Best mean-square approximation of functions defined on the real axis by entire functions of exponential type

Abstract

Exact constants in Jackson-type inequalities are calculated in the space L 2.(ℝ) in the case where the quantity of the best approximation A σ (f) is estimated from above by the averaged smoothness characteristic $$ {\varPhi_2}\left( {f,t} \right)=\frac{1}{t}\int\limits_0^t {\left\| {\Delta_h^2(f)} \right\|dh} $$ . We also calculate the exact values of the average v-widths of classes of functions defined by Φ2. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 64, No. 5, pp. 604–615, May, 2012.