×

zbMATH — the first resource for mathematics

Exact Jackson-Stechkin inequalities and diameters of classes of functions from \(L_{2} (\mathbb R^{2} ,e^{-x^2 -y^2})\). (English. Russian original) Zbl 1145.41003
Russ. Math. 51, No. 2, 1-7 (2007); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2007, No. 2, 3-9 (2007).
In [V. A. Abilov and M. V. Abilov, Math. Notes, 57, No. 1, 3–14 (1995; Zbl 0836.41006)] the order of the width of the class \(W^{\alpha}_2H_r^{\omega}\) for the case \(r=1,\omega(\frac{1}{n})=n^{-1/2},\) was found. Here the author calculated the exact value of the width for an arbitrary nonnegative function \(\omega(\delta)\) with any \(r,\alpha\geq 0.\) Moreover, a related exact Jackson-Stechkin inequality is proved.

MSC:
41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiń≠-type inequalities)
41A46 Approximation by arbitrary nonlinear expressions; widths and entropy
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] V. M. Tikhomirov, Some Questions in the Theory of Approximation (Mosk. Gos. Univ., Moscow, 1976) [in Russian].
[2] S. Z. Rafal’son, ”Approximation of Functions on Average by Fourier-Hermite Sums,” Iz. VUZ. Matematika 12(7), 78–84 (1968).
[3] V. A. Abilov and M. V. Abilov, ”Approximation of Functions in the Space L 2(R 2, exp(-|x|2),” Matem. Zametki 57(1), 3–19 (1995). · Zbl 0836.41006
[4] I. S. Gradshtein and I. M. Ryzhik, Tables of Integrals, Sums, Series, and Products (Fizmatgiz, Moscow, 1962) [in Russian].
[5] N. Ya. Vilenkin, Special Functions and the Theory of Group Representations (Nauka, Moscow, 1965) [in Russian].
[6] M. G. Esmaganbetov, E. S. Smailov, and G. A. Akishev, Theory of Approximation and Embedding of Classes of Functions of Many Variables (Karaganda, 1986) [in Russian].
[7] M. Zh. Baisalova and M. G. Esmaganbetov, Accuracy of Estimates Between the Mixed Module of Smoothness and Approximation by an ”Angle” of a Function from L 2[0, 2\(\pi\)]2, Available from KazNIINTI, No. 1722-KA87 (Karaganda, 1987).
[8] K. I. Babenko, ”Approximation of Periodic Functions of Many Variables by Trigonometric Polynomials,” Dokl. Akad. Nauk SSSR 132(2), 247–250 (1960). · Zbl 0099.28302
[9] M. G. Esmaganbetov, ”Diameters of Classes from L 2[0, 2\(\pi\)] and Minimization of Exact Constants in the Jackson Type Inequalities,” Matem. Zametki 65(6), 816–820 (1999). · doi:10.4213/mzm1117
[10] M. G. Esmaganbetov, ”Minimization of Exact Constants in the Jackson Type Inequalities and Diameters of Functions from L 2[0, 2\(\pi\)],” Fundamental’naya i Prikladnaya Matematika, Mosk. Univ. 7(1), 1–6 (2001). · Zbl 1049.42001
[11] M. G. Esmaganbetov, ”On diameters in L 2 of Classes of Differentiable Functions,” Proc. of A. Razmadze Math. Institute 121, 37–42 (1999). · Zbl 0959.42001
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.