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Akgün, R.

Polynomial approximation in Bergman spaces. (English) Zbl 1490.30019

Ukr. Math. J. 68, No. 4, 485-501 (2016) and Ukr. Mat. Zh. 68, No. 4, 435-448 (2016).
Summary: We obtain Jackson and converse inequalities for the polynomial approximation in Bergman spaces. Some known results presented for the moduli of continuity are extended to the moduli of smoothness. We also prove some simultaneous approximation theorems and deduce the Nikol’skii-Stechkin inequality for polynomials in these spaces.

MSC:

30E10 Approximation in the complex plane
30H20 Bergman spaces and Fock spaces
41A10 Approximation by polynomials

Keywords:

Jackson and converse inequalities; polynomial approximation; simultaneous approximation theorems; Nikolskii-Stechkin inequality
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References:

[1] S. Ja. Al’per, “Approximation of analytic functions over the region in the mean,” Dokl. Akad. Nauk SSSR, 136, 265-268 (1961).
[2] V. V. Arestov, “Integral inequalities for trigonometric polynomials and their derivatives,” Izv. Akad. Nauk SSSR, Ser. Mat., 45, No. 1, 3-22 (1981). · Zbl 0538.42001
[3] J. Burbea, “Polynomial approximation in Bers spaces of Carathéodory domains,” J. London Math. Soc. (2), 15, No. 2, 255-266 (1977). · Zbl 0365.30021
[4] R. A. DeVore and G. G. Lorentz, “Constructive approximation,” Fund. Princip. Math. Sci., Springer, Berlin (1993), 303. · Zbl 0797.41016
[5] P. Duren and A. Schuster, “Bergman spaces,” Math. Surv. Monogr., Amer. Math. Soc., Providence, RI (2004), 100. · Zbl 1059.30001
[6] D. Gaier, Lectures on Complex Approximation (1987). · Zbl 0612.30003
[7] Yu. V. Kryakin, “Jackson theorems in <Emphasis Type=”Italic“>H <Emphasis Type=”Italic“>p; 0 <Emphasis Type=”Italic“>< p < ∞<Emphasis Type=”Italic“>,” Izv. Vysch. Uchebn. Zaved. Mat., No. 8, 19-23 (1985). · Zbl 0582.30024
[8] Yu. Kryakin and W. Trebels, “<Emphasis Type=”Italic“>q-Moduli of continuity in <Emphasis Type=”Italic“>H <Emphasis Type=”Italic“>p(<Emphasis Type=”Italic“>D)<Emphasis Type=”Italic“>, p > 0<Emphasis Type=”Italic“>, and an inequality of Hardy and Littlewood,” J. Approxim. Theory, 115, No. 2, 238-259 (2002). · Zbl 1001.30031 · doi:10.1006/jath.2001.3656
[9] T. A. Metzger, “On polynomial approximation in <Emphasis Type=”Italic“>A <Emphasis Type=”Italic“>q(<Emphasis Type=”Italic“>D)<Emphasis Type=”Italic“>,” Proc. Amer. Math. Soc., 37, 468-470 (1973). · Zbl 0257.30034
[10] E. S. Quade, “Trigonometric approximation in the mean,” Duke Math. J., 3, No. 3, 529-543 (1937). · Zbl 0017.20501
[11] G. Ren and M. Wang, “Holomorphic Jackson’s theorems in polydisks,” J. Approxim. Theory, 134, No. 2, 175-198 (2005). · Zbl 1070.41007 · doi:10.1016/j.jat.2005.02.005
[12] M. Sh. Shabozov and O. Sh. Shabozov, “Best approximation and the values of widths for some classes of functions in the Bergman space <Emphasis Type=”Italic“>B <Emphasis Type=”Italic“>p, 1 ≤ <Emphasis Type=”Italic“>p ≤ ∞<Emphasis Type=”Italic“>,” Dokl. Akad. Nauk, 410, No. 4, 461-464 (2006). · Zbl 1134.32002
