On approximation by trigonometrical polynomials in $$$L_2$$$ space

O.V. Polyakov

doi:10.15421/241614

On approximation by trigonometrical polynomials in $$$L_2$$$ space

O.V. Polyakov (Oles Honchar Dnipropetrovsk National University)

Abstract

We obtain certain inequalities of Jackson type, connecting the value of the best approximation of periodic differentiable functions and the generalized modulus of continuity of the highest derivative.

Keywords

best approximation; modulus of continuity

Full Text:

PDF (Русский)

References

Korneichuk N.P. Extremum problems in approximation theory, Moscow, 1976; 320 p. (in Russian)

Chernykh N.I. "On Jackson inequalities in $$$L_2$$$ space", Trudy MIAN SSSR, 1967; 88: pp. 71-74. (in Russian)

Ligun A.A. "Some inequalities between the best approximations and moduli of continuity in $$$L_2$$$ space", Matem. zametki, 1978; 24(6): pp. 785-792. (in Russian) doi:10.1007/BF01140019

Doronin V.G., Ligun A.A. "Exact constant in Jackson inequality in $$$L_2$$$ space", Ukrainian Math. J., 1994; 46(9): pp. 1261-1265. (in Russian) doi:10.1007/BF01059429

Shalaev V.V. "On widths in $$$L_2$$$ of classes of differentiable functions, defined by moduli of continuity of the highest orders", Ukrainian Math. J., 1991; 43(1): pp. 125-129. (in Russian) doi:10.1007/BF01066914

Vakarchuk S.B., Zabutnaya V.I. "Inequalities of Jackson-Stechkin type for special moduli of continuity and widths of functional classes in $$$L_2$$$", Matem. zametki, 2012; 92(4): pp. 497-514. (in Russian)

Abilov V.A., Abilova F.V. "Some problems of approximation of $$$2\pi$$$-periodic functions by Fourier sums in $$$L_2(2\pi)$$$ space", Matem. zametki, 2004; 76(6): pp. 803-811. (in Russian) doi:10.1023/B:MATN.0000049674.45111.71

Vakarchuk S.B. "Exact constants in inequalities of Jackson type and exact values of widths of functional classes from $$$L_2$$$", Matem. zametki, 2006; 78(5): pp. 11-19. (in Russian) doi:10.1007/s11006-006-0102-y

Vakarchuk S.B. "Generalised characteristics of smoothness in inequalities of Jackson type and widths of classes of functions in $$$L_2$$$", Matem. zametki, 2015; 98(4): pp. 511-529. (in Russian) doi:10.1134/S0001434615090254

Shalaev V.V. "On one estimate of approximation of differentiable functions by trigonometrical polynomials", Researches on modern problems of summation and approximation of functions and their applications, Dnipropetrovsk, 1979; pp. 39-43 (in Russian)