Journal of Function Spaces and Applications

Volume 2012 (2012), Article ID 982360, 41 pages

http://dx.doi.org/10.1155/2012/982360

Research Article

## Approximating Polynomials for Functions of Weighted Smirnov-Orlicz Spaces

Department of Mathematics, Faculty of Art and Science, Balikesir University, 10145 Balikesir, Turkey

Received 9 July 2009; Accepted 13 December 2009

Academic Editor: V. M. Kokilashvili

Copyright © 2010 Ramazan Akgün. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Linked References

- G. M. Goluzin,
*Geometric Theory of Functions of a Complex Variable*, vol. 26 of*Translations of Mathematical Monographs*, American Mathematical Society, Providence, RI, USA, 1969. - J. L. Walsh and H. G. Russell, “Integrated continuity conditions and degree of approximation by polynomials or by bounded analytic functions,”
*Transactions of the American Mathematical Society*, vol. 92, pp. 355–370, 1959. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S. Ya. Al'per, “Approximation in the mean of analytic functions of class ${E}^{p}$,” in
*Investigations on the Modern Problems of the Function Theory of a Complex Variable*, pp. 273–286, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, Russia, 1960. View at Google Scholar - V. M. Kokilashvili, “Approximation in the mean of analytic functions of class ${E}_{p}$,”
*Soviet Mathematics. Doklady*, vol. 8, pp. 1393–1397, 1967. View at Google Scholar - W. Kokilashvili, “A direct theorem on mean approximation of analytic functions by polynomials,”
*Soviet Mathematics. Doklady*, vol. 10, pp. 411–414, 1969. View at Google Scholar · View at Zentralblatt MATH - J.-E. Andersson, “On the degree of polynomial approximation in ${E}^{p}(D)$,”
*Journal of Approximation Theory*, vol. 19, no. 1, pp. 61–68, 1977. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - D. M. Israfilov, “Approximate properties of the generalized Faber series in an integral metric,”
*Izvestiya Akademii Nauk Azerbaidzhanskoi SSR*, vol. 2, pp. 10–14, 1987 (Russian). View at Google Scholar · View at Zentralblatt MATH - A. Çavuş and D. M. Israfilov, “Approximation by Faber-Laurent rational functions in the mean of functions of class ${L}_{p}(\mathrm{\Gamma})$ with $1<p<\infty $,”
*Approximation Theory and its Applications*, vol. 11, no. 1, pp. 105–118, 1995. View at Google Scholar · View at Zentralblatt MATH - D. M. Israfilov, “Approximation by $p$-Faber polynomials in the weighted Smirnov class ${E}^{p}(G,\omega )$ and the Bieberbach polynomials,”
*Constructive Approximation*, vol. 17, no. 3, pp. 335–351, 2001. View at Publisher · View at Google Scholar · View at MathSciNet - D. M. Israfilov, “Approximation by $p$-Faber-Laurent rational functions in the weighted Lebesgue spaces,”
*Czechoslovak Mathematical Journal*, vol. 54, no. 3, pp. 751–765, 2004. View at Publisher · View at Google Scholar · View at MathSciNet - D. M. Israfilov and A. Guven, “Approximation in weighted Smirnov classes,”
*East Journal on Approximations*, vol. 11, no. 1, pp. 91–102, 2005. View at Google Scholar · View at Zentralblatt MATH - A. Guven and D. M. Israfilov, “Improved inverse theorems in weighted Lebesgue and Smirnov spaces,”
*Bulletin of the Belgian Mathematical Society. Simon Stevin*, vol. 14, no. 4, pp. 681–692, 2007. View at Google Scholar · View at Zentralblatt MATH - W. Kokilashvili, “On analytic functions of Smirnov-Orlicz classes,”
*Studia Mathematica*, vol. 31, pp. 43–59, 1968. View at Google Scholar - A. Guven and D. M. Israfilov, “Polynomial approximation in Smirnov-Orlicz classes,”
*Computational Methods and Function Theory*, vol. 2, no. 2, pp. 509–517, 2002. View at Google Scholar · View at Zentralblatt MATH - D. M. Israfilov, B. Oktay, and R. Akgun, “Approximation in Smirnov-Orlicz classes,”
*Glasnik Matematički III*, vol. 40, no. 1, pp. 87–102, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. Akgün and D. M. Israfilov, “Approximation by interpolating polynomials in Smirnov-Orlicz class,”
*Journal of the Korean Mathematical Society*, vol. 43, no. 2, pp. 413–424, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - D. M. Israfilov and R. Akgün, “Approximation in weighted Smirnov-Orlicz classes,”
*Journal of Mathematics of Kyoto University*, vol. 46, no. 4, pp. 755–770, 2006. View at Google Scholar - R. Akgün and D. M. Israfilov, “Polynomial approximation in weighted Smirnov-Orlicz space,”
*Proceedings of A. Razmadze Mathematical Institute*, vol. 139, pp. 89–92, 2005. View at Google Scholar - D. M. Israfilov and R. Akgün, “Approximation by polynomials and rational functions in weighted rearrangement invariant spaces,”
*Journal of Mathematical Analysis and Applications*, vol. 346, no. 2, pp. 489–500, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - R. Akgün and D. M. Israfilov, “Approximation and moduli of smoothness of fractional order in Smirov-Orlicz spaces,”
*Glasnik Matematicki III*, vol. 42, no. 1, pp. 121–136, 2008. View at Google Scholar - A. I. Shvai, “Approximation on analytic functions by Poisson polynomials,”
