On Convergence of Partial Derivatives of Multidimensional Convolution Operators

Gumrah Uysal; Başar Yılmaz

doi:10.36753/mathenot.763854

EN

On Convergence of Partial Derivatives of Multidimensional Convolution Operators

Abstract

In this paper, we prove some results on convergence properties of higher order partial derivatives of multidimensional convolution-type singular integral operators being applied to the class of functions which are integrable in the sense of Lebesgue.

Keywords

Fatou-type convergence, Approximation of partial derivatives, Integral operators

References

[1] Almali, S.E., Gadjiev, A.D.: On approximation properties of certain multidimensional nonlinear integrals. J. Nonlinear Sci. Appl. 9 (5) 3090–3097 (2016).
[2] Bardaro, C., Mantellini, I.: Voronovskaya-type estimates for Mellin convolution operators. Results Math. 50 (1-2) 1–16 (2007).
[3] Bardaro, C., Mantellini, I.: Bivariate Mellin convolution operators: Quantitative approximation theorems. Math. Comput. Modelling 53 (5-6) 1197–1207 (2011).
[4] Butzer, P.L., Nessel, R.J.: Fourier analysis and approximation Vol. 1: One-dimensional theory. Pure and Applied Mathematics Vol. 40. Academic Press: New York-London (1971).
[5] Fatou, P.: Séries trigonométriques et séries de Taylor (French). Acta Math. 30 (1) 335–400 (1906).
[6] Gadžiev, A.D.: The asymptotic value of the approximation of derivatives by the derivatives of a family of linear operators (Russian). Izv. Akad. Nauk Azerba˘ıdžan. SSR Ser. Fiz.-Mat. Tehn. Nauk 6 15–24 (1962).
[7] Gadžiev, A.D.: On the convergence of integral operators (Russian). Akad. Nauk Azerba˘ıdžan. SSR Dokl. 19 (12) 3–7 (1963).
[8] Gadžiev, A.D.: On the speed of convergence of a class of singular integrals (Russian). Izv. Akad. Nauk Azerbaıdžan. SSR Ser. Fiz.-Mat. Tehn. Nauk 6, 27–31 (1963).
[9] Gadžiev, A.D.: The order of convergence of singular integrals which depend on two parameters (Russian). In: Special problems of functional analysis and their applications to the theory of differential equations and the theory of functions (Russian), 40-44 (1968).
[10] Karsli, H., Ibikli, E.: Approximation properties of convolution type singular integral operators depending on two parameters and of their derivatives in L1(a; b). In: Proceedings of the 16th International Conference of the Jangjeon Mathematical Society, 66–76, Jangjeon Math. Soc.: Hapcheon (2005).

[11] Karsli, H.: Approximation properties of singular integrals depending on two parameters and their derivatives (Turkish). PhD Thesis, Ankara University, Ankara, 75 pp. (2006).
[12] Karsli, H.: Convergence of the derivatives of nonlinear singular integral operators. J. Math. Anal. Approx. Theory 2 (1) 26–34 (2007).
[13] Matsuoka, Y.: Asymptotic formula for Vallée Poussin’s singular integrals. Sci. Rep. Kagoshima Univ. 9, 25–34 (1960).
[14] Saks, S.: Theory of the integral. Monografie Matematyczne,Warszawa (1937).
[15] Siudut, S.: On the convergence of double singular integrals. Comment. Math. Prace Mat. 28 (1) 143–146 (1988).
[16] Siudut, S.: On the Fatou type convergence of abstract singular integrals. Comment. Math. Prace Mat. 30 (1) 171–176 (1990).
[17] Spivak, M.D.: Calculus. (3rd ed.), Publish or Perish, Incorporated: Houston-Texas (1994).
[18] Stein, E.M.: Singular integrals and differentiability properties of functions. Princeton Mathematical Series, No. 30. Princeton Univ. Press: Princeton-New Jersey (1970).
[19] Stein, E.M., Weiss, G.: Introduction to Fourier analysis on Euclidean spaces. Princeton Mathematical Series, No. 32. Princeton Univ. Press: Princeton-New Jersey (1971).
[20] Taberski, R.: Singular integrals depending on two parameters. Prace Mat. 7, 173–179 (1962).
[21] Taberski, R.: On double integrals and Fourier series. Ann. Polon. Math. 15, 97–115 (1964).
[22] Uysal, G., Yilmaz, M.M., Ibikli, E.: Approximation by radial type multidimensional singular integral operators. Palest. J. Math. 5 (2) 61–70 (2016).
[23] Yilmaz, B., Aral, A., Tunca, G.B.: Weighted approximation properties of generalized Picard operators. J. Comput. Anal. Appl. 13 (3) 499–513 (2011).
[24] Žornickaja, L.V.: The derivatives of some singular integrals (Russian). Izv. Vysš. Uˇcebn. Zaved. Matematika 29 (4) 62–72 (1962).

