Best approximations and widths of classes of convolutions of periodic functions of high smoothness

We consider classes of 2π-periodic functions that are represented in terms of convolutions with fixed kernels Ψ β - whose Fourier coefficients tend to zero at exponential rate. We determine exact values of the best approximations of these classes in the uniform and integral metrics. In several cases, we determine the exact values of the Kolmogorov, Bernstein, and linear widths for these classes in the metrics of the spaces C and L. © 2005 Springer Science+Business Media, Inc.