trigonometric Approximation of Functions in -Norm,” Journal of Mathematical Analysis and Applications, Vol. 275, No. 1, 2002, pp. 13-26. doi:10.1016/S0022-247X(02)00211-1

[8] L. Leindler, “Trigonometric Approximation in L

[9] V. N. Mishra, H. H. Khan and K. Khatri, “Degree of Approximation of Conjugate of Signals (Functions) by Lower Triangular Matrix Operator,” Applied Mathematics, Vol. 2, No. 12, 2011, pp. 1448-1452. doi:10.4236/am.2011.212206

[10] V. N. Mishra, “On the Degree of Approximation of Signals (Functions) Belonging to the Weighted W(L

[11] M. L. Mittal, U. Singh, V. N. Mishra, S. Priti and S. S. Mittal, “Approximation of functions belonging to

[12] M. L. Mittal, B. E. Rhoades and V. N. Mishra, “Approximation of Signals (Functions) Belonging to the Weighted W(L

[13] M. L. Mittal and V. N. Mishra, “Approximation of Signals (Functions) Belonging to the Weighted W(L

[14] B. E. Rhoades, K. Ozkoklu and I. Albayrak, “On Degree of Approximation to a Functions Belonging to the Class Lipschitz Class by Hausdroff Means of Its Fourier Series,” Applied Mathematics and Computation, Vol. 217, No. 16, 2011, pp. 6868-6871. doi:10.1016/j.amc.2011.01.034

[15] M. L. Mittal, B. E. Rhoades, V. N. Mishra and U. Singh, “Using Infinite Matrices to Approximate Functions of Class

[16] V. N. Mishra and L. N. Mishra, “Trigonometric Approximation of Signals (Functions) in L

[17] V. N. Mishra, “Some Problems on Approximations of Functions in Banach Spaces,” Ph.D. Thesis, Indian Institute of Technology, Roorkee, 2007.

[18] R. K. Shukla, “Certain Investigations in the theory of Summability and that of Approximation,” Ph.D. Thesis, V.B.S. Purvanchal University, Jaunpur, 2010.

[19] G. H. Hardy, “Divergent Series,” Oxford University Press, Oxford, 1949.

[20] A. Zygmund, “Trigonometric Series, Vol. I,” 2nd Edition, Cambridge University Press, Cambridge, 1959.