Approximation by entire functions on a countable union of segments on the real axis. III: Further generalization
This is a continuation of the authors' previous papers [Vestn. St. Petersbg. Univ., Math. 49, No. 4, 373--378 (2016); translation from Vestn. St-Peterbg. Univ., Ser. I, Mat. Mekh. Astron. 2016, No. 4, 644--650 (2016; Zbl 1393.30029); Vestn. St. Petersbg. Univ., Math. 50, No. 1, 35--43 (2017); translation from Vestn. St-Peterbg. Univ., Ser. I, Mat. Mekh. Astron. 62, No. 1, 53--63 (2017; Zbl 1382.41007)]. The problem is to approximate a smooth function \(f\) defined on a set \(E=\bigcup_{n\in\mathbb{Z}}[a_n,b_n]\), is a countable union of disjunct intervals on the real line, by functions of exponential type \(\sigma\geq 1\). In the previous papers \(f\) was taken from a Hölder class. In this paper the class is a significantly larger class \(\Lambda_M^{r+\omega}\) with \(r\geq 1\), where \(\Lambda_M^\omega\) consists of complex functions such that \(|f(x)|\leq M\), \(x\in E\) and \(|f(x_2)-f(x_1)|\leq c_f\omega(|x_2-x_1|)\) where \(\omega\) is a modulus of continuity satisfying \(\int_0^x \omega(t)/t\, dt +x\int_x^\infty \omega(t)/t^2\,dt\leq c\omega(x)\). As in the previous papers, also the intervals of \(E\) have to satisfy certain conditions but these are not changed. The main theorem proves the existence of an approximant \(F_\sigma\) such that \(|f(x)-F_\sigma(x)|\) is bounded in terms of \(\omega\) and the distance from \(x\) to the Joukowsky map of the closest interval \([a_n,b_n]\) from \(E\).
approximation on a subset of the real axis
scientific article; zbMATH DE number 6952192
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