Generalization of proximate order and applications

Variations on the proximate order

The concept of proximate order is widely used in the theories of entire, meromorphic, subharmonic and plurisubharmonic functions. We give a general interpretation of this concept as a proximate growth function relative to a model growth function. If a function is the proximate growth function with respect to the identity function on the positive semi-axis, then the logarithm of this function is the classical proximate order. Our definition uses only one condition. This form of definition is also new for the classical proximate order.

关于Polya频率序列的generating functions的近似增长阶

On the proximate order of growth of generating functions of Polya frequency sequences

我们研究了$PF_r$ g.f.的解析的单位圆盘，并描述这些功能的增长的近似阶。将经典函数论中关于增长阶的一些结果推广到近似阶的情形。

Entire functions with preassigned zero proximate order

It is known that if the proximate order $ρ(r)$ such that $\lim ρ(r) = ρ> 0 (r \to \infty)$, then there exists an entire function $f(z)$ of proximate order $ρ(r)$. In the case where $ρ= 0$ the question about the existence of such entire function remained open until now. This question is investigated in the paper.

Flat functions in Carleman ultraholomorphic classes via proximate orders

Whenever the defining sequence of a Carleman ultraholomorphic class (in the sense of H. Komatsu) is strongly regular and associated with a proximate order, flat functions are constructed in the class on sectors of optimal opening. As consequences, we obtain analogues of both Borel-Ritt-Gevrey theorem and Watson's lemma in this general situation.

Multisummability in Carleman ultraholomorphic classes by means of nonzero proximate orders

We introduce a general multisummability theory of formal power series in Carleman ultraholomorphic classes. The finitely many levels of summation are determined by pairwise comparable, nonequivalent weight sequences admitting nonzero proximate orders and whose growth indices are distinct. Thus, we extend the powerful multisummability theory for finitely many Gevrey levels, developed by J.-P. Ramis, J. Écalle and W. Balser, among others. We provide both the analytical and cohomological approaches, and obtain a reconstruction formula for the multisum of a multisummable series by means of iterate…

A Phragmén-Lindelöf theorem via proximate orders, and the propagation of asymptotics

We prove that, for asymptotically bounded holomorphic functions in a sector in $\mathbb{C}$, an asymptotic expansion in a single direction towards the vertex with constraints in terms of a logarithmically convex sequence admitting a nonzero proximate order entails asymptotic expansion in the whole sector with control in terms of the same sequence. This generalizes a result by A. Fruchard and C. Zhang for Gevrey asymptotic expansions, and the proof strongly rests on a suitably refined version of the classical Phragmén-Lindelöf theorem, here obtained for functions whose growth in a sector is spe…