Let σ > 0, and let (Formula Presented). The paper is devoted to approximation of classes of functions f for every ε > 0 representable as (Formula Presented) where Fε is an entire function of type not exceeding ε, (Formula Presented), and (Formula Presented) The approximating space SB consists of functions of the form (Formula Presented). Under some conditions on G = {Gε} and B, linear operators Xσ,G,B with values in SB are constructed for which (Formula Presented) the constant Kσ,G (it is an analog of the well-known Favard constant) cannot be reduced, even if one replaces the left-hand side by the best approximation by the space SB. The results of the paper generalize classical inequalities for approximations by entire functions of exponential type and by splines.