On the Degree of Approximation by theWoronoi-Norlund and Riesz Type Means in the GHM

Ugur Deger, Hilal Bayindir

Abstract


The rst results on approximation in the Holder metric is based on the study of
Prossdorf. In 1979, Leindler gave a paper on the generalizations of Prossdorf's theorems. Later Leindler studied a similar problem on approximation by the Woronoi-Norlund and the Riesz means that is more general than the Cesaro means in 2009 with respect to the generalized Holder metric(GHM) given by Das, Nath and Ray. In this paper, our aim is to give some results extending the results of Leindler on the degree of approximation in GHM by using the more general methods of means on the classes larger than classes of sequences used in Leindler's study.


Keywords


Norlund submethod, Riesz submethod, Holder metric, degree of approximation

References


G. Alexits, Convergence problems of orthogonal series, New York: Pergamon Press,

D H. Armitage, I J. Maddox, A new type of Cesaro mean, Analysis, 9, 1989, 195-204.

P. Chandra, On the generalized Fejer means in the metric of the Holder space, Math.

Nachr., 109, 1982, 39-45.

P. Chandra, Trigonometric approximation of functions in Lp-norm, Journal of Mathematical

Analysis and Applications, 275, 2002, 13-26.

G. Das, T. Ghosh and B K. Ray, Degree of approximation of functions by their

Fourier series in the generalized Holder metric, Proc. Indian Acad. Sci. (Math. Sci.),

(2), 1996, 139-153.

G. Das, A. Nath and B K. Ray, An estimate of the rate of convergence of Fourier

series in generalized Holder metric, Analysis and Applications (Ujjain,1999), Narosa

(New Delhi, 2002): 43-60.

U. Deger, _ I. Dagadur and M. Kucukaslan, Approximation by trigonometric polynomials

to functions in Lp norm, Proc. Jangjeon Math. Soc., 15, 2012, 203-213.

U. Deger, M. Kaya, On the approximation by Cesaro submethod, Palestine Journal

of Mathematics, 4(1), 2015, 44-56.

U. Deger, M. Kucukaslan, A generalization of deferred Cesaro means and some of

their applications, Journal of Inequalities and Applications, 14, 2015, 1-16.

U. Deger, A note on degree of approximation by matrix means in generalized Holder

metric, Ukr. Mat. Zh., 68(4), 2016, 485-494.

U. Deger, On approximation by Norlund and Riesz submethods variable exponent

Lebesgue spaces, Commun. Fac. Sci. Univ. Ank. Series A1, 67(1), 2018, 46-59.

L. Leindler, Generalizations of Prossdorf's theorems, Studia Sci. Math. Hung., 14,

, 431439.

L. Leindler, Trigonometric approximation in Lp-norm, Journal of Mathematical

Analysis and Applications, 302, 2005, 129-136.

L. Leindler, A relaxed estimate of the degree of approximation by Fourier series in

generalized Holder metric, Analysis Mathematica , 35, 2009, 51-60.

S M. Mazhar, V. Totik, Approximation of continuous functions by T- means of

Fourier series, J. Approx. Theory, 60(2), 1990, 174-182.

R N. Mohapatra, P. Chandra, Degree of approximation of functions in the Holder

metric, Acta Math. Hung., 41(1-2), 1983, 67-76.

R N. Mohapatra, B. Szal, On trigonometric approximation of functions in the Lp-

norm (2012) arXiv:1205.5869v1 [math.CA].

S. Prosdor, Zur Konvergenz der Fourier reihen Holder Stetiger Funktionen, Math.

Nachr., 69, 1975, 7-14.

J A. Osikiewicz, Equivalance results for Cesaro submethods, Analysis, 20, 2000, 35-

E S. Quade, Trigonometric approximation in the mean, Duke Mathematical Journal,

, 1937, 529-542.


Refbacks

  • There are currently no refbacks.


free counters