Direct and inverse approximation theorems of functions in the Orlicz type spaces 𝓢_M

References

[1] Akgün, R.: Approximating polynomials for functions of Weighted Smirnov-Orlicz spaces, J. Funct. Spaces Appl. 2012 (2012), 1–41.10.1155/2012/982360Search in Google Scholar

[2] Akgün, R.—Izrafilov, D.: Approximations in wieghted Orlicz spaces, Math. Slovaca 61(4) (2011), 601–618.10.2478/s12175-011-0031-4Search in Google Scholar

[3] Akhiezer, N. I.: Lectures on Approximation Theory, 2nd ed., Nauka, Moscow, 1965, (in Russian); Engl. transl. of the 1st ed., 1947: Theory of Approximation, Ungar, New York, 1956.Search in Google Scholar

[4] Bari, N. K.—Stechkin, S. B.: Best approximations and differential properties of two conjugate functions, Tr. Mosk. Mat. Obs. 5 (1956), 483–522, (in Russian).Search in Google Scholar

[5] Bernstein, S. N.: On the best approximation of continuous functions by polynomials of given degree, 1912, In: Collected Works 1, Acad. Nauk SSSR, Moscow, 1952, pp. 11–104, (in Russian).Search in Google Scholar

[6] Butzer, P.—Nessel, R.: Fourier Analysis and Approximation. One-Dimensional Theory, Birkhäuser, Basel, 1971.10.1007/978-3-0348-7448-9Search in Google Scholar

[7] Butzer, P. L.—Westphal, U.: An access to fractional differentiation via fractional difference quotients. In: Fractional Calculus and its Applications (B. Ross, ed.), Lecture Notes in Math. 457, Springer, Berlin, 1975, pp. 116–145.10.1007/BFb0067101Search in Google Scholar

[8] Chaichenko, S. O.: Inverse approximation theorems in the weighted Orlicz spaces, Bukovyn. Mat. Zh. 1(3-4) (2013), 148–157, (in Ukrainian).Search in Google Scholar

[9] DeVore, R. A.—Lorentz, G. G.: Constructive Approximation, Springer, Berlin, 1993.10.1007/978-3-662-02888-9Search in Google Scholar

[10] Dzyadyk, V. K.—Shevchuk, I. A.: Theory of Uniform Approximation of Functions by Polynomials, Walter de Gruyter GmbH & Co. KG, Berlin, 2008.10.1515/9783110208245Search in Google Scholar

[11] Garidi, W.: On approximation by polynomials in Orlicz spaces, Approx. Theory Appl. (N.S.) 7 (1991), 97–110.Search in Google Scholar

[12] Guven, A.—Israfilov, D. M.: On approximation in weigted Orlicz spaces, Math. Slovaca 62(1) (2012), 77–86.10.2478/s12175-011-0073-7Search in Google Scholar

[13] Israfilov, D. M.—Guven, A.: Approximation by trigonometric polynomials in weighted Orlicz spaces, Studia Math. 174 (2006), 147–168.10.4064/sm174-2-3Search in Google Scholar

[14] Jackson, D.: Über die Genauigkeit der Annäherung stetiger Funktionen durch ganze rationale Funktionen gegebenen Grades und trigonometrische Summen gegebener Ordnung, G öttingen, 1911, Thesis.Search in Google Scholar

[15] Jafarov, S. Z.: Approximation by rational functions in Smirnov-Orlicz classes, J. Math. Anal. Appl. 379 (2011), 870–877.10.1016/j.jmaa.2011.02.027Search in Google Scholar

[16] Jafarov, S. Z.: The inverse theorem of approximation of the function in Smirnov-Orlicz classes, Math. Inequal. Appl. 12(4) (2012) 835–844.Search in Google Scholar

[17] Jafarov, S. Z.: Approximation of conjugate functions by trigonometric polynomials in weighted Orlicz spaces, J. Math. Inequal. 7(2) (2013), 271–281.10.7153/jmi-07-25Search in Google Scholar

