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.

Bibliographie

[1] T. S. Davydova, N. A. Shirokov, “Priblizhenie funktsii iz klassa Geldera na poluosi”, Zap. nauchn. semin. POMI, 262, 1999, 127–137 | MR | Zbl

[2] N. A. Shirokov, “Priblizhenie mnogochlenami na kompaktakh s beskonechnosvyaznym dopolneniem”, Algebra i analiz, 10:1 (1998), 248–264 | MR | Zbl

[3] K. G. Mezhevich, N. A. Shirokov, “Polinomialnye priblizheniya na diz'yunktnykh otrezkakh”, Problemy matematicheskogo analiza, 1998, no. 18, 118–132 | Zbl

[4] E. M. Dynkin, “O ravnomernom priblizhenii funktsii v zhordanovykh oblastyakh”, Sib. mat. zh., 18:4 (1977), 775–786 | MR