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Sharp inequalities between the best root-mean-square approximations of analytic functions in the disk and some smoothness characteristics in the Bergman space. (English. Russian original) Zbl 1477.41006

Math. Notes 110, No. 2, 248-260 (2021); translation from Mat. Zametki 110, No. 2, 266-281 (2021).
Summary: In Jackson-Stechkin type inequalities for the smoothness characteristic \(\Lambda_m(f)\), \(m\in\mathbb{N}\), we find exact constants determined by averaging the norms of finite differences of \(m\)th order of a function \(f\in B_2\). We solve the problem of best joint approximation for a certain class of functions from \(B_2^{(r)}\), \(r\in\mathbb{Z}_+\) whose smoothness characteristic \(\Lambda_m(f)\) averaged with a given weight is bounded above by the majorant \(\Phi \). The exact values of \(n\)-widths of some classes of functions are also calculated.

MSC:

41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)
30E10 Approximation in the complex plane
41A25 Rate of convergence, degree of approximation
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[1] D. Jackson, Über die Genauigkeit der Annäherung stetiger Funktionen durch ganze rationale Funktionen gegebenen Grades und trigonometrischen Summen gegebener Ordnung, Dissertation (Göttingen, 1911). · JFM 42.0434.03
[2] Quade, E. S., Trigonometric approximation in the mean, Duke Math. J., 3, 529-543 (1937) · JFM 63.0224.01
[3] Akhiezer, N. I., Lectures in Approximation Theory (1965), Moscow: Nauka, Moscow
[4] Stechkin, S. B., On the order of the best approximations of continuous functions, Izv. Akad. Nauk SSSR Ser. Mat., 15, 3, 219-242 (1951) · Zbl 0042.30001
[5] Ivanov, V. I., Direct and inverse theorems in approximation theory for periodic functions in S. B. Stechkin’s papers and the development of these theorems, Proc. Steklov Inst. Math. (Suppl.), 273, 1, S1-S13 (2011) · Zbl 1227.49026
[6] Storozhenko, È. A.; Krotov, V. G.; Oswald, P., Direct and converse theorems of Jackson type in \(L^p\) spaces, \(0<p<1\), Math. USSR-Sb., 27, 3, 355-374 (1975) · Zbl 0372.41004
[7] Storozhenko, È. A.; Oswald, P., Jackson’s theorem in the spaces \(L^p(R^k), 0<p<1\), Siberian Math. J., 19, 4, 630-640 (1978) · Zbl 0414.41019
[8] Ivanov, V. I., Direct and converse theorems of the theory of approximation in the metric of \(L_p\) for \(0<p<1\), Math. Notes, 18, 5, 972-982 (1975) · Zbl 0349.42005
[9] Korneichuk, N. P., The exact constant in D. Jackson’s theorem on best uniform approximation of continuous periodic functions, Dokl. Akad. Nauk SSSR, 145, 3, 514-515 (1962) · Zbl 0136.36305
[10] Korneichuk, N. P., Precise constant in Jackson’s inequality for continuous periodic functions, Math. Notes, 32, 5, 818-821 (1982) · Zbl 0517.42002
[11] Chernykh, N. I., The best approximation of periodic functions by trigonometric polynomials in \(L^2\), Math. Notes, 2, 5, 803-808 (1967) · Zbl 0167.05002
[12] Ligun, A. A., Exact constants of approximation for differentiable periodic functions, Math. Notes, 14, 1, 563-569 (1973) · Zbl 0282.42002
[13] Chernykh, N. I., Exact Jackson inequality in \(L_p(0,2\pi)(1\le p<2)\), Proc. Steklov Inst. Math., 198, 223-231 (1994)
[14] Berdyshev, V. I., Jackson’s theorem in \(L_p\), Proc. Steklov Inst. Math., 88, 1-14 (1967) · Zbl 0187.01904
[15] Shabozov, M. Sh.; Yusupov, G. A., Best polynomial approximations in \(L_2\) of classes of \(2\pi \)-periodic functions and exact values of their widths, Math. Notes, 90, 5, 748-757 (2011) · Zbl 1284.42003
[16] M. Sh. Shabozov and A. A. Shabozova, “Sharp inequalities of Jackson-Stechkin type for periodic functions in \(L_2\) differentiable in the Weyl sense,” in Trudy Inst. Mat. i Mekh. UrO RAN (2019), Vol. 25, pp. 255-264.
[17] Vakarchuk, S. B.; Zabutnaya, V. I., Inequalities between best polynomial approximations and some smoothness characteristics in the space \(L_2\) and widths of classes of functions, Math. Notes, 99, 2, 222-242 (2016) · Zbl 1342.41033
[18] Smirnov, V. I.; Lebedev, N. A., Constructive Theory of Functions of a Complex Variable (1964), Moscow-Leningrad: Nauka, Moscow-Leningrad · Zbl 0164.37503
[19] Abilov, V. A.; Abilova, F. V.; Kerimov, M. K., Sharp estimates for the rate of convergence of Fourier series of functions of a complex variable in the space \(L_2(D,p(z))\), Comput. Math. Math. Phys., 50, 6, 946-950 (2010) · Zbl 1224.43006
[20] M. Sh. Shabozov and M. S. Saidusajnov, “Mean-square approximation of functions of a complex variable by Fourier sums in orthogonal systems,” in Trudy Inst. Mat. i Mekh. UrO RAN (2019), Vol. 25, pp. 258-272.
[21] Shabozov, M. Sh.; Saidusajnov, M. S., Upper bounds for the approximation of certain classes of functions of a complex variable by Fourier series in the space \(L_2\) and \(n\)-widths, Math. Notes, 103, 4, 656-668 (2018) · Zbl 1397.42001
[22] Shabozov, M. Sh.; Saidusaynov, M. S., Mean-square approximation of complex variable functions by Fourier series in the weighted Bergman space, Vladikavkaz. Mat. Zh., 20, 1, 86-97 (2018) · Zbl 1451.30077
[23] Runovskii, K. V., On approximation by families of linear polynomial operators in \(l_P\)-spaces, \(0<p<1\), Russian Acad. Sci. Sb. Math., 82, 2, 441-459 (1995) · Zbl 0841.42001
[24] Gradshtein, I. S.; Ryzhik, I. M., Tables of Integrals, Sums, Series, and Products (1965), New York-London: Academic Press, New York-London
[25] Tikhomirov, V. M., Certain Questions of Approximation Theory (1976), Moscow: Izd. Moskov. Univ., Moscow
[26] Pinkus, A., \(n\)-Widths in Approximation Theory (1985), Berlin: Springer- Verlag, Berlin · Zbl 0551.41001
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