Averaged modulus of smoothness and two-sided monotone approximation in Orlicz spaces
The paper deals with basic properties of averaged modulus of smoothness in Orlicz spaces $L^*_\varphi$. Some direct and inverse two-sided approximation problems in $L^*_\varphi$ are proved. In the last section, some inequalities concerning monotone two sided approximation by trigonometric polynomials in $L^*_\varphi$ are considered.
___
- Akgün, R. Inequalities for one sided approximation in Orlicz spaces, Hacet. J. Math. Stat.
40 (2), 231-240, 2011.
- Akgün, R., Approximating polynomials for functions of weighted Smirnov-Orlicz spaces,J.
Funct. Spaces Appl. 41, Article ID 982360, 2012.
- Akgün, R., Some inequalities of trigonometric approximation in weighted Orlicz spaces,
Math. Slovaca 66 (1), 217-234, 2016.
- Akgün, R. and Israfilov, D. M. Simultaneous and converse approximation theorems in
weighted Orlicz spaces, Bull. Belg. Math. Soc. Simon Stevin 17 (1), 13-28, 2010.
- Akgün, R. and Israfilov, D. M. Approximation in weighted Orlicz spaces, Math. Slovaca, 61
(4), 601-618, 2011.
- Akgün, R. and Koç, H. Approximation by interpolating polynomials in weighted symmetric
Smirnov spaces, Hacet. J. Math. Stat. 41 (5), 643-649, 2012.
- Akgün, R. and Koç, H. Simultaneous approximation of functions in Orlicz spaces with
Muckenhoupt weights, Complex Var. Elliptic Equ. 61 (8), 1107-1115, 2016.
- Babenko, V. F. and Ligun, A. A. The order of the best one-sided approximation by polyno-
mials and splines in the $L_p$-metric, Math. Notes 19 (3), 194-198, 1976.
- Cohen, E. On the degree of approximation of a function by the partial sums of its Fourier
series, Trans. Amer. Math. Soc. 235, 35-74, 1978.
- Ganelius, T. On one-sided approximation by trigonometric polynomials, Math. Scand., 4,
247-256, 1956.
- Israfilov, D. M. and Guven, A. Approximation by trigonometric polynomials in weighted
Orlicz spaces. Studia Math. 174 (2), 147-168, 2006.
- Jafarov, S. Z., Approximation by Fejér sums of Fourier trigonometric series in weighted
Orlicz spaces. Hacet. J. Math. Stat. 42 (3), 259-268, 2013.
- Jafarov, S. Z., Approximation by linear summability means in Orlicz spaces. Novi Sad J.
Math. 44 (2), 161-172, 2014.
- Jafarov, S. Z., Approximation of functions by de la Vallée-Poussin sums in weighted Orlicz
spaces. Arab. J. Sci. Eng. (Springer) 5 (3), 125-137, 2016.
- Jafarov, S. Z., Approximation of periodic functions by Zygmund means in Orlicz spaces. J.
Class. Anal. 9 (1) , 43-52, 2016.
- Koç, H. Simultaneous approximation by polynomials in Orlicz spaces generated by quasi-
convex Young functions, Kuwait J. Sci. 43 (4), 18-31, 2016.
- Kokilašvili, V. M. An inverse theorem of the constructive theory of functions in Orlicz
spaces, (Russian) Soobšč. Akad. Nauk Gruzin. SSR 37 (1965), 263-270.
- Krasnosel'ski , M. A. and Rutickii, Y. B. Convex functions and Orlicz spaces, Translated
from the First Russian edition by Leo F. Boron, Popko Noordhoff Ltd., Groningen.1961.
- Lorentz, G. G. and Golitschek, M. V. and Makovoz, Y. Constructive approximation: Ad-
vanced problems, Springer-Verlag, 1996.
- Ponomarenko, V. G. Approximation of periodic functions in an Orlicz space, (Russian)
Sibirsk. Mat. Zh. 7, 1337-1346, 1966.
- Ramazanov, A.-R. K. On approximation by polynomials and rational functions in Orlicz
spaces, Anal. Math. 10 (2), 117-132, 1984.
- Sendov, B. and Popov,V. A. The averaged moduli of smoothness, Pure Appl. Math.,(New
York), Wiley, Chichester, 1988.
- Shadrin, A. Yu. Monotone approximation of functions by trigonometric polynomials, Mat.
Zametki, 34 (3), 375-386, 1983.
- Shadrin, A.Yu., Orders of one sided approximations of functions in $L_p$-metric, Anal. Math.
12, 175-184, 1986.
- Shadrin, A.Yu., Jackson type theorems for monotone approximation of functions by trigono-
metric polynomials, Mat. Zametki, 42 (6), 790-809, 1987.
- Tsyganok, I. I. A generalization of a theorem of Jackson, Mat. Sbornik 71 (113), 257-260,
1966.