[18] Krasnosel’skii, M. A.—Rutickii, Ya. B.: Convex Functions and Orlicz Spaces, P.Noordhoff Ltd, Groningen, 1961.Search in Google Scholar

[19] Peetre, J.: A theory of interpolation of normed spaces. Notes, Brasilia, 1963.Search in Google Scholar

[20] Ramazanov, A. R-K.: On approximation by polynomials and rational functions in Orlicz spaces, Anal. Math. 10 (1984), 117–132.10.1007/BF02350522Search in Google Scholar

[21] Runovski, K.: On Jackson type inequality in Orlicz classes, Rev. Mat. Complut. 14 (2001), 395–404.10.5209/rev_REMA.2001.v14.n2.16986Search in Google Scholar

[22] Shidlich, A. L.—Chaichenko S. O.: Some extremal problems in the Orlicz spaces, Mat. Stud. 42(1) (2014), 21–32, (in Ukrainian).Search in Google Scholar

[23] Shidlich, A. L.—Chaichenko S. O.: Approximative properties of diagonal operators in Orlicz spaces, Numer. Funct. Anal. Optim. 36(10) (2015), 1339–1352.10.1080/01630563.2015.1066387Search in Google Scholar

[24] Stechkin, S. B.: On the order of the best approximations of continuous functions, Izv. Akad. Nauk SSSR, Ser. Mat. 15(3) (1951), 219–242, (in Russian).Search in Google Scholar

[25] Stepanets, A. I.: Approximation characteristics of the spacesSφp, Ukrainian Math. J. 53(3) (2001), 446–475.10.1023/A:1012348506457Search in Google Scholar

[26] Stepanets, A. I.: Methods of Approximation Theory, VSP, Leiden-Boston, 2005.10.1515/9783110195286Search in Google Scholar

[27] Stepanets, A. I.: Problems of approximation theory in linear spaces, Ukrainian Math. J. 58(1) (2006), 54–102.10.1007/s11253-006-0052-2Search in Google Scholar

[28] Stepanets, A. I.—Serdyuk, A. S.: Direct and inverse theorems in the theory of the approximation of functions in the space 𝓢^p, Ukrainian Math. J. 54(1) (2002), 126–148.10.1023/A:1019701805228Search in Google Scholar

[29] Sterlin, M. D.: Exact constants in inverse theorems of approximation theory, Dokl. Akad. Nauk SSSR 202 (1972), 545–547, (in Russian).Search in Google Scholar

[30] Timan, A. F.: Theory of approximation of functions of a real variable, Fizmatgiz, Moscow, 1960, (in Russian); Engl. transl. by J. Berry, International Series of Monographs on Pure and Applied Mathematics 34, Pergamon Press and MacMillan, Oxford, 1963.Search in Google Scholar

[31] Timan, M. F., Inverse theorems of the constructive theory of functions inL_p spaces (1 ≤ p ≤ ∞), Mat. Sb. 46(1) (1958), 125–132.Search in Google Scholar

[32] Timan, M. F.: Approximation and Properties of Periodic Functions, Nauk. dumka, Kiev, 2009, in Russian.Search in Google Scholar

[33] Vakarchuk, S. B.: Jackson-type inequalities and exact values of widths of classes of functions in the spaces S^p, 1 ≤ p < ∞, Ukrainian Math. J. 56(5) (2004), 718–729.10.1007/s11253-005-0070-5Search in Google Scholar

[34] Vallée-Poussin, C.-J.: Leşons sur l’Approximation des Functions d’une VariableRéelle, Gauthier-Villars, Paris, 1919.Search in Google Scholar

[35] Zygmund, A.: On the continuity module of the sum of the series conjugate to a Fourier series, Prace Mat.-Fiz. 33 (1924), 25–132, (in Polish).Search in Google Scholar