zbMATH — the first resource for mathematics

Two related extremal problems for entire functions of several variables. (English. Russian original) Zbl 0976.41015
Math. Notes 66, No. 3, 271-282 (1999); translation from Mat. Zametki 66, No. 3, 336-350 (1999).
From the abstract: The author establishes a relationship between the Logan problem for functions whose Fourier transform is supported in a centrally symmetric convex closed subset of \(\mathbb R^m\) and whose mean value on \(\mathbb R^m\) is nonnegative and the Chernykh problem on the optimal point for the Jackson inequality in \(L_2(\mathbb R^m)\), which relates the best approximation of a function by the class of entire functions of exponential type to the first modulus of continuity.

41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)
Full Text: DOI
[1] B. F. Logan, ”Extremal problems for positive-definite bandlimited functions. II. Eventually negative functions,”SIAM J. Math. Anal.,14, No. 2, 253–257 (1983). · Zbl 0513.42013 · doi:10.1137/0514023
[2] N. I. Chernykh, ”Jackson’s inequality inL 2,”Trudy Mat. Inst. Steklov [Proc. Steklov Inst. Math.],88, 71–74 (1967).
[3] N. I. Chernykh, ”The best approximation of periodic functions by trigonometric polynomials inL 2,”Mat. Zametki [Math. Notes],2, No. 5, 513–522 (1967).
[4] V. A. Yudin, ”The multidimensional Jackson theorem inL 2,”Mat. Zametki [Math. Notes],29, No. 2, 309–315 (1981).
[5] V. V. Arestov and N. I. Chernykh, ”On theL 2-approximation of periodic functions by trigonometric polynomials,” in:Approximation and Function Spaces (Gdansk, 1979), North-Holland, Amsterdam (1981), pp. 25–43.
[6] A. G. Babenko, ”An extremal problem for polynomials,”Mat. Zametki [Math. Notes],35, No. 3, 349–356 (1984). · Zbl 0538.41043
[7] E. E. Berdysheva, ”An extremal problem for entire functions of exponential type with nonnegative mean value,”East J. Approx.,3, No. 4, 393–402 (1997). · Zbl 0908.30030
[8] E. Stein and G. Weiss,Introduction to Fourier Analysis on Euclidean Spaces, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton (1971). · Zbl 0232.42007
[9] S. M. Nikol’skii,Approximation of Functions of Several Variables and Embedding Theorems [in Russian], 2d ed., Nauka, Moscow (1977). English transl. of the 1st ed. in: Die Grundlehren der Mathematischen Wissenschaften, Band 205, Springer-Verlag, New York-Heidelberg (1975).
[10] L. V. Kantorovich and G. P. Akilov,Functional Analysis in Normed Spaces [in Russian], Gosudarstv. Izdat. Fis.-Mat. Lit., Moscow (1959). English transl. in: International Series of Monographs in Pure and Applied Mathematics, Vol. 46, Macmillan, New York (1964). · Zbl 0127.06102
[11] B. F. Logan, ”Extremal problems for positive-definite bandlimited functions. I. Eventually positive functions with zero integral,”SIAM J. Math. Anal. 14, No. 2, 249–252 (1983). · Zbl 0513.42012 · doi:10.1137/0514022
[12] R. Courant and D. Hilbert,Methods of Mathematical Physics. Vol. I, Interscience, New York, (1953). · Zbl 0051.28802
[13] A. G. Babenko, ”Exact Jackson-Stechkin inequality in the spaceL 2 of functions on a multidimensional sphere,”Mat. Zametki [Math. Notes],60, No. 3, 333–355 (1996). · Zbl 0903.41014
[14] I. I. Ibragimov and F. G. Nasibov, ”The estimation of the best approximation of a summable function on the real axis by means of entire functions of finite degree,”Dokl. Akad. Nauk SSSR [Soviet Math. Dokl.],194, No. 5, 1013–1016 (1970). · Zbl 0224.41004
[15] V. Yu. Popov, ”Best mean square approximations by entire functions of exponential type,”Izv. Vyssh. Uchebn. Zaved. Mat. [Soviet Math. (Iz. VUZ)], No. 6(399), 65–73 (1972).
[16] A. V. Moskovskii, ”Jackson theorems in the spacesL p (\(\mathbb{R}\) n ) andL p,\(\lambda\)(\(\mathbb{R}\)+),”Izv. Tulskogo Gos. Univ., Ser. Matem., Mekh., Inform.,3, No. 1, 44–70 (1997).
[17] A. G. Babenko, ”The exact constant in the Jackson inequality inL 2,”Mat. Zametki [Math. Notes],39, No. 5, 651–664 (1986).
[18] V. V. Arestov and V. Yu. Popov, ”Jackson inequalities on a sphere inL 2,”Izv. Vyssh. Uchebn. Zaved. Mat. [Russian Math. (Iz. VUZ)], No. 8(399), 13–20 (1995). · Zbl 0860.41015
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.