Abstract
Let L ⊂ C be a regular Jordan curve. In this work, the approximation properties of the p-Faber-Laurent rational series expansions in the ω weighted Lebesgue spaces L p(L, ω) are studied. Under some restrictive conditions upon the weight functions the degree of this approximation by a kth integral modulus of continuity in L p(L, ω) spaces is estimated.
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S. Y. Alper: 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. Gos. Izdat. Fiz.-Mat. Lit., Moscow, 1960, pp. 272–286. (In Russian.)
J. E. Andersson: On the degree of polynomial approximation in E p (D). J. Approx. Theory 19 (1977), 61–68.
A. Çavuş and D. M. Israfilov: Approximation by Faber-Laurent rational functions in the mean of functions of the class L p (Γ) with 1 < p < 1 ∞. Approximation Theory App.11 (1995), 105–118.
G. David: Operateurs integraux singulers sur certaines courbes du plan complexe. Ann. Sci. Ecol. Norm. Super. 4 (1984), 157–189.
P. L. Duren: Theory of H p-Spaces. Academic Press, 1970.
E. M. Dyn'kin and B. P. Osilenker:Weighted estimates for singular integrals and their applications. In: Mathematical analysis, Vol. 21. Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1983, pp. 42–129. (In Russian.)
D. Gaier: Lectures on Complex Approximation. Birkhäuser-Verlag, Boston-Stuttgart, 1987.
G. M. Golusin: Geometric Theory of Functions of a Complex Variable. Translation of Mathematical Monographs, Vol. 26, AMS, 1969.
E. A. Haciyeva: Investigation of the properties of functions with quasimonotone Fourier coefficients in generalized Nikolsky-Besov spaces. Author's summary of candidates dis-sertation. Tbilisi. (In Russian.)
I. I. Ibragimov and D. I. Mamedhanov: A constructive characterization of a certain class of functions. Dokl. Akad. Nauk SSSR 223 (1975), 35–37; Soviet Math. Dokl. 4 (1976), 820–823.
D. M. Israfilov: Approximate properties of the generalized Faber series in an integral metric. Izv. Akad. Nauk Az. SSR, Ser. Fiz.-Tekh. Math. Nauk 2 (1987), 10–14. (In Russian.)
D. M. Israfilov: Approximation by p-Faber polynomials in the weighted Smirnov class E p (G; ω) and the Bieberbach polynomials. Constr. Approx. 17 (2001), 335–351.
V. M. Kokilashvili: A direct theorem on mean approximation of analytic functions by polynomials. Soviet Math. Dokl. 10 (1969), 411–414.
A. I. Markushevich: Theory of Analytic Functions, Vol. 2. Izdatelstvo Nauka, Moscow, 1968.
B. Muckenhoupt: Weighted norm inequalites for Hardy maximal functions. Trans. Amer. Math. Soc. 165 (1972), 207–226.
P. K. Suetin: Series of Faber Polynomials. Nauka, Moscow, 1984; Cordon and Breach Publishers, 1998.
J. L. Walsh and H. G. Russel: Integrated continuity conditions and degree of approxi-mation by polynomials or by bounded analytic functions. Trans. Amer. Math. Soc. 92 (1959), 355–370.
M. Wehrens: Best approximation on the unit sphere in R n. Funct. Anal. and Approx. Proc. Conf. Oberwolfach. Aug. 9-16, 1980, Basel. 1981, pp. 233–245.
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Israfilov, D.M. Approximation by p-Faber-Laurent Rational Functions in the Weighted Lebesgue Spaces. Czechoslovak Mathematical Journal 54, 751–765 (2004). https://doi.org/10.1007/s10587-004-6423-7
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DOI: https://doi.org/10.1007/s10587-004-6423-7