On Convergence of Partial Derivatives of Multidimensional Convolution Operators

Year 2021, Volume: 9 Issue: 1, 9 - 21, 01.03.2021

Gumrah Uysal Başar Yılmaz

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

Cited By: 1

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).