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Jafarov, Sadulla Z.

On moduli of smoothness of functions in Orlicz spaces. (English) Zbl 1435.41012

Tbil. Math. J. 12, No. 3, 121-129 (2019).
Summary: In this work the estimate about modulus of smoothness of the derivatives of the functions is obtained in Orlicz spaces. The relations between the modulus of smoothness of the functions with \(n\)th partial and Vallée-Poussin sums of the Fourier series in Orlicz spaces are studied.

Cited in 2 Documents

MSC:

41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)
41A25 Rate of convergence, degree of approximation
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)

Keywords:

Orlicz space; modulus of smoothness; Vallée-Poussin sum; polynomial approximation
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Full Text: DOI Euclid

References:

[1] R. Akgün and D. M. Israfilov, Polynomial approximation in weighted Smirnov-Orlicz space, Proc. A. Razmadze Math. Inst. 139 (2005), 89-92. · Zbl 1102.30033
[2] R.Akgün and D.M. Israfilov, Approximation and moduli of fractional orders in Smirnov-Orlicz classes, Glas.Mat. Ser. III 43 (63) (2008), no.1, 121-136. · Zbl 1152.30038 · doi:10.3336/gm.43.1.09
[3] R. Akgün, Improved converse theorems and fractional moduli of smoothness in Orlicz spaces, Bull. Malays. Math. Sci. Soc.(2) 36 (1), (2013), 49-62. · Zbl 1268.41013
[4] R. Akgün and D. M. Israfilov, Simultaneous and converse approximation theorems in weighted Orlicz spaces, Bull. Belg. Math. Soc. Simon Stevin 17 (2010), 13-28. · Zbl 1192.46025 · doi:10.36045/bbms/1267798496
[5] R. Akgün, Approximating polynomials for functions of weighted Smirnov- Orlicz spaces, J. Funct. Spaces Appl. 2012 Article ID 982360, 41 pages, 2012, do: 10. 1155/2012/982360. · Zbl 1250.41003
[6] P. L. Butzer, H. Dyckhoof, E. Cörlich and R. L. Stens, Best trigonometric approximation, fractional order derivatives and Lipschitz classes, Canad. J. Math. 29 (1977), no.4, 781-799. · Zbl 0357.26003
[7] D.W. Boyd, Indices for the Orlicz spaces, Pacific J.Math. 38 (1971). · Zbl 0227.46039 · doi:10.2140/pjm.1971.38.315
[8] R. A. DeVore and G. G. Lorentz, Constructive Approximation, Springer, New York, 1993. · Zbl 0797.41016
[9] Z.Ditzian, V.Totik, \(K-\) functionals and best polynomial approximation in weighted \(L^p(R)\), J.Approx. Theory 46 (1986), 38-41. · Zbl 0606.41010 · doi:10.1016/0021-9045(86)90084-5
[10] Z. Ditzian, V.Totik, Moduli of Smoothness, Springer Ser. Comput. Math., Springer, New York, 1987. · Zbl 0666.41001
[11] Z. Ditzian, V. H. Hristov, K.G. Ivanov, Moduli of smoothness and \(K\)-functionals in \(L_p, 0<p<1\). Constr. Approx. 11 (1995), 67-83. · Zbl 0822.46031
[12] Z. Ditzian, and S. Tikhonov, Ul’yanov and Nikolskii-type inequalities, J. Approx. Theory 133 (2005), 100-133. · Zbl 1069.41014 · doi:10.1016/j.jat.2004.12.008
[13] Z. Ditzian, S.Tikhnov, Moduli of Smoothness of functions and their derivatives, Studia Mathematica 180 (2) (2007), 143-160. · Zbl 1141.26005 · doi:10.4064/sm180-2-4
[14] F. Dai, Z. Ditzian and S. Tikhonov, Sharp Jackson inequalities, J. Approx. Theory 151 (2008), no. 1, 86-112. · Zbl 1136.41003
[15] P. L.Duren, Theory of \(H^p\) spaces, Academic Press, 1970. · Zbl 0215.20203
[16] D. M. Israfilov, A. Guven, Approximation by trigonometric polynomials in weighted Orlicz spaces, Studia Mathematica, 174 (2) (2006) 147-168. · Zbl 1127.42003 · doi:10.4064/sm174-2-3
