Approximation of a function and its derivatives by entire functions (Q2788791)

Let \(m\) be a non-negative integer, \(I=(a,b)\) (\(-\infty\leq a\leq b\leq\infty\)). Denote by \(H(U)\) the family of functions holomorphic on \(U\), and denote by \(C^+(I)\) the positive continuous functions on \(I\). In the paper, the following result is proved. If \(f\in C^{(m)}(I)\), and \(\epsilon\in C^+(I)\), then there exists a function \(g\in H(\mathbb{C}\setminus I^c)\) such that \(|f^{(i)}(x)-g^{(i)}(x)|<\epsilon(x)\), \(x\in I\), \(i=0,1,\dots,m\).