Approximation by entire functions on subsets of a ray

O. V. Sil'vanovich ; N. A. Shirokov

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 21, Tome 337 (2006), pp. 233-237

Résumé

Let $E\subset\mathbb R^+$ be a set consisting of finitely many intervals and a ray $[a,\infty)$, and let $H_\omega^r(E)$ be the set of functions defined on $E$ for which $$ |f^{(r)}(x)-f^{(r)}(y)|\le c_f\omega(|x-y|), $$ where the continuity module $\omega(x)$ satisfies the condition $$ \int_0^y\frac{\omega(x)}{x}dx+y\int_y^\infty\frac{\omega(x)}{x^2}dx\le C_0\omega(y), \quad y>0. $$ Let $C_\sigma^{(r,\omega)}$, $\sigma>0$, denote the class of entire functions $F$ of order 1/2 and of type $\sigma$ such that $$ \sup_{z\in\mathbb C\setminus\mathbb R^+}\frac{|F(z)|e^{-\sigma|\operatorname{Im}\sqrt{z}|}}{1+|z|^r\omega(|z|)+\sigma^{-2r}\omega(\sigma^{-2})}\infty. $$ In the paper, given a function $f\in H_\omega^r(E)$, we construct approximating functions $F$ in the class $C_\sigma^{(r,\omega)}$. Approximation by such functions on the set $E$ is analogous to approximation by polynomials on compacts. The analogy involves constructing a scale for measuring approximations and providing a constructive description of the class $H_\omega^r(E)$ in terms of the approximation rate, similar to that of polynomial approximation. Bibliography: 4 titles.