[17] D. M.Israfilov, B. Oktay and R. Akgün, Approximation in Smirnov-Orlicz classes, Glas. Mat. Ser. III 40 (60) (2005), no.1, 87-102. · Zbl 1084.30041 · doi:10.3336/gm.40.1.09
[18] S. Z. Jafarov, On moduli of smoothness in Orlicz classes, Proc.of IMM of NAS of Azerbaijan, vol. XXXIII (2000), 95-100. · Zbl 1231.41016
[19] S. Z. Jafarov, Approximation by rational functions in Smirnov-Orlicz classes, Journal of Mathematical Analysis and Applications 379 (2011) 870-877. · Zbl 1214.30023 · doi:10.1016/j.jmaa.2011.02.027
[20] S. Z.Jafarov, J.I. Mamedkhanov, On approximation by trigonometric polynomials in Orlicz spaces, Georgian Math. J. 19 (2012), 687-695. · Zbl 1259.42002 · doi:10.1515/gmj-2012-0043
[21] M. A.Krasnoselskii and Ya.B.Rutickii, Convex Functions and Orlicz Spaces, Norrdhoff Ltd. (1961). · Zbl 0095.09103
[22] V.M. Kokilashvili, The converse theorem of constructive theory of functions in Orlicz spaces, Soobshch Akad. Nauk Gruzin. SSR 37 (1965), 263-270,(In Russian). · Zbl 0133.01801
[23] V. M. Kokilashvili, On approximation of periodic functions, Trudy Tbiliss. Mat. Inst. Razmadze Akad. Nauk. Gruzin. SSR 34 (1968), 51-81. · Zbl 0205.37301
[24] L. Maligranda, Indices and interpolation, Dissertation Math. 234 (1985). · Zbl 0566.46038
[25] W.Matuszewska and W.Orlicz, On certain properties of \(\phi \)-functions, Bull. Acad. Polon. Sci., Ser. Sci. Math. Astronom. Phys. 8 (1960), pp.439-443. · Zbl 0101.09001
[26] R. Ryan, Conjugate functions in Orlicz spaces, Pacific J. Math. 13 (1963),1371-1377. · Zbl 0133.37301 · doi:10.2140/pjm.1963.13.1371
[27] A.-R. K. Ramazanov, On approximation by polynomials and rational functions in Orlicz spaces, Analysis Mathematica, 10 (1984) pp.117-132. · Zbl 0566.41015 · doi:10.1007/BF02350522
[28] M. M. Rao and Z. D. Ren, Applications of Orlicz spaces, Marcel Dekker Inc. (2002). · Zbl 0997.46027
[29] S. B. Stechkin, The approximation of periodic functions by Fejér sums, Trudy Math. Inst. Steklov, G2 (1961), 522-523 (in Russian)
[30] B. V. Simonov and S. Yu. Tikhonov, On embeddings of function classes defined by constructive characteristics, Approximation and probability (Bedlewo, Poland 2004), Banach Center Publ. vol. 72, Polish Acad. Sci., Warsaw 2006, pp.285-307. · Zbl 1115.46029
[31] B. V. Simonov, S.Yu. Tikhonov, Embedding theorems in the constructive theory of approximations, Mat. Sb. (2008), 199(9), 107-148 (in Russian); translation in Sb. Math. 199 (2008), no. 9-10, 1367-1407. · Zbl 1172.46023
[32] S. Yu. Tikhonov, Moduli of smoothness and the interrelation of some classes of functions, Function spaces, interpolation theory and related topics (Lond. Sweden 2000), de Gruyter, Berlin 2002, pp.413-423. · Zbl 1129.46307
[33] S.Yu. Tikhonov, On moduli of smoothness of fracional order, Real Ana. Exchange 30 (2) (2005), 507-518. · Zbl 1108.26004
[34] A. F. Timan, Theory of Approximation of Functions of a Real Variable, Pergamon Press and MacMillan, 1963; Russian orginal published by Fizmatgiz, Moscow, 1960.
[35] R. M. Trigub and E. S. Belinsky, Fourier Analysis and Approximation of Functions, Kluwer, Dordrecht, 2004. · Zbl 1063.42001
[36] A. Zygmund, Trigonometric series, vol. I and II, Cambridge, 1959. · Zbl 0085.05601
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