On the approximation of positive functions by power series. II
Let \(h: \mathbb{R}_0^+\to \mathbb{R}^+\) be a rapidly growing function and let \(N(x)= \sum c_nx^n\), \(c_n\geq 0\) be a power series such that \(d_1\leq h(x)/N(x)\leq d_2\), \(x\geq 0\), where \(d_1\) and \(d_2\) are some constants from \(\mathbb{R}^+\). One writes in this case \(h\sim N\). The author is preoccupied with the question of the existence of the power series approximating given \(h\). In a previous paper under the same title [J. Approximation Theory 83, No. 3, 342-346 (1995; Zbl 0835.41009)] \textit{U. Schmid} proved the following result: \[ h\sim N \quad\text{iff} \quad h(x)/\{\max \{x^n/ \sup\{x^n/ h(x),\;x\geq 0\}\}\}\leq c \] for some constant \(c\in \mathbb{R}^+\) (Theorem 1). In the present paper ``methods are discussed which allow for a given function \(h\) the construction of the coefficients \(c_n\) \dots{} and to find suitable constants \(d_1\) and \(d_2\)'' (Theorem 4). On the other hand the author emphasizes the natural connection between the function \(P(x)= \max\{x^n/ \sup\{x^n/ h(x)\), \(x\geq 0\}\}\) and the power series \(H(x)= \sum x^n/ \sup\{x^n/h(x)\), \(x\geq 0\}\) which is now taken for approximating \(h\). Namely the author proves the following result in this sense: \(h\sim H\) iff there exists a positive constant \(d<1\) and an integer \(z\) such that \(u_{n+1}/u_n\leq du_{n+z+1}/ u_{n+z}\) (theorem 2). But this fact is not true for every \(h\) satisfying the condition from theorem 1 (see above) and the author proves one more result: if \(h\sim H\), then there exist constants \(a>1\) and \(b>0\) with \(h(x)= x^{b\ln x}\), \(x\geq 0\), that is the author establishes a limit of growth for all functions satisfying \(h\sim H\). In this way the author rounds off his interesting study.
scientific article; zbMATH DE number 1197043
scientific article