Sharp Jackson-type Inequalities for Approximations of Classes of Convolutions by Entire Functions of Exponential Type

In this paper, a new method is introduced for the proof of sharp Jacksontype inequalities for approximation of convolution classes of functions defined on the real line. These classes are approximated by linear operators with values in sets of entire functions of exponential type. In particular, a sharp Jackson-type inequality for the even-order derivatives of the conjugate function is proved. For the uniform and the integral norm, the estimates are sharp even if their left-hand sides are replaced by the best approximation. Sharp inequalities for approximations of periodic functions by trigonometric polynomials and of almost-periodic functions by generalized trigonometric polynomials are special cases of the inequalities mentioned above. §