International Journal of Nonlinear Analysis and Applications
On the degree of approximation of certain continuous bivariate functions by double matrix means of a double Fourier series
Document Type : Research Paper
Author
Department of Mathematics and Informatics, University of Prishtina, Hasan Prishtina, Prishtina, Kosovo
Abstract
In this paper, we have studied the degree of approximation of certain bivariate functions by double factorable matrix means of a double Fourier series. Four theorems are proved using single rest bounded variation sequences, single head bounded variation sequences, double rest bounded variation sequences, and two non-negative mediate functions. These results expressed in terms of two functions of modulus type and two non-negative mediate functions, imply many particular results as shown at last section of this paper.
Keywords
[1] C. R. Adams, On summability of double series, Trans. Amer. Math. Soc. 34(2) (1932) 215–230.
[2] C. R. Adams, On non-factorable transformations of double sequences, Proc. Natl. Acad. Sci. USA 19 (1933) 564–567.
[3] N. A. Bokayev and Zh. B. Mukanov, Weighted integrability of double trigonometric series and of double series with respect to multiplicative systems with coefficients of class R+0 BV S2, Translation of Mat. Zametki 91 (2012), no. 4, 617-620. Math. Notes 91 (2012), no. 3-4, 575–578.
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[9] E. Hille and J. D. Tamarkin, On the summability of Fourier series, Trans. Amer. Math. Soc. 32 (1932) 757–783.
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[12] Xh. Z. Krasniqi, On the degree of approximation of continuous functions by matrix means related to partial sums of a Fourier series, Comment. Math. 52(2) (2012) 207–215.
[13] Xh. Z. Krasniqi, Approximation of continuous functions by generalized deferred Voronoi-N¨orlund means of partial sums of their Fourier series, An. S¸tiint¸. Univ. Al. I. Cuza Ia¸si. Mat. (N.S.) 66(1) (2020) 37–53.
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[15] Xh. Z. Krasniqi, On the degree of approximation of conjugate functions of periodic continuous functions, Poincare J Anal. Appl. 7(2) (2020) 175–184.
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[17] Xh. Z. Krasniqi, Applications of the deferred generalized de la Vall´ee Poussin means in approximation of continuous functions, Studia Univ. Babe¸s-Bolyai Math.(accepted).
[18] S. Lal and V. N. Tripathi, On the study of double Fourier series by double matrix summability method, Tamkang J. Math. 34(1) (2003) 1–16.
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[31] B. N. Sahney and D. S. Goel, On the degree of approximation of continuous functions, Ranchi Univ. Math. J. 4 (1973) 50–53.
[2] C. R. Adams, On non-factorable transformations of double sequences, Proc. Natl. Acad. Sci. USA 19 (1933) 564–567.
[3] N. A. Bokayev and Zh. B. Mukanov, Weighted integrability of double trigonometric series and of double series with respect to multiplicative systems with coefficients of class R+0 BV S2, Translation of Mat. Zametki 91 (2012), no. 4, 617-620. Math. Notes 91 (2012), no. 3-4, 575–578.
[4] P. Chandra, On the degree of approximation of functions belonging to the Lipschitz class, Nanta Math. 8(1) (1975) 88–91.
[5] P. Chandra, On the degree of approximation of a class of functions by means of Fourier series, Acta Math. Hungar. 52 (1988) 199–205.
[6] P. Chandra, A note on the degree of approximation of continuous functions, Acta Math. Hungar. 62 (1993) 21–23.
[7] Y. S. Chow, On the Ces`aro summability of double Fourier series, Tohoku Math. J. 5 (1953) 277–283.
[8] J. G. Herriot, N¨orlund summability of double Fourier series, Trans. Amer. Math. Soc. 52 (1942) 72–94.
[9] E. Hille and J. D. Tamarkin, On the summability of Fourier series, Trans. Amer. Math. Soc. 32 (1932) 757–783.
[10] A. S. B. Holland, B. N. Sahney, J. Tzimbalario, On degree of approximation of a class of functions by means of Fourier series, Acta Sci. Math. (Szeged) 38(1-2) (1976) 69–72.
[11] Xh. Z. Krasniqi, On the degree of approximation of continuous functions that pertains to the sequence-to-sequence transformation, Aust. J. Math. Anal. Appl. 7(2) (2011) 1–10.
[12] Xh. Z. Krasniqi, On the degree of approximation of continuous functions by matrix means related to partial sums of a Fourier series, Comment. Math. 52(2) (2012) 207–215.
