For any nontrivial linear combinations of finitely many Poisson kernels
\(\begin{array}{ccccc}{P}_{{q}_{i},\beta }\left(t\right)={\sum }_{k=0}^{\infty }{q}_{i}^{k}\text{cos}\left(kt-\frac{\beta \pi }{2}\right),& \beta \in {\mathbb{R}},& {q}_{i}\in \left(\text{0,1}\right),& i=\stackrel{-}{1,m},& m\in {\mathbb{N}},\end{array}\)
we establish the validity of the Nagy condition \({N}_{n}^{*}\) for all n starting from a certain number n0. In addition, for any \(n\in {\mathbb{N}}\) , we prove the existence of linear combinations m ( \(m\in {\mathbb{N}}\) \ {1}) of Bernoulli kernels
\(\begin{array}{ccccc}{D}_{{r}_{i}}\left(t\right)={\sum }_{k=1}^{\infty }{\left(-1\right)}^{\frac{{r}_{i}-1}{2}}\frac{\text{sin}kt}{{k}^{{r}_{i}}},& {r}_{i}=2{l}_{i}-1,& {l}_{i}\in {\mathbb{N}},& i=\stackrel{-}{1,m},& m\in {\mathbb{N}} \left\{1\right\},\end{array}\)
where ri ≠ rj for i ≠ j, as well as linear combinations m of conjugate Poisson kernels
\(\begin{array}{cccc}{P}_{{q}_{i},1}\left(t\right)={\sum }_{k=1}^{\infty }{q}_{i}^{k}\text{sin}kt,& {q}_{i}\in \left(\text{0,1}\right),& i=\stackrel{-}{1,m},& m\in {\mathbb{N}} \left\{1\right\},\end{array}\)
where qi ≠ qj for i ≠ 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 the average, of these linear combinations by the trigonometric polynomials of orders not greater 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.
