Degree of Approximation of Conjugate of Signals (Functions) by Lower Triangular Matrix Operator

ABSTRACT

In the present paper, an attempt is made to obtain the degree of approximation of conjugate of functions (signals) belonging to the generalized weighted W(LP, ξ(t)), (p ≥ 1)-class, by using lower triangular matrix operator of conjugate series of its Fourier series.

In the present paper, an attempt is made to obtain the degree of approximation of conjugate of functions (signals) belonging to the generalized weighted W(LP, ξ(t)), (p ≥ 1)-class, by using lower triangular matrix operator of conjugate series of its Fourier series.

KEYWORDS

Conjugate Fourier Series, Generalized Weighted W(LP, ξ(t))-Class, Degree of Approximation and Lower Triangular Matrix Means

Conjugate Fourier Series, Generalized Weighted W(LP, ξ(t))-Class, Degree of Approximation and Lower Triangular Matrix Means

Cite this paper

nullV. Mishra, 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.

nullV. Mishra, H. Khan and K. Khatri, "Degree of Approximation of Conjugate of Signals (Functions) by Lower Triangular Matrix Operator,"

References

[1] A. Zygmund, “Trigonometric Series Vol. I,” Cambridge University Press, Cambridge, 1959.

[2] O. T?eplitz, “ über Allgemeine Lineare Mittelbildungen,” Prace Matematyczno—Fizyczne Journal, Vol. 22, 1913, pp. 113-119. http://www.zentralblatt-math.org/zmath/en/journals/search/?an=00003590

[3] L. McFadden, “Absolute N?rlund Summability,” Duke Mathematical Journal, Vol. 9, 1942, pp. 168-207. doi:10.1215/S0012-7094-42-00913-X

[4] H. H. Khan, “On the Degree of Approximation to a Function Belonging to Weighted (LP, ξ(t)) Class,” Aligarh Bulletin of Mathematics, Vol. 3-4, 1973-1974, pp. 83-88.

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

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

[7] K. Qureshi, “On the Degree of Approximation of a Function Belonging to Lipα,” Indian Journal of Pure and Applied Mathematics, Vol. 13, No. 8, 1982, pp. 898-903.

[8] K. Qureshi, “On the Degree of Approximation of a Function Belonging to the Class Lip(α,p),” Indian Journal of Pure and Applied Mathematics, Vol. 13, No. 4, 1982, pp. 466-470.

[9] 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, 1975, pp. 1473-1478.

[10] P. Chandra, “Trigonometric Approximation of Functions in LP-Norm,” Journal of Mathematical Analysis and Applications, Vol. 275, 2002, pp. 13-26. doi:10.1016/S0022-247X(02)00211-1

[11] L. Leindler, “Trigonometric Approximation in LP-Norm,” Journal of Mathematical Analysis and Applications, Vol. 302, 2005, pp. 129-136. doi:10.1016/j.jmaa.2004.07.049

[12] K. Qureshi, “On the Degree of Approximation of Conjugate of Function Belonging to the Lipschitz Class by Means of Conjugate Series,” Indian Journal of Pure and Applied Mathematics, Vol. 12, No. 9, 1981, pp. 1120-1123.

[13] K. Qureshi, “On the Degree of Approximation of Conjugate of Function Belonging to the Class Lip(α,p) by Means of Conjugate Series,” Indian Journal of Pure and Applied Mathematics, Vol. 13, No. 5, 1982, pp. 560-563.

[14] S. Lal, and J. K. Kushwaha, “Approximation of Conjugate of Functions Belonging to the Generalized Lipschitz class by Lower Triangular Matrix Means,” International Journal of Mathematical Analysis, Vol. 3, No. 21, 2009, pp. 1031-1041.

[1] A. Zygmund, “Trigonometric Series Vol. I,” Cambridge University Press, Cambridge, 1959.

[2] O. T?eplitz, “ über Allgemeine Lineare Mittelbildungen,” Prace Matematyczno—Fizyczne Journal, Vol. 22, 1913, pp. 113-119. http://www.zentralblatt-math.org/zmath/en/journals/search/?an=00003590

[3] L. McFadden, “Absolute N?rlund Summability,” Duke Mathematical Journal, Vol. 9, 1942, pp. 168-207. doi:10.1215/S0012-7094-42-00913-X

[4] H. H. Khan, “On the Degree of Approximation to a Function Belonging to Weighted (LP, ξ(t)) Class,” Aligarh Bulletin of Mathematics, Vol. 3-4, 1973-1974, pp. 83-88.

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

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

[7] K. Qureshi, “On the Degree of Approximation of a Function Belonging to Lipα,” Indian Journal of Pure and Applied Mathematics, Vol. 13, No. 8, 1982, pp. 898-903.

[8] K. Qureshi, “On the Degree of Approximation of a Function Belonging to the Class Lip(α,p),” Indian Journal of Pure and Applied Mathematics, Vol. 13, No. 4, 1982, pp. 466-470.

[9] 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, 1975, pp. 1473-1478.

[10] P. Chandra, “Trigonometric Approximation of Functions in LP-Norm,” Journal of Mathematical Analysis and Applications, Vol. 275, 2002, pp. 13-26. doi:10.1016/S0022-247X(02)00211-1

[11] L. Leindler, “Trigonometric Approximation in LP-Norm,” Journal of Mathematical Analysis and Applications, Vol. 302, 2005, pp. 129-136. doi:10.1016/j.jmaa.2004.07.049

[12] K. Qureshi, “On the Degree of Approximation of Conjugate of Function Belonging to the Lipschitz Class by Means of Conjugate Series,” Indian Journal of Pure and Applied Mathematics, Vol. 12, No. 9, 1981, pp. 1120-1123.

[13] K. Qureshi, “On the Degree of Approximation of Conjugate of Function Belonging to the Class Lip(α,p) by Means of Conjugate Series,” Indian Journal of Pure and Applied Mathematics, Vol. 13, No. 5, 1982, pp. 560-563.

[14] S. Lal, and J. K. Kushwaha, “Approximation of Conjugate of Functions Belonging to the Generalized Lipschitz class by Lower Triangular Matrix Means,” International Journal of Mathematical Analysis, Vol. 3, No. 21, 2009, pp. 1031-1041.