Approximation by Some Singular Operators Type of Functions from a Generalized Zygmund Space

Xhevat Krasniqi; Ram Mohapatra

doi:10.52737/18291163-2025.17.5-1-14

Authors

Xhevat Krasniqi University of Prishtina
Ram Mohapatra University of Central Florida

DOI:

https://doi.org/10.52737/18291163-2025.17.5-1-14

Keywords:

Degree of Approximation, Singular Integrals, Zygmund Modulus of Continuity, Generalized Zygmund Class, Generalized Minkowski Inequality

Abstract

This study continues previous research on the approximation of functions by means of singular integrals. We begin by introducing the truncated Picard singular integral. Subsequently, using this integral along with the classical Picard--Cauchy and Gauss--Weierstrass singular integrals, we establish the orders of approximation for functions belonging to a generalized Zygmund space, both in the $L^p$-norm and in the corresponding norm of the generalized Zygmund space.

References

G.A. Anastassiou and A. Aral, Generalized Picard singular integrals. Comput. Math. Appl., 57 (2009), no. 5, pp. 821-830.

A. Bogalska, E. Gojtka, M. Gurdek and L. Rempulska, The Picard and the Gauss-Weierstrass singular integrals of functions of two variables. Matematiche (Catania), 52 (1997), no. 1, pp. 71-85 (1998).

G. Das, A.K. Ojha and B.K. Ray, Degree of approximation of functions associated with Hardy-Littlewood series in the generalized Hölder metric. Proc. Indian Acad. Sci. Math. Sci., 108 (1998), no. 2, pp. 109-120.

G. Das, T. Ghosh and B.K. Ray, Degree of approximation of functions by their Fourier series in the generalized Hölder metric. Proc. Indian Acad. Sci. Math. Sci., 106, (1996), no. 2, pp. 139-153.

E. Deeba, R.N. Mohapatra and R.S. Rodriguez, On the degree of approximation of some singular integrals. Rend. Mat. Appl. (7) 8 (1988), no. 3, pp. 345-355 (1989).

B. Firlejy and L. Rempulska, On some singular integrals in Hölder spaces. Math. Nachr., 170 (1994), pp. 93-100.

S. G. Gal, Remark on the degree of approximation of continuous functions by singular integrals. Math. Nachr., 164 (1993), pp. 197-199.

G.H. Hardy, J.E. Littlewood and G. Pólya, Inequalities. Cambridge University Press, London, 1967.

H.H. Khan and G. Ram, On the degree of approximation by Gauss Weierstrass integrals. Int. J. Math. Math. Sci., 23 (2000), no. 9, pp. 645-649.

J. Kim, Degree of approximation in the space Hpω by the even-type delayed arithmetic mean of Fourier series. Georgian Math. J., 28 (2021), no. 5, pp. 747-753.

Xh.Z. Krasniqi, On approximation by Abel-Poisson and conjugate Abel-Poisson means in Hölder metric. Afr. Mat., 35 (2024), no. 2, Paper No. 51.

Xh.Z. Krasniqi, On approximation of bivariate functions by Abel-Poisson and conjugate Abel-Poisson means, J. Anal., 32 (2024), no. 3, pp. 1591-1617.

Xh.Z. Krasniqi, Approximation of functions in a certain Banach space by some generalized singular integrals. AIMS Math., 9 (2024), no. 2, pp. 3386-3398.

Xh.Z. Krasniqi, P. K'orus and B. Szal, Approximation by double second type delayed arithmetic mean of periodic functions in Hp(ω, ω) space. Bol. Soc. Mat. Mex. (3) 29 (2023), no. 1, Paper No. 21.

Xh.Z. Krasniqi, Effectiveness of the even-type delayed mean in approximation of conjugate functions. J. Contemp. Math. Anal., 58 (2023), no. 5, pp. 315-329; translated from Izv. Nats. Akad. Nauk Armenii Mat, 58 (2023), no. 5, pp. 17-32.

Xh.Z. Krasniqi, W. Łenski and B. Szal, Seminormed approximation by deferred matrix means of integrable functions in Hp(ω) space. Results Math., 77 (2022), no. 4, Paper No. 145, 16 pp.

S. Lal and Shireen, Best approximation of functions of generalized Zygmund class by Matrix-Euler summability mean of Fourier series. Bull. Math. Anal. Appl., 5 (2013), no. 4, pp. 1-13.

L. Leindler, A relaxed estimate of the degree of approximation by Fourier series in generalized Hölder metric. Anal. Math. 35 (2009), no. 1, pp. 51-60.

R.A. Mezei, Applications and Lipschitz results of approximation by smooth Picard and Gauss-Weierstrass type singular integrals, Cubo, 13 (2011), no. 3, pp. 17-48.

R.N. Mohapatra and R.S. Rodriguez, On the rate of convergence of singular integrals for Hölder continuous functions. Math. Nachr., 149 (1990), pp. 117-124.

L. Nayak, G. Das and B.K. Ray, An estimate of the rate of convergence of Fourier series in the generalized Hölder metric by deferred Cesàro mean. J. Math. Anal. Appl., 420 (2014), no. 1, pp. 563-575.

H.K. Nigam et al., Approximation of function using generalized Zygmund class. Adv. Difference Equ., 2021 (2021), Paper No. 34, 22 pp.

S. Prössdorf, Zur Konvergenz der Fourierreihen Hölderstetiger Funktionen (in German). Math. Nachr., 69 (1975), pp. 7-14.

L. Rempulska and K. Tomczak, On some properties of the Picard operators. Arch. Math. (Brno), 45 (2009), no. 2, pp. 125-135.