The best approximation by trigonometric polynomials of classes of convolutions generated by some linear combinations of periodic kernels

For arbitrary nontrivial linear combinations of a finite number of Poisson kernels, the fulfillment of the Nagy condition is established for all numbers n, starting from some number. It is also proved for any n the existence of linear combinations of m Bernoulli kernels and linear combinations of m conjugate Poisson kernels that satisfy the Nikolsky condition and at the same time do not satisfy the Nagy condition.