Ultrapolynomial

In mathematics, an ultrapolynomial is a power series in several variables whose coefficients are bounded in some specific sense.

Definition

Let [math]\displaystyle{ d \in \mathbb{N} }[/math] and [math]\displaystyle{ K }[/math] a field (typically [math]\displaystyle{ \mathbb{R} }[/math] or [math]\displaystyle{ \mathbb{C} }[/math]) equipped with a norm (typically the absolute value). Then a function [math]\displaystyle{ P: K^d \rightarrow K }[/math] of the form [math]\displaystyle{ P(x) = \sum_{\alpha \in \mathbb{N}^d} c_\alpha x^\alpha }[/math] is called an ultrapolynomial of class [math]\displaystyle{ \left\{ M_p \right\} }[/math], if the coefficients [math]\displaystyle{ c_\alpha }[/math] satisfy [math]\displaystyle{ \left| c_\alpha \right| \leq C L^{\left| \alpha \right|}/M_\alpha }[/math] for all [math]\displaystyle{ \alpha \in \mathbb{N}^d }[/math], for some [math]\displaystyle{ L\gt 0 }[/math] and [math]\displaystyle{ C\gt 0 }[/math] (resp. for every [math]\displaystyle{ L\gt 0 }[/math] and some [math]\displaystyle{ C(L)\gt 0 }[/math]).

References

• Lozanov-Crvenković, Z.; Perišić, D. (5 Feb 2007). "Kernel theorem for the space of Beurling - Komatsu tempered ultradistibutions". arXiv:math/0702093.
• Lozanov-Crvenković, Z (October 2007). "Kernel theorems for the spaces of tempered ultradistributions". Integral Transforms and Special Functions 18 (10): 699–713. doi:10.1080/10652460701445658. 
• Pilipović, Stevan; Pilipović, Bojan; Prangoski, Jasson (2021). "Infinite order $$\Psi $$DOs: Composition with entire functions, new Shubin-Sobolev spaces, and index theorem". Analysis and Mathematical Physics 11 (3). doi:10.1007/s13324-021-00545-w.