Best approximation by trigonometric polynomials of convolution classes generated by some linear combinations of periodic kernels - Serdyuk, A.; Sorych, V.; Sorych, N. (Ukrains’kyi Matematychnyi Zhurnal, 2025)
UDC 517.5For any nontrivial linear combinations of finitely many Poisson kernels $P_{q_i,β}(t)=\displaystyle\sum olimits^∞_{k=0}{q^k_i{\cos \left(kt-\frac{βπ}{2}\right)}},$ $β\in {\mathbb R},$ $q_i\in (0,1),$ $i=\overline{1,m},\ m\in\mathbb{N},$ we establish the Nagy condition $N^*_n$ for all numbers $n$ starting from a certain number $n_0.$ In addition, for any $n\in {\mathbb N},$ we prove the existence of linear combinations $m\ (m\in\mathbb{N}\setminus\{1\})$ of Bernoulli kernels $D_{r_i}(t)=\displaystyle\sum olimits_{k=1}^∞{(-1)}^{\frac{r_i-1}{2}} \dfrac{{\sin k }t}{k^{r_i}},$ $r_i=2l_i-1,\ l_i\in {\mathbb N},$ $i=\overline{1,m},\ m\in\mathbb{N}\setminus\{1\},$ where $r_i e r_j$ for $i e j,$ as well as linear combinations $m$ of conjugate Poisson kernels $P_{q_i,1}(t)=\displaystyle\sum olimits^{∞ }_{k=1}q^k_i{\sin k}t,$ $ q_i\in (0,1),$ $i=\overline{1,m},\ m\in\mathbb{N}\setminus\{1\},$ 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 $L.$