Abstract
We obtain exact estimates for the approximation of functions defined on a sphere in the metrics of C and L 2 by linear methods of summation of Fourier series in spherical harmonics in the case where differential and difference properties of these functions are defined in the space L 2.
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REFERENCES
D. Jackson, Uber die Genauigkeit der Annaherung stetiger Funktionen durch ganze rationale Funktionen gegebenen Grades und trigonometrische Summen gegebener Ordnung, Thesis, Gottingen (1911).
S. B. Stechkin, “On the order of the best approximations of continuous functions,” Izv. Akad. Nauk SSSR, Ser. Mat., 15, 219–242 (1951).
N. P. Korneichuk, “Exact constant in the Jackson theorem on the best uniform approximation of continuous periodic functions,” Dokl. Akad. Nauk SSSR, 145, No.3, 514–515 (1962).
N. P. Korneichuk, “On exact constant in the Jackson inequality for continuous periodic functions,” Mat. Zametki, 32, No.6, 669–674 (1982).
V. V. Zhuk, “Some exact inequalities for uniform approximations of periodic functions,” Dokl. Akad. Nauk SSSR, 201, 263–266 (1967).
A. A. Ligun, “Some inequalities for upper bounds of seminorms on classes of periodic functions,” Mat. Zametki, 13, No.5, 647–654 (1973).
A. A. Ligun, “On exact constants in Jackson-type inequalities,” Mat. Zametki, 39, No.5, 248–256 (1985).
V. V. Zhuk, “On the approximation of periodic functions by linear methods of summation of Fourier series,” Sib. Mat. Zh., 9, No.3, 717–718 (1968).
V. V. Shalaev, “On the approximation of continuous periodic functions by trigonometric polynomials,” in: Investigations on Contemporary Problems of Summation and Approximation of Functions and Their Applications [in Russian], nepropetrovsk (1977), pp. 39–43.
N. P. Korneichuk, Exact Constants in Approximation Theory [in Russian], Nauka, Moscow (1987).
N. I. Chernykh, “On the Jackson inequality in L 2,” Tr. Mat. Inst. Akad. Nauk SSSR, 88, 71–74 (1967).
N. I. Chernykh, “On the best approximation of periodic functions by trigonometric polynomials in L 2,” Mat. Zametki, 2, No.5, 513–522 (1967).
A. G. Babenko, “On the exact constant in a Jackson-type inequality in L 2,” Mat. Zametki, 39, No.5, 651–664 (1986).
L. V. Taikov, “Inequalities containing the best approximations and a modulus of continuity in L 2,” Mat. Zametki, 20, No.3, 433–438 (1976).
A. A. Ligun, “Some inequalities for the best approximations and moduli of continuity in the space L 2,” Mat. Zametki, 19, No.3, 353–364 (1976).
L. V. Taikov, “Best approximation of differentiable functions in the metric of the space L 2,” Mat. Zametki, 22, No.4, 535–542 (1977).
V. A. Yudin, “Diophantine approximations in extremal problems in L 2,” Dokl. Akad. Nauk SSSR, 251, No.1, 54–57 (1980).
A. A. Ligun, “Exact Jackson-type inequalities for periodic functions in the space L 2,” Mat. Zametki, 43, No.6, 757–768 (1988).
A. A. Ligun, “Jackson's type inequalities,” East J. Approxim., 2, No.2 (1996).
A. A. Ligun, “Exact constants in Jackson-type inequalities,” in: A. A. Ligun, V. E. Kapustyan, and Yu. I. Volkov, Special Problems in the Theory of Approximations and Optimal Control of Distributed Resources [in Russian], Vyshcha Shkola, Kiev (1990), pp. 3–75.
V. G. Doronin and A. A. Ligun, “On exact constants in Jackson's type inequalities in the space L 2,” East J. Approxim., 1, No.2, 189–197 (1995).
V. V. Volchkov, “On exact constants in Jackson-type inequalities in the space L 2,” Ukr. Mat. Zh., 47, No.1, 108–110 (1995).
N. I. Chernykh, “Jackson inequality in L p (0, 2π) (1 ≤ p < 2),” Tr. Mat. Inst. Ross. Akad. Nauk, 198, 232–241 (1992).
V. A. Yudin, “Multidimensional Jackson theorem in L 2,” Mat. Zametki, 29, No.2, 309–315 (1981).
V. V. Shalaev, “Exact estimates for the approximation of functions continuous on a sphere by linear convolution-type operators,” Ukr. Mat. Zh., 43, No.4, 565–567 (1991).
V. V. Arestov and V. Yu. Popov, “Jackson inequality on a sphere in L 2,” Izv. Vyssh. Uchebn. Zaved., Ser. Mat., No. 8, 13–20 (1995).
A. G. Babenko, “Exact Jackson inequality in the space L 2 with Jacobi weight,” in: Proceedings of the International Conference and Chebyshev Lectures Dedicated to the 175th Anniversary of P. L. Chebyshev's Birthday [in Russian], Vol. 1, Moscow University, Moscow (1996), pp. 40–43.
A. G. Babenko, “Exact Jackson-Stechkin inequality in the space L 2 of functions on a multidimensional sphere,” Mat. Zametki, 60, No.3, 333–355 (1996).
A. G. Babenko, “Exact Jackson-Stechkin inequality in the space L 2(R m),” in: Tr. Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk, 5, 183–198 (1998).
V. Yu. Popov, “Multidimensional approximations in L 2(T m ),” in: Theory of Functions and Approximations. Proceedings of the Third Saratov Winter School (January 27–February 7, 1986) [in Russian], Part 3, Saratov University, Saratov (1998), pp. 22–25.
D. V. Gorbachev, “Exact Jackson inequality in the space L p on a sphere,” Mat. Zametki, 66, No.1, 50–62 (1999).
E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton, NJ (1971).
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Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 3, pp. 291–304, March, 2005.
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Babenko, V.F., Doronin, V.G., Ligun, A.A. et al. On Jackson-Type Inequalities for Functions Defined on a Sphere. Ukr Math J 57, 347–363 (2005). https://doi.org/10.1007/s11253-005-0195-6
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DOI: https://doi.org/10.1007/s11253-005-0195-6