M.Sh. Shabozov, O.A. Dzhurakhonov. Approximation in the mean of some classes of bivariate functions by Fourier–Chebyshev sums ... P. 268-278

In space $L_{2,\rho}$ of bivariate functions summable with square on set $Q=[-1,1]^2$ with weight $\rho(x,y)={1}/{\sqrt{(1-x^{2})(1-y^{2})}}$ the sharp inequalities of Jackson--Stechkin type in which the best polynomial approximation estimated above by Peetre $\mathcal{K}$-functional were obtained. We also find the exact values of various widths of classes of functions defined by generalized modulus of continuity and $\mathcal{K}$-functionals. Also the exact upper bounds for modules of coefficients of Fourier--Tchebychev on considered classes of functions were calculated.

Keywords: approximation, generalized modulus of continuity, Fourier–Chebyshev double series, generalized translation operator

Received August 08, 2020

Revised November 16, 2020

Accepted November 23, 2020

Mirgand Shabozovich Shabozov, Dr. Sci. (Phys.-Math.), Prof., member of Academy of NAN Tajikistan, Tajik National University, Dushanbe, 734025 Republic of Tajikistan, e-mail: shabozov@mail.ru

Olimjon Akmalovich Jurakhonov, Tajik National University, Dushanbe, Associate Professor of the Department of Functional Analysis and Differential Equations, 734025 Republic of Tajikistan, e-mail: olim1974@mail.ru


1.   Paszkowski S. Vychislitel’nye primenenija mnogochlenov i rjadov Chebyshjova [Numerical applications of Chebyshev polynomials and series], translation from Polish to Russian, Moscow: Nauka Publ., 1983, 384 p.

2.   Vasil’ev N.I., Klokov Yu.A., Shkerstena A.Ya. Primenenie polinomov Chebysheva v chislennom analize [The application of Chebyshev polynomials in numerical analysis]. Riga: Zinatne Publ., 1984. 240 p.

3.   Beerends R.I. Chebyshev polynomials in several variables and the radial part of the Laplace — Beltrami operator. Trans. Amer. Math. Soc., 1991, vol. 328, no. 2, pp. 1951–1961.

doi: 10.1090/S0002-9947-1991-1019520-3 

4.   Lidl R. Tschebyscheff polynome in mehreren Variablen. J. reine und angew. Math., 1975, vol. 273, pp. 178–198. doi: 10.1515/crll.1975.273.178 

5.   Ricci P.E. I polynomi di Tchbycheff in piu variabli. Rend. Math. Appl., 1978, vol. 11, no. 2, pp. 295–327.

6.   Suetin P.K. Ortogonalnye mnogochleny po dvum peremennym [Orthogonal polynomials in two variables]. Moscow: Nauka Publ., 1988, 384 p.

7.   Abilov V.A., Kerimov M.K. Estimates of residual terms of multiple Fourier–Chebyshev series and the Chebyshev type cubature formulas. Comput. Math. Math. Phys., 2003, vol. 43, no. 5, pp. 613–632.

8.   Jurakhonov O.A. Approximation of bivariate functions by Fourier–Tchebychev “circular” sums in $L_{2,\rho}$. Vladikavkaz. Math. J. , 2020, vol. 22, no. 2, pp. 5–17. doi: 10.46698/n6807-7263-4866-r 

9.   Nikol’skii S.M. Approximation of functions of several variables and embedding theorems. Berlin; N Y: Springer-Verlag, 1975, 420 p. doi: 10.1007/978-3-642-65711-5 . Original Russian text (2nd ed.) published in Nikol’skii S.M. Priblizhenie funktsii mnogikh peremennykh i teoremy vlozheniya, Moscow: Nauka Publ., 1977, 480 p.

10.   Vakarchuk S.V., Shvachko A.V. On the approximation in the mean with the Chebyshev — Hermite weight by algebraic polynomials on the real axis. Ukrainian Math. J., 2013, vol. 65, no. 12, pp. 1774–1792. doi: 10.1007/s11253-014-0897-8 

11.   Vakarchuk S.V. 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 

12.   Pinkus A. n-Widths in approximation theory. Berlin: Springer–Verlag, 1985, 294 p. doi: 10.1007/978-3-642-69894-1 

13.   Tichomirov V.M. Nekotorye voprosy teorii priblizhenii [Some questions in approximation theory]. Moscow: Izdat. Moskov. Univ., 1976, 304 p.

14.   Shevchuk I.A. Priblizhenie mnogochlenami i sledy nepreryvnykh na otrezke funktsii [Approximation by Polynomials and Traces of the Functions Continuous on an Interval]. Kiev: Naukova Dumka Publ., 1992, 225 p.

Cite this article as: M.Sh. Shabozov, O.A. Dzhurakhonov. Approximation in the mean of some classes of bivariate functions by Fourier–Chebyshev sums, Trudy Instituta Matematiki i Mekhaniki URO RAN, 2020, vol. 26, no. 4, pp. 268–278.