Sharp inequalities for approximations of convolution classes on the real line as the limit case of inequalities for periodic convolutions

We establish sharp estimates for the best approximations of convolution classes by entire functions of exponential type. To obtain these estimates, we propose a new method for testing Nikol’ski?-type conditions which is based on kernel periodization with an arbitrarily large period and ensuing passage to the limit. As particular cases, we obtain sharp estimates for approximation of convolution classes with variation diminishing kernels and generalized Bernoulli and Poisson kernels.