On an approximation of functions regular in convex polygons by linear means of their exponent series (Q2739957)

Let \(\overline M\) be a closed convex polygon with vertexes \(\gamma_1,\ldots, \gamma_{N}, N\geq 3\); let \(AC(\overline M)\) be a Banach space of regular in \(M\) and continuous in \(\overline M\) functions with norm \(\|f\|_{AC(\overline M)}= \max_{z\in M}|f(z)|\); \(H^{\omega}(\overline M)\) be a class of functions \(f\in AC(\overline M)\) satisfying the condition \(|f(z_1)-f(z_2)|\leq A\omega(|z_1-z_2|), z_1,z_2\in\overline M\); \(\omega(h)\) be the continuity modulus; \(W^{r}H^{\omega}(\overline M)\) be the class of regular in \(M\) functions such that \(f^{(r)}\in H^{\omega}(\overline M)\). The author proves that if \(f\in W^{r}H^{\omega}(\overline M)\), \(\int_{0}^{h} \omega(t)/t dt+h\int_{h}^{2\pi}\omega(t)/t^2 dt\leq A_1 \omega(h)\) and \(\sum_{k=1}^{N}d_{k}f^{(s)}(\gamma_{k})=0, 0\leq s\leq r\), then \(\|f-P_{n}(f)\|_{AC(\overline M)}\leq A_2 \sum_{j=1}^{N}\epsilon_{n_{j}}\), \(A_1, A_2=\text{ const}, n=(n_1,\ldots,n_{N})\). Here \(P_{n}(f)\) is the linear average of the series of exponents corresponding to \(f\); \(\epsilon_{n}\to 0\) as \(n\to\infty\) is a sequence of values defined in the article.