An exact constant on the estimation of the approximation of classes of periodic functions of two variables by Ceśaro means

O. G. Rovenska

doi:10.30970/ms.57.1.3-9

O. G. Rovenska Donbas State Engineering Academy, Kramatorsk

DOI: https://doi.org/10.30970/ms.57.1.3-9

Keywords: Cesaro means, exact inequality, Lipschitz class

Abstract

In the present work, we study problem related to the approximation of continuous $2\pi$-periodic functions by linear means of their Fourier series. The simplest example of a linear approximation of periodic function is the approximation of this function by partial sums of the Fourier series. However, as well known, the sequence of partial Fourier sums is not uniformly convergent over the class of continuous $2\pi$-periodic functions. Therefore, a significant number of papers is devoted to the research of the approximative properties of different approximation methods, which are generated by some transformations of the partial sums of the Fourier series. The methods allow us to construct sequence of trigonometrical polynomials that would be uniformly convergent for all functions $f \in C$. Particularly, Ceśaro means and Fejer sums have been widely studied in past decades.
One of the important problems in this field is the study of the exact constant in an inequality for upper bounds of linear means deviations of the Fourier sums on fixed classes of periodic functions. Methods of investigation of integral representations for trigonometric polynomial deviations are generated by linear methods of summation of the Fourier series. They were developed in papers of Nikolsky, Stechkin, Nagy and others.

The paper presents known results related to the approximation of classes of continuous functions by linear means of the Fourier sums and new facts obtained for some particular cases.
In the paper, it is studied the approximation by the Ceśaro means of Fourier sums in Lipschitz class. In certain cases, the exact inequalities are found for upper bounds of deviations in the uniform metric of the second order rectangular Ceśaro means on the Lipschitz class of periodic functions in two variables.

References

Bugaets V.P., Martynyuk V.T. Exact constants of approximation of continuous functions by Jackson integrals// Ukr. Math. J. – 1974. – V.26. – P. 357–364.

Bugaets V.P., Martynyuk V.T. Exact constant for approximation of continuous functions by summation operators of Jackson type// Ukr. Math. J. – 1977. – V.29. – P. 586–590.

Falaleev L.P. On exact constants for matrix summation methods// Sib. Math. J. – 1995. – V.36. – P. 800–806.

Gavrilyuk, V.T. Approximation of continuous periodic functions of one or two variables by Rogozinski polynomials of interpolation type// Ukr. Math. J. – 1973. – V.25. – P. 530–537.

Gradshtein I.S., Ryzhik I.M. Tables of integrals, sums, series and productions. – M.: Fizmatgiz, 1971.

Zastavnyi V.P. Exact estimation of an approximation of some classes of differentiable functions by convolution operators// J. Math. Sci., New York – 2011. – V.175, №2. – P. 192–210.

Martynyuk V.T. Best constants for approximations of periodic functions by Fejйr operators// Ukr. Math. J. – 1990. – V.42. – P. 66–74.

Moricz F., Shi X. Approximation to Continuous Functions by Cesaro Means of Double Fourier Series and Conjugate Series// J. of Approx. Theory. – 1987. – V.49. – P. 346–317.

Nagy Sz.B. Approximation der Funktionen durch die arithmetischen Mittel ihrer Fourierschen Reihen// Acta scient. math. – 1946. – Bd.11, №1–2. – S. 71–84.

Nagy B. Fuggvenyek megkozelitese Fourier-sorok szamtani kozepeivel// Math. Fiz. Lapok. – 1942. – Bd. 49. – S. 123–138.

Natanson B.P. Constructive theory of functions. – M.: Gostekhteoretizdat, 1949.

Nikolsky S.M. On the asymptotic behavior of the remainder at approximation of functions satisfying the Lipschitz condition by the Fejer sums// Bull. Acad. Sci. URSS. Ser. Math. – 1940. – V.4. – P. 501–508.

Nikolsky S. M. Estimations of the remainder of Fejer’s sum for periodical functions possessing a bounded derivative// C. R. (Dokl.) Acad. Sci. URSS, n. Ser. – 1941. – V.31. – P. 210–214.

Ryzhankova G.I. Approximation of differentiable functions by Cesaro sums// Sib. Math. J. – 1976. – V.17. – P. 181–184.

Schurer F., Steutel F. On the Degree of Approximation by the Operators of de la Vallee Poussin// Monatshefte fur Mathematik. – 1979. – V.87. – P. 53–64.

Schurer F., Steutel F. On the Degree of Approximation of Functions in C1 2π with Operators of the Jackson Type // J. of Approx. Theory. – 1979. – V.27. – P. 153–178.

Stechkin S.B. The approximation of continuous periodic functions by Favard sums// Proc. Steklov Inst. Math. – 1971. – №109. – P. 28–38.

Wang Xing-hua. The exact constant of approximation of continuois functions by the Jackson singular integral// Acta Math. Sinica. – 1964. – V.14, №2. – P. 231–237.

Zigmund A. Trigonometric series. M.: Mir, 1965. Vol. 1.