*Ukrainian Mathematical Journal*, vol. 25, no. 6, pp. 710–713, 1973. View at Google Scholar - A. Gogatishvili and V. Kokilashvili, “Criteria of weighted inequalities in Orlicz classes for maximal functions defined on homogeneous type spaces,”
*Georgian Mathematical Journal*, vol. 1, no. 6, pp. 641–673, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - I. Genebashvili, A. Gogatishvili, V. Kokilashvili, and M. Krbec,
*Weight Theory for Integral Transforms on Spaces of Homogeneous Type*, vol. 92 of*Pitman Monographs and Surveys in Pure and Applied Mathematics*, Longman, Harlow, UK, 1998. - R. Akgün and D. M. Israfilov, “Simultaneous and converse approximation theorems in weighted Orlicz spaces,”
*Bulletin of the Belgian Mathematical Society. Simon Stevin*, vol. 17, no. 1, pp. 13–28, 2010. View at Google Scholar · View at Zentralblatt MATH - E. A. Gadjieva,
*Investigation of the Properties of Functions with Quasimonotone Fourier Coeffcients in Generalized Nikolskii-Besov Spaces*, Author's Summary of Dissertation, Tbilisi, Georgia, 1986. - N. X. Ky, “On approximation by trigonometric polynomials in ${L}_{u}^{p}$-spaces,”
*Studia Scientiarum Mathematicarum Hungarica*, vol. 28, no. 1-2, pp. 183–188, 1993. View at Google Scholar - N. X. Ky, “Moduli of mean smoothness and approximation with ${A}_{p}$-weights,”
*Annales Universitatis Scientiarum Budapestinensis de Rolando Eötvös Nominatae. Sectio Mathematica*, vol. 40, pp. 37–48, 1997. View at Google Scholar - L. Ephremidze, V. Kokilashvili, and Y. E. Yildirir, “On the inverse inequalities for trigonometric polynomial approximations in weighted Lorentz spaces,”
*Proceedings of A. Razmadze Mathematical Institute*, vol. 144, pp. 132–136, 2007. View at Google Scholar · View at Zentralblatt MATH - V. Kokilashvili and Y. E. Yildirir, “On the approximation in weighted Lebesgue spaces,”
*Proceedings of A. Razmadze Mathematical Institute*, vol. 143, pp. 103–113, 2007. View at Google Scholar · View at Zentralblatt MATH - G. Mastroianni and V. Totik, “Best approximation and moduli of smoothness for doubling weights,”
*Journal of Approximation Theory*, vol. 110, no. 2, pp. 180–199, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z. Ditzian and D. S. Lubinsky, “Jackson and smoothness theorems for Freud weights in ${L}_{p}(0<p\le \infty )$,”
*Constructive Approximation*, vol. 13, no. 1, pp. 99–152, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z. Ditzian and V. Totik,
*Moduli of Smoothness*, vol. 9 of*Springer Series in Computational Mathematics*, Springer, New York, NY, USA, 1987. - H. N. Mhaskar,
*Introduction to the Theory of Weighted Polynomial Approximation*, vol. 7 of*Series in Approximations and Decompositions*, World Scientific, River Edge, NJ, USA, 1996. - A. Zygmund,
*Trigonometric Series*, vol. 1-2, Cambridge University Press, New York, NY, USA, 2nd edition, 1959. - S. G. Samko, A. A. Kilbas, and O. I. Marichev,
*Fractional Integrals and Derivatives*, Gordon and Breach Science Publishers, Yverdon, Switzerland, 1993. - I. Joó, “Saturation theorems for Hermite-Fourier series,”
*Acta Mathematica Hungarica*, vol. 57, no. 1-2, pp. 169–179, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Khabazi, “The mean convergence of trigonometric Fourier series in weighted Orlicz classes,”
*Proceedings of A. Razmadze Mathematical Institute*, vol. 129, pp. 65–75, 2002. View at Google Scholar · View at Zentralblatt MATH - P. K. Suetin,
*Series of Faber Polynomials*, vol. 1 of*Analytical Methods and Special Functions*, Gordon and Breach Science Publishers, Amsterdam, The Netherlands, 1998. - D. Gaier,
*Lectures on Complex Approximation*, Birkhäuser, Boston, Mass, USA, 1987. - V. I. Smirnov and N. A. Lebedev,
*Functions of a Complex Variable: Constructive Theory*, Translated from the Russian by Scripta Technica, The M.I.T. Press, Cambridge, Mass, USA, 1968. - X. C. Shen and L. Zhong, “On Lagrange Interpolation in ${E}^{p}(D)$ for $1<p<\infty $,”
*Advanced Mathematics*, vol. 18, pp. 342–345, 1989 (Chinese). View at Google Scholar - L. Y. Zhu, “A new class of interpolation nodes,”
*Advances in Mathematics*, vol. 24, no. 4, pp. 327–334, 1995 (Chinese). View at Google Scholar - L. Zhong and L. Zhu, “Convergence of interpolants based on the roots of Faber polynomials,”
*Acta Mathematica Hungarica*, vol. 65, no. 3, pp. 273–283, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Warschawski, “Über das randverhalten der ableitung der abbildungsfunktion bei konformer abbildung,”
*Mathematische Zeitschrift*, vol. 35, no. 1, pp. 321–456, 1932. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. M. Israfilov and A. Guven, “Approximation by trigonometric polynomials in weighted Orlicz spaces,”
*Studia Mathematica*, vol. 174, no. 2, pp. 147–168, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - B. Muckenhoupt, “Weighted norm inequalities for the Hardy maximal function,”
*Transactions of the American Mathematical Society*, vol. 165, pp. 207–226, 1972. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - W. Rudin,
*Real and Complex Analysis*, McGraw-Hill, New York, NY, USA, 3rd edition, 1987. - P. L. Duren,
*Theory of H*, vol. 38 of^{p}Spaces*Pure and Applied Mathematics*, Academic Press, New York, NY, USA, 1970.