Approximation in weighted Orlicz spaces. (English) Zbl 1265.41029
All the classical theorems of approximation theory (Jackson type-estimates, converse results, Bernstein-type inequality, equivalence of a modulus of smoothness with a \(K\)-functional) are proven in a very general setting in weighted Orlicz spaces (both in the trigonometric and in the algebraic case). In these problems, a weight function introduces a serious additional problem, so the results are very far from trivial extensions of earlier works. The modulus of smoothness is defined in terms of the averaging Steklov operator, and it corresponds to \(2r\)th classical moduli, and the results are matching the classical results for \(2r\)th moduli.
Reviewer: Vilmos Totik (Hungary)
MSC:
| 41A25 | Rate of convergence, degree of approximation |
| 41A27 | Inverse theorems in approximation theory |
References:
| [1] | BUTZER, P. L.– NESSEL, R. J.: Fourier Analysis and Approximation. Vol. 1: Onedimensional Theory, Birkhauser, Basel-Stuttgart, 1971. · Zbl 0217.42603 |
| [2] | DE BONIS, M. C.– MASTROIANNI, G.– RUSSO, M. G.: Polynomial approximation with special doubling weights, Acta Sci. Math. (Szeged) 69 (2003), 159–184. · Zbl 1027.41006 |
| [3] | DE VORE, R. A.– LORENTZ, G. G.: Constructive approximation. Grundlehren Math. Wiss. 303, Springer, Berlin, 1993. |
| [4] | DITZIAN, Z.– TOTIK, V.: 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 |
| [5] | DITZIAN, Z.– TOTIK, V.: Moduli of Smoothness. Springer Ser. Comput. Math. 9, Springer, New York, 1987. · Zbl 0666.41001 |
| [6] | DITZIAN, Z.– TOTIK, V.: K-functionals and weighted moduli of smoothness, J. Approx. Theory 63 (1990), 3–29. · Zbl 0715.46043 · doi:10.1016/0021-9045(90)90110-C |
| [7] | GARIDI, W.: On approximation by polynomials in Orlicz spaces, Approx. Theory Appl. (N.S.) 7 (1991), 97–110. · Zbl 0761.41046 |
| [8] | GENEBASHVILI, I.– GOGATISHVILI, A.– KOKILASHVILI, V. M.– KRBEC, M.: Weight theory for integral transforms on spaces of homogeneous type, Addison Wesley Longman, Harlow, 1998. · Zbl 0955.42001 |
| [9] | HACIYEVA, E. A.: Investigation of the properties of functions with quasimonotone Fourier coefficients in generalized Nikolskii-Besov spaces. Author’s summary of Dissertation, Tbilisi, 1986 (Russian). |
| [10] | ISRAFILOV, D. M.: Approximation by p-Faber polynomials in the weighted Smirnov class E p (G, {\(\omega\)}) and the Bieberbach polynomials, Constr. Approx. 17 (2001), 335–351. · Zbl 0987.41016 · doi:10.1007/s003650010030 |
| [11] | ISRAFILOV, D. M.: Approximation by p-Faber-Laurent rational functions in the weighted Lebesgue spaces, Czechoslovak Math. J. 54 (2004), 751–765. · Zbl 1080.41500 · doi:10.1007/s10587-004-6423-7 |
| [12] | ISRAFILOV, D. M.– AKGÜN, R.: Approximation in weighted Smirnov-Orlicz classes, J. Math. Kyoto Univ. 46 (2006), 755–770. · Zbl 1140.30013 · doi:10.1215/kjm/1250281602 |
| [13] | ISRAFILOV, D. M.– GUVEN, A.: Approximation by trigonometric polynomials in weighted Orlicz spaces, Studia Math. 174 (2006), 147–168. · Zbl 1127.42003 · doi:10.4064/sm174-2-3 |
| [14] | KHABAZI, M.: The mean convergence of trigonometric Fourier series in weighted orlicz classes, Proc. A. Razmadze Math. Inst. 129 (2002), 65–75. · Zbl 1084.42005 |
| [15] | KOKILASHVILI, V. M.: On analytic functions of Smirnov-Orlicz classes, Studia Math. 31 (1968), 43–59. · Zbl 0182.40103 |
| [16] | KOKILASHVILI, V. M.: A direct theorem on mean approximation of analytic functions by polynomials, Dokl. Akad. Nauk SSSR 10 (1969), 411–414. · Zbl 0212.09901 |
| [17] | KRASNOSELSKII, M. A.– RUTICKII, YA. B.: Convex Functions and Orlicz Spaces, P. Noordhoff Ltd., Groningen, 1961. |
| [18] | KY, N. X.: On approximation by trigonometric polynomials in L u p -spaces, Studia Sci. Math. Hungar. 28 (1993), 183–188. · Zbl 0797.42002 |
| [19] | MASTROIANNI, G.– TOTIK, V.: Jackson type inequalities for doubling and A p weights. In: Proc. Third International Conference on Functional Analysis and Approximation Theory, Vol. 1 (Acquafredda di Maratea, 1996). Rend. Circ. Mat. Palermo (2) Suppl. 52 (1998), 83–99. · Zbl 0908.41005 |
| [20] | MASTROIANNI, G.– TOTIK, V.: Weighted polynomial inequalities with doubling and A weights, Constr. Approx. 16 (2000), 37–71. · Zbl 0956.42001 · doi:10.1007/s003659910002 |
| [21] | MASTROIANNI, G.– TOTIK, V.: Best approximation and moduli of smoothness for doubling weights, J. Approx. Theory 110 (2001), 180–199. · Zbl 0981.41016 · doi:10.1006/jath.2000.3546 |
| [22] | MHASKAR, H. N.: Introduction to the theory of weighted polynomial approximation. Series in Approximation and Decompositions 7, World Sci., River Edge, NJ, 1996. · Zbl 0948.41500 |
| [23] | MUCKENHOUPT, B.: Two weight function norm inequalities for the Poisson integral, Trans. Amer. Math. Soc. 210 (1975), 225–231. · Zbl 0311.42008 · doi:10.1090/S0002-9947-1975-0374790-0 |
| [24] | RAMAZANOV, A. R-K.: On approximation by polynomials and rational functions in Orlicz spaces, Anal. Math. 10 (1984), 117–132. · Zbl 0566.41015 · doi:10.1007/BF02350522 |
| [25] | RAO, M. M.– REN, Z. D.: Theory of Orlicz Spaces, Marcel Dekker, New York, 1991. · Zbl 0724.46032 |
| [26] | RUNOVSKI, K.: On Jackson type inequality in Orlicz classes, Rev. Mat. Complut. 14 (2001), 395–404. · Zbl 1004.42002 · doi:10.5209/rev_REMA.2001.v14.n2.16986 |
| [27] | TIMAN, A. F.: Theory of approximation of functions of a real variable. International Series of Monographs on Pure and Applied Mathematics 34, Pergamon Press and MacMillan, Oxford, 1963 [Russian original published by Fizmatgiz, Moscow, 1960]. |
| [28] | ZYGMUND, A.: Trigonometric series, Vol. I and II, Cambridge University Press, Cambridge, 1959. · Zbl 0085.05601 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
