BEST MEAN SQUARE APPROXIMATION BY ENTIRE FUNCTIONS AND VALUES OF THE AVERAGE WIDTHS OF SOME FUNCTIONAL CLASSES
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
References:
1. Tukhliev K. O nailuchshih priblizhenijah celymi funkcijami v prostranstve L2 (R). I [Best approximations of entire functions in the space L2 (R). I]. Izvestiya AN RT, otd. fiz.-mat., him., geol. i teh. n., 2013, no. 3 (152), pp. 19–29 (in Russian).
2. Vakarchuk S. B. O nekotrykh ekstremal’nykh zadachakh teorii approksimatsii funktsiy na veshchestvennoy osi. II [Some extreme problems in the theory of approximation of functions on real axis. II]. Ukr. matem. visnik, 2012, vol. 9, no. 4, pp. 578–602 (in Russian).
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Issue: 2, 2015
Series of issue: Issue 2
Rubric: INTERDISCIPLINARY STUDIES
Pages: 229 — 231
Downloads: 1311