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Direct and inverse theorems on the approximation of \(2\pi\)-periodic functions by Taylor-Abel-Poisson operators. (English. Ukrainian original) Zbl 1499.41036

Ukr. Math. J. 69, No. 5, 766-781 (2017); translation from Ukr. Mat. Zh. 69, No. 5, 657-669 (2017).
Summary: We prove direct and inverse theorems on the approximation of \(2\pi\)-periodic functions by Taylor-Abel-Poisson operators in the integral metric.

MSC:

41A27 Inverse theorems in approximation theory

References:

[1] V. P. Zastavnyi and V. V. Savchuk, “Approximation of the classes of convolutions by linear operators of special form,” Mat. Zametki, 90, No. 3, 351-361 (2011). · Zbl 1286.41007 · doi:10.4213/mzm8545
[2] V. V. Savchuk, “Approximation of holomorphic functions by Taylor-Abel-Poisson means,” Ukr. Mat. Zh., 59, No. 9, 1253-1260 (2007); English translation:Ukr. Math. J., 59, No. 9, 1397-1407 (2007). · Zbl 1150.30031
[3] V. V. Savchuk and A. L. Shidlich, “Approximation of functions of several variables by linear methods in the space <Emphasis Type=”Italic“>S <Emphasis Type=”Italic“>p <Emphasis Type=”Italic“>,” Acta Sci. Math., 80, No. 3-4, 477-489 (2014). · Zbl 1340.42018
[4] C. K. Chui and A. S. B. Holland, “On the order of approximation by Euler and Taylor means,” J. Approxim. Theory, 39, No. 1, 24-38 (1983). · Zbl 0525.41018 · doi:10.1016/0021-9045(83)90066-7
[5] A. S. B. Holland, B. N. Sahney, and R. N. Mohapatra, “<Emphasis Type=”Italic“>L <Emphasis Type=”Italic“>p approximation of functions by Euler means,” Rend. Mat., 3(7), No. 2, 341-355 (1983). · Zbl 0536.42005
[6] R. N. Mohapatra, A. S. B. Holland, and B. N. Sahney, “Functions of class Lip(α<Emphasis Type=”Italic“>, p) and their Taylor mean,” J. Approxim. Theory, 45, No. 4, 363-374 (1985). · Zbl 0603.42009 · doi:10.1016/0021-9045(85)90032-2
[7] P. Chandra and R. N. Mohapatra, “Approximation of functions by (<Emphasis Type=”Italic“>J, q <Emphasis Type=”Italic“>n) means of Fourier series,” Approxim. Theory Appl., 4, No. 2, 49-54 (1988). · Zbl 0673.42002
[8] R. Leis, “Approximationssätze für stetige Operatoren,” Arch. Math., 14, 120-129 (1963). · Zbl 0196.43303 · doi:10.1007/BF01234932
[9] P. L. Butzer and G. Sunouchi, “Approximation theorems for the solution of Fourier’s problem and Dirichlet’s problem,” Math. Ann., 155, 316-330 (1964). · Zbl 0178.12603 · doi:10.1007/BF01354864
[10] P. Butzer and R. Nessel, Fourier Analysis and Approximation. One-Dimensional Theory, Birkhäuser, Basel (1971). · Zbl 0217.42603 · doi:10.1007/978-3-0348-7448-9
[11] R. A. de Vore and G. G. Lorentz, Constructive Approximation, Springer, Berlin (1993). · Zbl 0797.41016
[12] R. M. Trigub and E. S. Bellinsky, Fourier Analysis and Approximation of Functions, Kluwer AP, Dordrecht (2004). · Zbl 1063.42001 · doi:10.1007/978-1-4020-2876-2
[13] P. L. Butzer and H. G. Tillmann, “Approximation theorems for semigroups of bounded linear transformations,” Math. Ann., 140, 256-262 (1960). · Zbl 0094.30602 · doi:10.1007/BF01361149
[14] P. L. Butzer, “Beziehungen zwischen den Riemannschen, Taylorschen und gew¨ohnlichen Ableitungen reellwertiger Funktionen,” Math. Ann., 144, 275-298 (1961). · Zbl 0106.27002 · doi:10.1007/BF01470502
[15] W. Rudin, Function Theory in Polydiscs [Russian translation], Mir, Moscow (1974). · Zbl 0177.34101
[16] N. K. Bari and S. B. Stechkin, “Best approximations and differential properties of two conjugate functions,” Tr. Mosk. Mat. Obshch., 5, 483-522 (1956). · Zbl 0072.05702
[17] J. Prestin, V. V. Savchuk, and A. L. Shidlich, “Approximation of 2π-periodic functions by Taylor-Abel-Poisson operators in the integral metric,” Dop. Nats. Akad. Nauk Ukr., No. 1, 17-20 (2017). · Zbl 1374.42006
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