Abstract
An arbitrary operator A on a Banach space
. such that either A or iA generates the Co-group with certain growth condition at infinity is considered. The direct and inverse theorems on connection between the degree of smoothness of a vector
with respect to the operator A, the rate of convergence to zero of the best approximation of x by exponential type entire vectors for the operator A, and the k-module of continuity are established. These results allow to obtain Jackson-type and Bernstein-type inequalities in weighted L p spaces.
This work was partially supported by the Ukrainian State Foundation for Fundamental Research (project N14.1/003).
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References
N.P. Kupcov, Direct and inverse theorems of approximation theory and semigroups of operators. Uspekhi Mat.Nauk. 23 (1968), No. 4, 118–178. (Russian)
A.P. Terehin, A bounded group of operators and best approximation. Differencial’nye Uravneniya i Vyçisl.Mat., Vyp. 2, 1975, 3–28. (Russian)
G.V. Radzievskii, On the best approximations and the rate of convergence of decompositions in the root vectors of an operator. Ukrain. Math. Zh. 49 (1997), no. 6, 754–773. (Russian); English trans. in Ukrainian Math. J. 49 (1997), no. 6, 844–864.
G.V. Radzievskii, Direct and converse theorems in problems of approximation by vectors of finite degree. Math. Sb. 189 (1998), no. 4, 83–124.
M.L. Gorbachuk and V.I. Gorbachuk, On the approximation of smooth vectors of a closed operator by entire vectors of exponential type. Ukrain. Mat. Zh. 47 (1995), no. 5, 616–628. (Ukrainian); English transl. in Ukrainian Math. J. 47 (1995), no. 5, 713-726.
M.L. Gorbachuk and V.I. Gorbachuk, Operator approach to approximation problems. St. Petersburg Math. J. 9 (1998), no. 6, 1097–1110.
M.L. Gorbachuk, Ya.I. Grushka and S.M. Torba, Direct and inverse theorems in the theory of approximations by the Ritz method. Ukrain. Mat. Zh. 57 (2005), no. 5, 633–643. (Ukrainian); English transl. in Ukrainian Math. J. 57 (2005), no. 5, 751-764.
Ju.I. Ljubic and V.I. Macaev, Operators with separable spectrum (Russian). Mat. Sb. 56(98) (1962), no. 4, 433–468. (Russian)
M.L. Gorbachuk, On analytic solutions of operator-differential equations. Ukrain. Mat. Zh. 52 (2000), no. 5, 596–607. (Ukrainian); English transl. in Ukrainian Math. J. 52 (2000), no. 5, 680-693.
V.A. Marchenko, On some questions of the approximation of continuous functions on the whole real axis. Zap. Mat. Otd. Fiz-Mat. Fak. KhGU i KhMO 22 (1951), no. 4, 115–125. (Russian)
O.I. Inozemcev and V.A. Marchenko, On majorants of genus zero. Uspekhi Mat. Nauk 11 (1956), 173–178. (Russian)
Walter Rudin, Real and Complex Analysis. McGrow-Hill, New York, 1970.
Ya. Grushka and S. Torba, Direct theorems in the theory of approximation of vectors in a Banach space with exponential type entire vectors. Methods Func. Anal. Topology 13 (2007), no. 3, pp. 267–278.
S. Torba, Inverse theorems in the theory of approximation of vectors in a Banach space with exponential type entire vectors. Methods Func. Anal. Topology 15 (2009), no. 4 (to appear).
Ganzburg M.I., Limit theorems and best constants in approximation theory // Anas-tassiou, George (ed.), Handbook of analytic-computational methods in applied mathematics. Boca Raton, FL: Chapman & Hall/CRC, 2000, pp. 507–569.
S.N. Bernstein, Collected Works, vol. I. Akad. Nauk SSSR, Moscow, 1952.
S.N. Bernstein, Collected Works, vol. II. Akad. Nauk SSSR, Moscow, 1954.
N.I. Akhiezer, Lectures on Approximation Theory, 2nd edition. Nauka, Moscow, 1965.
M. Reed, B. Simon, Methods of Modern Mathematical Physics: Functional Analysis I, Revised and enlarged edition. Academic Press, 1980.
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Grushka, Y., Torba, S. (2009). Direct and Inverse Theorems in the Theory of Approximation of Banach Space Vectors by Exponential Type Entire Vectors. In: Adamyan, V.M., et al. Modern Analysis and Applications. Operator Theory: Advances and Applications, vol 190. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-9919-1_17
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