[13] Sh.-X. Chang and F.-Ch. Xing, “The problem of the best approximation by polynomials in the spaces <Emphasis Type=”Italic“>H <Emphasis Type=”Italic“>q <Emphasis Type=”Italic“>p (0 <Emphasis Type=”Italic“>< p < 1<Emphasis Type=”Italic“>, q > 1)<Emphasis Type=”Italic“>,” J. Math. Res. Expos., 9, No. 1, 107-114 (1989). · Zbl 0687.41024
[14] E. A. Storozhenko, “Jackson-type theorems in <Emphasis Type=”Italic“>H <Emphasis Type=”Italic“>p <Emphasis Type=”Italic“>, 0 <Emphasis Type=”Italic“>< p < 1<Emphasis Type=”Italic“>,” Izv. Akad. Nauk SSSR, Ser. Mat., 44, No. 4, 946-962 (1980). · Zbl 0455.42002
[15] E. A. Storozhenko, “On a Hardy-Littlewood problem,” Mat. Sb. (N. S.), 119(161), No. 4, 564-583 (1982).
[16] E. A. Storozhenko, “The Nikol’skiĭ-Stechkin inequality for trigonometric polynomials in <Emphasis Type=”Italic“>L0 <Emphasis Type=”Italic“>,” Mat. Zametki, 80, No. 3, 421-428 (2006). · Zbl 1117.41017 · doi:10.4213/mzm2828
[17] P. K. Suetin, Series of Faber Polynomials, Reading (1994).
[18] S. B. Vakarchuk, “On the best polynomial approximation in some Banach spaces for functions analytic in the unit disk,” Math. Notes, 55, No. 4, 338-343 (1994). · Zbl 0822.41006 · doi:10.1007/BF02112471
[19] M.-Zh. Wang and G.-B. Ren, “Jackson’s theorem on bounded symmetric domains,” Acta Math. Sinica (Engl. Ser.), 23, No. 8, 1391-1404 (2007). · Zbl 1148.41013 · doi:10.1007/s10114-007-0953-5
[20] L. V. Taikov, “Best approximation in the mean for some classes of analytic functions,” Mat. Zametki, 1, No. 2, 155-162 (1967). · Zbl 0168.10903
[21] F.-Ch. Xing, “Bernstein-type inequality in Bergman spaces <Emphasis Type=”Italic“>B <Emphasis Type=”Italic“>q <Emphasis Type=”Italic“>p (<Emphasis Type=”Italic“>p > 0<Emphasis Type=”Italic“>, q > 1)<Emphasis Type=”Italic“>,” Acta Math. Sinica (Chin. Ser.), 49, No. 2, 431-434 (2006). · Zbl 1115.30043
[22] F.-Ch. Xing, “Inverse theorem for the best approximation by polynomials in Bergman spaces <Emphasis Type=”Italic“>B <Emphasis Type=”Italic“>q <Emphasis Type=”Italic“>p (<Emphasis Type=”Italic“>p > 0<Emphasis Type=”Italic“>, q > 1)<Emphasis Type=”Italic“>,” Acta Math. Sinica (Chin. Ser.), 49, No. 3, 519-524 (2006). · Zbl 1117.30031
[23] F.-Ch. Xing and Ch.-L. Su, “On some properties of <Emphasis Type=”Italic“>H <Emphasis Type=”Italic“>q <Emphasis Type=”Italic“>p space,” J. Xinjiang Univ. Nat. Sci., 3, No. 3, 49-56 (1986).
[24] F.-Ch. Xing and Z. Su, “Inverse theorems to the best approximation by polynomials in <Emphasis Type=”Italic“>H <Emphasis Type=”Italic“>q <Emphasis Type=”Italic“>p (<Emphasis Type=”Italic“>p ≥ 1<Emphasis Type=”Italic“>, q > 1) spaces,” J. Xinjiang Univ. Natur. Sci., 6, No. 4, 36-46 (1989). · Zbl 1056.30501
[25] L.-F. Zhong, “Interpolation polynomial approximation in Bergman spaces,” Acta Math. Sinica, 34, No. 1, 56-66 (1991). · Zbl 0735.30035
[26] K. Zhu, “Translating inequalities between Hardy and Bergman spaces,” Amer. Math. Mon., 111, No. 6, 520-525 (2004). · Zbl 1072.46017 · doi:10.2307/4145071
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