Exact values of Kolmogorov widths of classes of Poisson integrals

We prove that the Poisson kernel Pq,β(t)=∑k=1∞qkcos(kt−βπ2), q∈(0,1), β∈R, satisfies Kushpel's condition Cy,2n beginning with a number nq where nq is the smallest number n≥9, for which the following inequality is satisfied: 4310(1−q)qn+16057(n−n)q(1−q)2≤(12+2q(1+q2)(1−q))(1−q1+q)41−q2. As a consequence, for all n≥nq we obtain lower bounds for Kolmogorov widths in the space C of classes Cβ,∞q of Poisson integrals of functions that belong to the unit ball in the space L∞. The obtained estimates coincide with the best uniform approximations by trigonometric polynomials for these classes. As a result, we obtain exact values for widths of classes Cβ,∞q and show that subspaces of trigonometric polynomials of order n−1 are optimal for widths of dimension 2n.