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ON SOME EXTREMAL PROBLEMS OF THE BEST APPROXIMATION BY ENTIRE FUNCTIONS // Tomsk State Pedagogical University Bulletin. 2015. Issue 2 (155). P. 213-220

In this paper a number of extremal problems of approximation theory of square summable functions on the whole line R : = (–∞,+∞) by entire functions of exponential type. In the space L2(R) of the exact constants of Jackson-Stechkin type inequalities were calculated. Found There was found the upper bounds approximation of classes of functions L2(R), defined with the help of the average modulus of continuity of m-th order, where instead of the shift operator ( , ): ( ) h T f x = f x + h is used Steklov’s operator Sh ( f ). Similar smoothness characteristics for solving the extremal problems of approximation theory for periodic functions in L2[0,2π] were previously considered in the works by V. A. Abilov, F. I. Abilova, S. B. Vakarchuk, M. Sh. Shabozov and others. It is proved that the obtained results in this paper are ultimate does not approving.

Keywords: the best approximation, modulus of continuity of m-order, Jakson-Stechkin type inequality, entire function of exponential type, operator of Steklova

BEST MEAN SQUARE APPROXIMATION BY ENTIRE FUNCTIONS AND VALUES OF THE AVERAGE WIDTHS OF SOME FUNCTIONAL CLASSES // Tomsk State Pedagogical University Bulletin. 2015. Issue 2 (155). P. 229-231

We solve number of extremal problems on the best mean square approximation of functions defined on the whole line R := (−∞,+∞) by entire functions of exponential type σ > 0 . Calculated exact inequalities between the best approximations of the value of 2 f ∈L (R) and integrals containing special moduli of continuity of the m-th order associated with the operator Steklov introduced in V. A. Abilova and F. V. Abilovoy. For the widths were calculated the exact mean values formulated by G. G. Magaril-Ilyaev for the classes functions ( ) 2 f ∈L r (R) satisfying the condition – generalized modulus of continuity m order derivative ( ) – the arbitrary increasing function Φ(0) = 0. Similar problems for periodic functions in the space 2L [0,2π ] previously considered works of V. A. Abilova, F. I. Abilovoy, S. B. Vakarchuk, M. Sh. Shabozova and others.

Keywords: the best approximation, Fourier transform, modulus of continuity of m-order, the characteristic function, entire function of exponential type, the mean of ν-widths