Some saturation classes for deferred Riesz and deferred Nörlund means
DOI:
https://doi.org/10.24193/subbmath.2025.2.08Keywords:
Lipschitz class, Fourier series, Deferred Riesz Means, Deferred N\"{o}rlund MeansAbstract
One of main problem in approximation theory is determination a saturation class for given method. The problem of determining a saturation class has been considered by Zamanski, Sunouchi and Watari and others. Mohaparta and Russel have considered some direct and inverse theorems in approximation of functions. Sunouchi and Watari have studied the Riesz means of type \(n\). In [5], Goel et al. have extended these results by considering Nörlund means.
In this paper, we examine some direct and inverse theorems in approximation of functions under weaker conditions by considering Deferred Riesz means and Deferred Nörlund means. Also, we extent above mentioned results.
References
[1] Agnew, R.P., On Deferred Ces aro means, Ann. of Math., 33(1932), no. 2, 413-421.
[2] Da gadur, _I., C atal, C., On convergence of deferred Norlund and deferred Riesz means of Mellin-Fourier series, Palest. J. Math., 8(2019), 127-136.
[3] Favard, J., Sur l'approximation des fonction d'une variable r eelle, Colloques Internationaux du Centre National de la Recherche Scienti que, 15(1949), 97-110.
[4] Favard, J., Sur la saturation des proc ed es de summation, J. Math. Pures Appl., 36(1957), 359-372.
[5] Goel, D.S., Holland, A.S.B., Nasim, C., Sahney, B.N., Best approximation by a saturation class of polynomial operators, Paci c J. Math., 55(1974), 149-155.
[6] Hardy, G.H., Littlewood, J.E., Some properties of fractional integral I, Math. Z., 27(1928), 565-600.
[7] Hardy, G.H., Littlewood, J.E., Polya, G., Inequalities, Cambridege Univ. Press, Cambridge, New York, 1939.
[8] Kuttner, B., Mohapatra R.N., Sahney, B.N., Saturation results for a class of linear operators, Math. Proc. Cambridge Philos. Soc., 94(1983), 133-148.
[9] Kuttner, B., Sahney, B.N., On non-uniqueness of the order of saturation, Math. Proc. Cambridge Philos. Soc., 84(1978), 113-116.
[10] Mohapatra, R.N., Russell, D.C., Some direct and inverse theorems in approximation of functions, J. Aust. Math. Soc., 34(1981), 143-154.
[11] Saini, K., Raj, K., Applications of statistical convergence in complex uncertain sequences via deferred Riesz mean, Internat. J. Uncertain. Fuzziness Knowledge-Based Systems, 29(2021), 337-351.
[12] Srivastava, H.M., Jena, B.B., Paikray, S.K., Misra, U.K., Generalized equi-statistical convergence of the deferred Norlund summability and its applications to associated approximation theorems, Rev. R. Acad. Cienc. Exactas F s. Nat. Ser. A Mat. RACSAM,
112(2018), 1487-1501.
[13] Sunouchi, G., Watari, C., On determination of the class of saturation in the theory of approximation of functions, Proc. Japan Acad. Ser. A Math. Sci., 34(1958), 477-481.
[14] Zamanski, M., Clases de saturation de certaines proces d'approximation des series de Fourier des functions continues, Ann. Sci. Ec. Norm. Sup er., 66, 19-93.
[15] Zymund, A., Trigonometric Series, Cambridege Univ. Press, Cambridge, New York, 1939.
