We construct the best linear methods of approximation in the Hardy and Bergman spaces of the classes of functions holomorphic in a unit disk, which are the convolutions of unit balls of the Hardy and Bergman spaces with certain reproducing kernel. We obtain the exact values for the n-widths, the best polynomial approximations, and the best linear approximations of the mentioned functional classes. We find the necessary and sufficient conditions, under which the generalized Bernstein inequalities for algebraic polynomials are hold true in the metrics of the Hardy and Bergman spaces.