Uniform and tangential approximation on a strip by entire functions having optimal growth (Q1884437)

For a functions \(f\), continuous on a closed strip \(S_h:=\mathbb R\times[-h,h]\), \(h>0\), and holomorphic on its interior, the authors discuss the problem of uniform and tangential approximation on \(S_h\) by entire functions with simultaneous estimation of their growth. For the optimization of this growth (depending on the growth of \(f\) on \(S_h\) and on the differential properties of \(f\) on \(\partial S_h\)) it is developed an adequate approximation technique. By \(\mathcal A'(E)\), \(\mathcal A''(E)\) and \(\mathcal A^p(E)\), we denote the classes of functions \(E\to\mathbb C\) which are one, two and \(p\) times respectively continuously differentiable on \(E\) in the sense of \(\mathbb C\). Optimal uniform approximation on a strip \(S_h\) is realized in two steps. First, a function \(f\in\mathcal A''(S_h)\) is approximated by a function \(F\in\mathcal A(S_{2h})\) which is holomorphic in a larger strip, then the function \(F\) is approximated on \(S_h\) by entire functions. The main problem is to ensure that one can realise each step using approximating functions having optimal growth. Runge-Jackson type approximation on a strip is stated: For any \(F\in\mathcal A(S_{h_1})\) with \(h_1 = h(1 +\delta)\), \(\delta > 0\), and for any \(\varepsilon > 0\) and \(p > 1\) there is an entire function \(G\) such that \[ \| F-G\|_{S_h}<\varepsilon,\quad \log\frac{| G(z)| }{\varepsilon}<c(p)\left(1+(1+ \delta)\frac{| \text{Im}z| }{h\delta}\right)\left(1+\delta+\log^+ \frac{M_f(p| z| )}{\delta\varepsilon}\right). \] Let \(f\in\mathcal A''(S_h)\) and \(\varphi\in\mathcal A(S_{2h})\). There is an entire function \(G\) such that \[ \| \varphi f-G\|_{S_h}<\varepsilon, \] \[ \log\frac{| G(z)| }{\varepsilon}<c(p)\left(1+\frac{| \text{Im}z| }{h}\right) \left(1+ \log^+\frac{M_f(r)M_\varphi(r)}{\varepsilon}+\frac{h^2}{\varepsilon} M_{f''}(r,\partial S_h)M_\varphi(r)\right) \] where \(r= p| z| + 2h\). Using these results, it is obtained optimal tangential approximation on \(S_h\) of the form \(\omega_{2, f_\delta}(\delta)\leq\omega_{2,f}(\delta)\), \(\delta>0\), by entire functions.