On Some Bivariate Gauss-Weierstrass Operators

Year 2019, Volume: 2 Issue: 2, 57 - 63, 01.06.2019

Grazyna Krech , Ireneusz Krech

https://doi.org/10.33205/cma.518582

Cited By: 1

Abstract

The aim of the paper is to investigate the approximation properties of bivariate generalization of Gauss-Weierstrass operators associated with the Riemann-Liouville operator. In particular, the approximation error will be estimated by these operators in the space of functions defined and continuous in the half-plane $(0, \infty) \times \mathbb{R}$, and bounded by certain exponential functions.

Keywords

Gauss-Weierstrass operator, Linear operators, Approximation order

References

• [1] G. A. Anastassiou and A. Aral: On Gauss-Weierstrass type integral operators. Demonstratio Math. 43(4) (2010), 841– 849.
• [2] G. A. Anastassiou and O. Duman: Statistical approximation by double complex Gauss-Weierstrass integral operators. Appl. Math. Letters 24(4) (2011), 438–443.
• [3] G. A. Anastassiou and O. Duman: Statistical Lp-approximation by double Gauss-Weierstrass singular integral operators. Comput. Math. Appl. 59(6) (2010), 1985–1999.
• [4] G. A. Anastassiou and R. A. Mezei: Global smoothness and uniform convergence of smooth Gauss-Weierstrass singular operators. Math. Comput. Modelling 50(7-8) (2009), 984–998.
• [5] A. Aral: On a generalized $\lambda$-Gauss Weierstrass singular integral. Fasc. Math. 35 (2005), 23–33.
• [6] A. Aral: On the generalized Picard and Gauss Weierstrass singular integrals, J. Comput. Anal. Appl. 8(3) (2006), 246– 261.
• [7] A. Aral: Pointwise approximation by the generalization of Picard and Gauss-Weierstrass singular integrals. J. Concr. Appl. Math. 6 (2008), 327–339.
• [8] A. Aral and S. G. Gal: q-generalizations of the Picard and Gauss-Weierstrass singular integrals. Taiwanese J. Math. 12(9) (2008), 2501–2515.
• [9] B. Armi and L. T. Rachdi: The Littlewood-Paley g-function associated with the Riemann-Liouville operator. Ann. Univ. Paedagog. Crac. Stud. Math. 12 (2013), 31–58.
• [10] C. Baccar, N. B. Hamadi and L. T. Rachdi: Inversion formulas for Riemann-Liouville transform and its dual associated with singular partial differential operators. Int. J. Math. Math. Sci. (2006), Art. ID 86238, 26.
• [11] K. Bogalska, E. Gojka, M. Grudek and L. Rempulska: The Picard and the Gauss-Weierstrass singular integrals of function of two variables. Le Mathematiche LII (1997), 71–85.
• [12] L. R. Bragg: The radial heat polynomials and related functions. Trans. Amer. Math. Soc. 119 (1965), 270–290.
• [13] P. L. Butzer and R. J. Nessel: Fourier Analysis and Approximation. Vol 1, Birkhauser, Basel and Academic Press, New York 1971.
• [14] B. Firlej and L. Rempulska: On some singular integrals in Hölder spaces. Mat. Nachr. 170 (1994), 93–100.
• [15] F. H. Jackson: On a q-definite integrals. Quart. J. Pure Appl. Math. 41 (1910), 193–203.
• [16] A. Khan and S. Umar: On the order of approximation to a function by generalized Gauss-Weierstrass singular integrals. Commun. Fac. Sci. Univ. Ank., Series A1 30 (1981), 55–62.
• [17] R. A. Mezei: Applications and Lipschitz results of approximation by smooth Picard and Gauss-Weierstrass type singular integrals. Cubo 13(3) (2011), 17–48.
• [18] R. N. Mohapatra and R. S. Rodriguez: On the rate of convergence of singular integrals for Hölder continuous functions. Math. Nachr. 149 (1990), 117–124.
• [19] L. Rempulska and Z.Walczak: On modified Picard and Gauss-Weierstrass singular integrals. Ukrainian Math. J. 57(11) (2005), 1844-1852.
• [20] E. Wachnicki: On a Gauss-Weierstrass generalized integral. Rocznik Naukowo-Dydaktyczny Akademii Pedagogicznej w Krakowie, Prace Matematyczne 17 (2000), 251–263.