Best approximation by trigonometric polynomials of convolution classes generated by some linear combinations of periodic kernels

• Datum: 2026
• Hauptverfasser: Serdyuk, A., Sorych, V., Sorych, N., Сердюк, Анатолій, Сорич, Віктор, Сорич, Ніна
• Format: Artikel
• Sprache: Ukrainisch
• Veröffentlicht: Institute of Mathematics, NAS of Ukraine 2026
• Online Zugang: https://umj.imath.kiev.ua/index.php/umj/article/view/8934
• Назва журналу: Ukrains’kyi Matematychnyi Zhurnal

UDC 517.5 For any nontrivial linear combinations of finitely many Poisson kernels $P_{q_i,\beta}(t)=\displaystyle\sum\nolimits^\infty_{k=0}{q^k_i{\cos \left(kt-\frac{\beta\pi}{2}\right)}},$ $\beta\in {\mathbb R},$ $q_i\in (0,1),$ $i=\overline{1,m},\ m\in\mathbb{N},$ 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 $D_{r_i}(t)=\displaystyle\sum\nolimits_{k=1}^\infty{(-1)}^{\frac{r_i-1}{2}} \dfrac{{\sin k }t}{k^{r_i}},$ $r_i=2l_i-1,\ l_i\in {\mathbb N},$ $i=\overline{1,m},\ m\in\mathbb{N}\setminus\{1\},$ where $r_i\ne r_j$ for $i\ne j,$ as well as linear combinations $m$ of  conjugate Poisson kernels $P_{q_i,1}(t)=\displaystyle\sum\nolimits^{\infty }_{k=1}q^k_i{\sin k}t,$ $ q_i\in (0,1),$ $i=\overline{1,m},\ m\in\mathbb{N}\setminus\{1\},$ where $q_i\ne q_j$ for $i\ne j,$ 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 $L.$