We derive exact inequalities of Jackson--Stechkin type between the value $E_{n-1}(f^{(s)})_{2}$ of the best mean-square approximation on~$\mathbb{R}$ with the weight $\rho(x)=e^{-x^2}$ of successive derivatives $f^{(s)}$, $s=0,1,...,r$, of functions $f\in L_{2,\rho}^{(r)}(\mathbb{R})$ and average values of $m$th-order generalized moduli of continuity of the $r$th derivatives. The exact values of some extremal approximation characteristics in the space $L_{2,\rho}(\mathbb{R})$ are found for classes of functions defined in terms of these moduli of continuity.
Keywords: best approximations, algebraic polynomial, Jackson--Stechkin inequalities, $m$th-order modulus of continuity, Chebyshev--Hermite polynomial.
REFERENCES
1. Rafal’son S.Z. The approximation of functions in the mean by Fourier–Hermite sums. Izv. Vyssh. Uchebn. Zaved. Mat., 1968, no. 7, pp. 78–84 (in Russian).
2. Suetin P.K. Klassicheskie ortogonal’nye mnogochleny [Classical orthogonal polynomials]. Moscow: Nauka Publ., 1979, 416 p. ISBN (3rd ed.): 978-5-9221-0406-7 .
3. Abilov V.A., Abilova F.V. Some problems of the approximation of functions by Fourier–Hermite sums in the space L2(R;e-x2 ). Russian Math. (Iz. VUZ), 2006, vol. 50, no. 1, pp. 1–10.
4. Vakarchuk S.B., Vakarchuk M.B. Mean approximation of functions by algebraic polynomials on real axis with Chebyshev–Hermite weight. Dnipr. Univ. Math. Bull., 2011, vol. 19, no. 6/1, pp. 28–31 (in Russian).
5. Vakarchuk S.B., Shvachko A.V. On the approximation in the mean with the Chebyshev-Hermite weight by algebraic polynomials on the real axis. Transactions of Institute of Mathematics of NAS of Ukraine, 2013, vol. 10, no. 1, pp. 28–38 (in Russian).
6. Vakarchuk S.B. Mean approximation of functions on the real axis by algebraic polynomials with Chebyshev–Hermite weight and widths of function classes. Math. Notes, 2014, vol. 95, no. 5, pp. 599–614. doi: 10.1134/S0001434614050046 .
Received August 28, 2019
Revised March 16, 2020
Accepted March 23, 2020
K. Tukhliev, Dr. Phys.-Math Sci., Prof., Khujand State University named after acad. B. Gafurov, Khujand, 735700, Republic of Tajikistan, e-mail: kamaridin.t54@mail.ru
A.M. Tuychiev, Khujand State University named after acad. B. Gafurov, Khujand, 735700, Republic of Tajikistan, e-mail: t-87yil@mail.ru
Cite this article as: K.Tukhliev, A.M.Tuichiev. Mean-square approximation of functions on the whole axis by algebraic polynomials with the Chebyshev–Hermite weight, Trudy Instituta Matematiki i Mekhaniki URO RAN, 2020, vol. 26, no. 2, pp. 270–277.