Details

Primary Language

English

Subjects

Mathematical Sciences

Journal Section

Research Article

Authors

Gumrah Uysal
0000-0001-7747-1706
Türkiye

Başar Yılmaz ^*
0000-0003-3937-992X
Türkiye

Publication Date

March 1, 2021

Submission Date

July 3, 2020

Acceptance Date

December 4, 2020

Published in Issue

Year 2021 Volume: 9 Number: 1

DOI

https://doi.org/10.36753/mathenot.763854

IZ

https://izlik.org/JA27DB62RM

APA

Uysal, G., & Yılmaz, B. (2021). On Convergence of Partial Derivatives of Multidimensional Convolution Operators. Mathematical Sciences and Applications E-Notes, 9(1), 9-21. https://doi.org/10.36753/mathenot.763854

AMA

1.Uysal G, Yılmaz B. On Convergence of Partial Derivatives of Multidimensional Convolution Operators. Math. Sci. Appl. E-Notes. 2021;9(1):9-21. doi:10.36753/mathenot.763854

Chicago

Uysal, Gumrah, and Başar Yılmaz. 2021. “On Convergence of Partial Derivatives of Multidimensional Convolution Operators”. Mathematical Sciences and Applications E-Notes 9 (1): 9-21. https://doi.org/10.36753/mathenot.763854.

EndNote

Uysal G, Yılmaz B (March 1, 2021) On Convergence of Partial Derivatives of Multidimensional Convolution Operators. Mathematical Sciences and Applications E-Notes 9 1 9–21.

IEEE

[1]G. Uysal and B. Yılmaz, “On Convergence of Partial Derivatives of Multidimensional Convolution Operators”, Math. Sci. Appl. E-Notes, vol. 9, no. 1, pp. 9–21, Mar. 2021, doi: 10.36753/mathenot.763854.

ISNAD

Uysal, Gumrah - Yılmaz, Başar. “On Convergence of Partial Derivatives of Multidimensional Convolution Operators”. Mathematical Sciences and Applications E-Notes 9/1 (March 1, 2021): 9-21. https://doi.org/10.36753/mathenot.763854.

JAMA

1.Uysal G, Yılmaz B. On Convergence of Partial Derivatives of Multidimensional Convolution Operators. Math. Sci. Appl. E-Notes. 2021;9:9–21.

MLA

Uysal, Gumrah, and Başar Yılmaz. “On Convergence of Partial Derivatives of Multidimensional Convolution Operators”. Mathematical Sciences and Applications E-Notes, vol. 9, no. 1, Mar. 2021, pp. 9-21, doi:10.36753/mathenot.763854.

Vancouver

1.Gumrah Uysal, Başar Yılmaz. On Convergence of Partial Derivatives of Multidimensional Convolution Operators. Math. Sci. Appl. E-Notes. 2021 Mar. 1;9(1):9-21. doi:10.36753/mathenot.763854

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