Exact constants in Jackson-type inequalities are calculated in the space L 2.(ℝ) in the case where the quantity of the best approximation A σ (f) is estimated from above by the averaged smoothness characteristic
.
We also calculate the exact values of the average v-widths of classes of functions defined by Φ2.
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 64, No. 5, pp. 604–615, May, 2012.
Similar content being viewed by others
References
S. N. Bernstein, “On the best approximation of continuous functions on the entire real axis with the use of entire functions of given power,” in: S. N. Bernstein, Collected Works [in Russian], Vol. 2, Academy of Sciences of the USSR, Moscow (1952), pp. 371–375.
N. I. Akhiezer, Lectures on Approximation Theory [in Russian], Nauka, Moscow (1947).
A. F. Timan, Approximation Theory of Functions of a Real Variable [in Russian], Fizmatgiz, Moscow (1960).
A. F. Timan, “Approximation of functions defined on the entire real axis by entire functions of exponential type,” Izv. Vyssh. Uchebn. Zaved., Ser. Mat., No. 2, 89–101 (1968).
S. M. Nikol’skii, Approximation of Functions of Many Variables and Imbedding Theorems [in Russian], Nauka, Moscow (1969).
I. I. Ibragimov and F. G. Nasibov, “On an estimate for the best approximation of summable functions on the real axis by entire functions of finite degree,” Dokl. Akad. Nauk SSSR, 194, No. 5, 1013–1016 (1970).
V. Yu. Popov, “On the best mean-square approximations by entire functions of exponential type,” Izv. Vyssh. Uchebn. Zaved., Ser. Mat., No. 6, 65–73 (1972).
V. K. Dzyadyk, “On exact upper bounds for the best approximations on certain classes of continuous functions defined on the real axis,” Dopov. Akad. Nauk Ukr. RSR, Ser. A, No. 7, 589–592 (1975).
A. I. Stepanets, Methods of Approximation Theory [in Russian], Vol. 2, Institute of Mathematics, Ukrainian National Academy of Sciences, Kiev (2002).
S. B. Vakarchuk, “Exact constant in an inequality of Jackson type for L 2-approximation on the line and exact values of mean widths of functional classes,” East J. Approxim., 10, No. 1-2, 27–39 (2004).
A. A. Ligun and V. G. Doronin, “Exact constants in Jackson-type inequalities for L 2-approximation on an axis,” Ukr. Mat. Zh., 61, No. 1, 92–98 (2009); English translation: Ukr. Math. J., 61, No. 1, 112–120 (2009).
S. B. Vakarchuk and V. G. Doronin, “Best mean square approximations by entire functions of finite degree on a straight line and exact values of mean widths of functional classes,” Ukr. Mat. Zh., 62, No. 8, 1032–1043 (2010); English translation: Ukr. Math. J., 62, No. 8, 1199–1212 (2010).
N. N. Pustovoitov, “Estimates for the best approximations of periodic functions by trigonometric polynomials in terms of averaged differences and the multidimensional Jackson theorem,” Mat. Sb., 188, No. 10, 95–108 (1997).
S. B. Vakarchuk and V. I. Zabutnaya, “On Jackson-type inequalities and widths of classes of periodic functions in the space L 2;” in: Theory of Approximation of Functions and Related Problems, Institute of Mathematics, Ukrainian National Academy of Sciences, Kiev, (2008), pp. 37–48.
G. G. Magaril-Il’yaev, “Mean dimension and widths of classes of functions on a straight line,” Dokl. Akad. Nauk SSSR, 318, No. 1, 35–38 (1991).
G. G. Magaril-Il’yaev, “Mean dimension, widths, and optimal restoration of Sobolev classes of functions on a straight line,” Mat. Sb., 182, No. 11, 1635–1656 (1991).
A. I. Shevchuk, Approximation by Polynomials and Traces of Functions Continuous on a Segment [in Russian], Naukova Dumka, Kiev (1992).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Vakarchuk, S.B. Best mean-square approximation of functions defined on the real axis by entire functions of exponential type. Ukr Math J 64, 680–692 (2012). https://doi.org/10.1007/s11253-012-0671-8
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-012-0671-8