Abstract
We consider an extremal problem for even positive definite entire functions of exponential type with zero mean with power weight on the semiaxis. This problem is related to the multidimensional Jackson-Stechkin theorem in the space L 2(ℝn).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Bibliography
N. I. Akhiezer, Lectures on Approximation Theory [in Russian], Nauka, Moscow, 1965.
H. Bateman and A. Erdélyi, Higher Transcendental Functions, vol. 2, McGraw-Hill, New York-Toronto-London, 1953; Russian transl.: Nauka, Moscow, 1966.
B. M. Levitan and I. S. Sargsyan, Introduction to Spectral Theory [in Russian], Nauka, Moscow, 1970.
D. V. Gorbachev, Selected Problems of Function Theory and Approximation Theory and Their Applications, Isd. Tulskogo Univ., Tula, 2004.
N. I. Chernykh, “On Jackson’s inequality in L 2,” Trudy Mat. Inst. Steklov [Proc. Steklov Inst. Math.], 88 (1967), 71–74.
B. F. Logan, “Extremal problems for positive-definite bandlimited functions. II. Eventually negative functions,” SIAM J. Math. Anal., 14 (1983), no. 2, 253–257.
V. A. Yudin, “The multidimensional Jackson theorem in L 2,” Mat. Zametki [Math. Notes], 29 (1981), no. 2, 309–315.
A. V. Moskovskii, “The Jackson theorems in the spaces L p (ℝn) and L p,λ(ℝ+),” Izv. TulGU. Ser. Mat., 3 (1997), no. 1, 44–70.
A. G. Babenko, “A sharp Jackson-Stechkin inequality in the space L 2(ℝm),” Trudy Inst. Mat. Mekh. URO RAN, 5 (1998), 182–198.
N. I. Chernykh and V. V. Arestov, “On the L 2-approximation of periodic functions by trigonometric polynomials,” in: Approximation and Function Spaces, Proc. Inter. Conf. (Gdansk, 1979), North-Holland, Amsterdam, 1981, pp. 25–43.
V. V. Arestov and A. G. Babenko, “On the optimal point in Jackson’s inequality in L 2(−∞, ∞) with the second modulus of continuity,” East J. Approximation, 10 (2004), no. 1–2, 201–214.
A. G. Babenko, Direct Theorems of Approximation Theory in L 2 and Related Extremal Problems for Positive Definite Functions [in Russian], Doctorate thesis in the physico-mathematical sciences, Ekaterinburg, 2004.
E. E. Berdysheva, “Two interconnected extremal problems for entire functions of several variables,” Mat. Zametki [Math. Notes], 66 (1999), no. 3, 336–350.
E. E. Berdysheva, “An extremal problem for entire functions of exponential type with non-negative mean value,” East J. Approximation, 3 (1997), no. 4, 393–401.
N. I. Chernykh, “On the best approximation of periodic functions by trigonometric polynomials in L 2,” Mat. Zametki [Math. Notes], 2 (1967), no. 5, 513–522.
S. N. Vasil’ev, Approximation of Functions by Trigonometric Polynomials in L 2 and Fractal Functions in C [in Russian], Kandidat thesis in the physico-mathematical sciences, Ekaterinburg, 2002.
Author information
Authors and Affiliations
Additional information
__________
Translated from Matematicheskie Zametki, vol. 80, no. 5, 2006, pp. 712–717.
Original Russian Text Copyright © 2006 by D. V. Gorbachev, S. A. Strankovskii.
Rights and permissions
About this article
Cite this article
Gorbachev, D.V., Strankovskii, S.A. An extremal problem for even positive definite entire functions of exponential type. Math Notes 80, 673–678 (2006). https://doi.org/10.1007/s11006-006-0188-2
Received:
Issue Date:
DOI: https://doi.org/10.1007/s11006-006-0188-2