Best approximation by trigonometric polynomials of convolution classes generated by some linear combinations of periodic kernels
UDC 517.5 For any nontrivial linear combinations of finitely many Poisson kernels P_q_i,β(t)=olimits^∞_k=0q^k_i(kt-βπ2) , βR, q_i(0,1), i=1,m, mN, we establish the Nagy condition N^*_n for all numbers n starting from a certain number n_0. In addition, for any nN, we prove the existence of linear combinations m (mN{1}) of Bernoulli kernels D_r_i(t)=olimits_k=1^∞(-1)^r_i-12 k tk^r_i , r_i=2l_i-1, l_iN, i=1,m, mN{1}, where r_i e r_j for i e j, as well as linear combinations m of conjugate Poisson kernels P_q_i,1(t)=olimits^∞ _k=1q^k_ikt, q_i(0,1), i=1,m, mN{1}, where q_i e q_j for i e j, which satisfy the Nikolsky condition A^*_n but do not satisfy the Nagy condition N^*_n. As a result, in each analyzed case, we determine the exact values of the best approximations, on average, of these linear combinations by the trigonometric polynomials of orders not higher than n-1 and compute the exact values of the best approximations for the classes of convolutions generated by the indicated linear combinations in metrics of the spaces C and L. , number=5, journal=Ukrains’kyi Matematychnyi Zhurnal, author=Serdyuk, A. and Sorych, V. and Sorych, N., year=2025, month=Jul., pages=349–363