Polynomial approximation of functions on a quasi-smooth arc with Hermitian interpolation
Let \(L\) be a Jordan arc in the complex plane \(\mathbb C\) so that \(\Omega = \overline{\mathbb C} \setminus L\) is a simply connected domain. \(L\) is assumed to be quasi-smooth (in the sense of Lavrentiev), i. e., length \(L(z_1,z_2) \leq C_1|z_1 - z_2|\), \(C_1 = C(L) \geq 1\), for any \(z_1,\, z_2 \in L\), where \(L(z_1,z_2)\) is the subarc of \(L\) connecting the points \(z_1\) and \(z_2\). The goal of the paper is to give a theorem on polynomial approximation of functions \(f\) that are smooth on a quasi-smooth arc which simultaneously interpolate \(f\) and its derivative at certain points of \(L\). Some notation is need before the statement: \(D(z;\delta) = \{ \zeta : |z - \zeta| \leq \delta \}\), \(E_m(f,B) : = \inf_{f \in \mathbf{P}_m} \|f - p \|_B\), where \(B \subset L\), \(m \in \mathbb{N}_0\) and \(\mathbf{P}_m\) represent the class of polynomials of degree at most \(m\). The \(k\)th local modulus of smoothness is \[ \omega_{f,k,z,L}(\delta) : = E_{k -1}(f, L \cap D(z, \delta)) \] and the \(kth\) global modulus of smoothness of \(f\) on \(L\) is defined as \[ \omega_{f,k,L}(\delta) : = \sup_{z \in L}\omega_{f,k,z,L}(\delta). \] Let be \(\Omega = \overline{\mathbb{C}} \setminus L\), \(\Delta = \{ z : |z|> 1\}\) and \(\Phi\) the conformal map of \(\Omega\) onto \(\Delta \) such that \(\Phi(\infty) = \infty\) and \(\Phi '(\infty) = \lim_{z \to \infty} \frac{\Phi(z)}{z} > 0\). Set \(\Psi = \Phi^{-1}\) and note that \(\Phi\) can be continuously extended to the endpoints \(\zeta_1\) and \(\zeta_2\) of \(L\), and let be \(w_j : \Phi(\zeta_j)\), \(j= 1,2\). Let \(L_{\delta} = \{ \zeta : |\Phi(\zeta)| = 1 + \delta \}\). Put \(\Delta_1 : = \{ w: |w| > 1, \arg w_1 < \arg w < \arg w_2 \}\), \(\Delta_ 2 = \Delta \setminus \overline{\Delta_1}\), \(\Omega_j : = \Psi(\Delta_j)\), \(\rho_{\delta }^{(j)} = d(z, L_{\delta} \cap \Omega_j)\), and \(\rho_{\delta}^*(z) = \max(\rho_{\delta}^{(1)}(z), \rho_{\delta}^{(2)}(z))\). For \(0 < \delta < \rho\), let \(\Omega_{f,k, L}(\delta, \rho) = \omega_{f,1,L}(\delta)\) if \(k= 1\), and \(\Omega_{f,k, L}(\delta, \rho) = \delta \int_{\delta}^{2\rho}\frac{\omega_{f,k,L}(t)}{t^2}dt\) if \(k> 1\). The main result is the following: Theorem. Suppose that \(z_1, \dots , z_N\) are distint point on the quasi-smooth arc \(L\). Then, for any \(r \in \mathbb{N}_0\) and \(k \in \mathbb{N}\), there exist a positive constant \(c = c(r,k,L,z_1,\dots, z_N)\) and an integer \(n_0 = n_0(r,k,L,z_1,\dots, z_N)\) such that, for any \(f \in {\mathcal{C}}^{r}(L)\) and any \(n \geq n_0\), it can be constructed a polynomial \(p_n \in \mathbf{P}_n\) satisfying, for \(l = 0,1,2, \dots r\) and \(z \in L\), \[ |f^{(l)}(z) - p_n^{(l)}(z)| \leq c(\rho_{1/n}^{*}(z))^{r - l} \omega_{f^{(r)},k,L}(\rho_{1/n}^{*}(z)). \] Moreover, for \(j = 1,\dots, N\) and \(z \in L \cap D(z_j, \rho_{1/n}^{*}(z))\), \[ |f^{(l)}(z) - p_n^{(l)}(z)| \leq c|z - z_j|^{r - l}\Omega_{f^{(r)},k,L}(|z - z_j|, \rho^{*}_{1/n}(z_j)) . \] The structure of the proof can be described as follows: The anti-derivate \(F\) of \(f\) along \(L\) is extended to the complex plane, then a construction due to \textit{E. M. Dyn'kin} [Sib. Math. J. 18, 548--557 (1978; Zbl 0411.30024), translation from Sib. Mat. Zh. 18, 774--786 (1977; Zbl 0364.30034)] is used together with a polynomial approximation of the Cauchy kernel by polynomial kernels of the form \[ K_n(\zeta, z) = \sum_{j=0}^n a_j(\zeta)z^j,\qquad z \in L, \quad \zeta \in \overline{\mathbb{C}}, \] where the coefficients \(a_j(\zeta)\) are summable in \(\Omega^{*} = \{ \zeta : |\Phi(\zeta)| < 3 \}.\)
scientific article; zbMATH DE number 5600306
scientific article