[13] Xh. Z. Krasniqi, Approximation of continuous functions by generalized deferred Voronoi-N¨orlund means of partial sums of their Fourier series, An. S¸tiint¸. Univ. Al. I. Cuza Ia¸si. Mat. (N.S.) 66(1) (2020) 37–53.
[14] Xh. Z. Krasniqi, On the degree of approximation of continuous functions by a specific transform of partial sums of their Fourier series, Acta Comment. Univer. Tartuensis de Math. (accepted)
[15] Xh. Z. Krasniqi, On the degree of approximation of conjugate functions of periodic continuous functions, Poincare J Anal. Appl. 7(2) (2020) 175–184.
[16] Xh. Z. Krasniqi, On the degree of approximation of continuous functions by a linear transformation of their Fourier series, Commun. Math. (accepted).
[17] Xh. Z. Krasniqi, Applications of the deferred generalized de la Vall´ee Poussin means in approximation of continuous functions, Studia Univ. Babe¸s-Bolyai Math.(accepted).
[18] S. Lal and V. N. Tripathi, On the study of double Fourier series by double matrix summability method, Tamkang J. Math. 34(1) (2003) 1–16.
[19] S. Lal and H. P. Singh, Double matrix summability of double Fourier series, Int. J. Math. Anal. (Ruse) 3(33-36) (2009) 1669–1681.
[20] L. Leindler, On the uniform convergence and boundedness of a certain class of sine series, Anal. Math. 27(4) (2001) 279–285.
[21] L. Leindler, On the degree of approximation of continuous functions, Acta Math. Hungar. 104 (2004) 105–113.
[22] V. N. Mishra, S. K. Paikray, P. Palo,; P. N. Samanta, M. Misra, U. K. Misra, On double absolute factorable matrix summability, Tbilisi Math. J. 10(4) (2017) 29–44.
[23] N. L. Mittal and B. E. Rhoades, Approximation by matrix means of double Fourier series to continuous functions in two variables, Rad. Mat. 9(1) (1999) 77–99.
[24] F. M´oricz and B. E. Rhoades, Approximation by N¨orlund means of double Fourier series to continuous functions in two variables, Constr. Approx. 3(3) (1987) 281–296.
[25] F. M´oricz and X. L. Shi, Approximation to continuous functions by Ces`aro means of double Fourier series and conjugate series, J. Approx. Theory 49(4) (1987) 346–377.
[26] F. M´oricz and B. E. Rhoades, Approximation by N¨orlund means of double Fourier series for Lipschitz functions, J. Approx. Theory 50(4) (1987) 341–358.
[27] J. N´emeth, A remark on the degree of approximation of continuous functions, Acta Math. Hungar. 106(1-2) (2005) 83–88.
[28] A. Rathore, U. Singh, Approximation of certain bivariate functions by almost Euler means of double Fourier series, J. Inequal. Appl. 2018, Paper No. 89, 15 pp.
[29] B. E. Rhoades, On absolute normal double matrix summability methods, Glas. Mat. Ser. III 38(58) (2003) 57–73.
[30] M. G. Robison, Divergent double sequences and series, Trans. Amer. Math. Soc. 28(1) (1926) 50–73.
[31] B. N. Sahney and D. S. Goel, On the degree of approximation of continuous functions, Ranchi Univ. Math. J. 4 (1973) 50–53.
[32] S¸. Sezgek and ˙I. Da˘gadur, Approximation by double Ces`aro submethods of double Fourier series for Lipschitz fuctions, Palest. J. Math. 8(1) (2019) 71–85.
[33] P. L. Sharma, On the harmonic summability of double Fourier series, Proc. Amer. Math. Soc. 91 (1958) 979–986.
[34] B. Wei and D. Yu, On the degree of approximation of continuous functions by means of Fourier series, Math. Commun. 17 (2012) 211–219.
[35] Y. Zhao and D. Yu, Approximation by T-transformation of double Walsh-Fourier series to multivariable functions, ISRN Math. Anal. 2014, Art. ID 713175, 14 pp.
[33] P. L. Sharma, On the harmonic summability of double Fourier series, Proc. Amer. Math. Soc. 91 (1958) 979–986.
[34] B. Wei and D. Yu, On the degree of approximation of continuous functions by means of Fourier series, Math. Commun. 17 (2012) 211–219.
[35] Y. Zhao and D. Yu, Approximation by T-transformation of double Walsh-Fourier series to multivariable functions, ISRN Math. Anal. 2014, Art. ID 713175, 14 pp.
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Volume 12, Issue 2
November 2021Pages 609-628
- Receive Date: 11 November 2019
- Revise Date: 10 January 2020
- Accept Date: 12 January 2020