Convolution-type integral operators in complex approximation
Let \(K_n(u)\) be a trigonometric polynomial of degree \(d_n\in\mathbb N\) with \(K_n(u)\geq 0\), for all \(u\in [0, 2\pi]\), \( n\in\mathbb N\). For a function \(f(z)\) analytic on the open unit disk \(\mathbb D\) and continuous on \(\overline{\mathbb D}\) and for all \(n\in\mathbb N\), let us define the operators \[ L_n(f)(z)=\frac{1}{c_n}\int_0^{2\pi} f(ze^{iu})K_n(u) du, \quad c_n=\int_0^{2\pi}K_n(u) du. \] The authors prove that \(L_n(f)(z)\) is a convolution-type operator and is a polynomial of degree at most \(d_n\) in \(z\); if \(f(z)\) is a polynomial of degree at most \(p\) in \(z\) then \(L_n(f)(z)\) is a polynomial of degree at most \(\min\{p, d_n\}\); \[ \omega_1(L_n(f); \delta)_{\overline{\mathbb D}}\leq \omega_1(f; \delta)_{\overline{\mathbb D}}\quad \text{for all}\quad n\in\mathbb N,\quad \delta\geq 0; \] if \(\int_0^\pi uK_n(u) du< M\) for all \(n\in\mathbb N\) then \[ |f(z) - L_n(f)(z)|\leq 2(1 + M) \omega_1\left(f; \frac{1}{c_n}\right)_{\overline{\mathbb D}}\quad\text{for all}\quad n\in\mathbb N,\quad z\in{\overline{\mathbb D}}, \] that is, if \(\lim_{n\to\infty}c_n = +\infty\) then \(\lim_{n\to\infty}L_n(f)(z) = f(z)\) uniformly in \(\overline{\mathbb D}\). Approximation and global smoothness preservation properties for several choices of the kernel \(K_n(u)\) (Fejér kernel, Jackson kernel and some other ones) are studied, and some geometric properties of the Jackson-type convolution operators are presented.
scientific article; zbMATH DE number 1851118