Direct approximation theorems for Dirichlet series in the norm of uniform convergence
Let \(D \subset {\mathbb C}\) be a convex polygon with vertices at points \(a_1,\dots,a_N\), \(N \geq 3\), and \(0 \in D\). Let \(L(z) = \sum_{k=1}^{N} d_k\,e^{a_kz}\), \(d_k \in {\mathbb C}\setminus \{0\}\), be a quasipolynomial. Denote by \(\Lambda\) the set of zeros of \(L\). For \(f \in AC(\bar D)\), consider its Dirichlet series \[ f(z) \sim \sum_{\lambda \in \Lambda} k_f(\lambda) \, \frac{e^{\lambda z}}{L'(\lambda)}. \] The subject of the paper is direct theorems of Jackson type for approximations of \(f \in AC(\bar{D})\) in the uniform metric by partial sums of Dirichlet series weighted with the generalized Jackson kernel. Let \(\omega_{r, \bar D}(f,t)\) be Tamrazov modulus of smoothness of order \(r \geq 1\), which was introduced by \textit{P. M. Tamrazov} [Smoothness and polynomial approximations (Russian) (Naukova Dumka, Kiev) (1975; Zbl 0351.41004)]. Denote by \(AW^{q}H_{r}^{\varphi}(\bar D)\) the class of functions \(f\) regular in \(D\) such that \(f^{(q)} \in AC(\bar D)\) and \(\omega_{r,\bar D}(f^{(q)},t) \leq \text{const} \cdot \varphi(t)\). Direct approximation theorems are proved for \( f \in AW^{q}H_{r}^{\varphi}(\bar D)\), \(r \geq 1\), under some natural restrictions on \(\varphi\). This extends the result of \textit{Yu. I. Mel'nik} [Ukr. Mat. Zh. 40, No. 4, 446--452 (1988; Zbl 0656.30002)] who proved such estimates in the case \(r=1\). Also, the author obtains a refinement of Mel'nik's result in the case \(r=1\), \(q \neq 0\).
scientific article; zbMATH DE number 2136629
scientific article