Ultrapolynomial

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

Definition

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

References

{{cite arXiv |last1=Lozanov-Crvenković |first1=Z. |last2=Perišić |first2=D. |eprint=math/0702093 |title=Kernel theorem for the space of Beurling - Komatsu tempered ultradistibutions |class= |date=5 Feb 2007 }}
{{cite journal |author=Lozanov-Crvenković, Z |title= Kernel theorems for the spaces of tempered ultradistributions |journal= Integral Transforms and Special Functions |pages=699–713 |date=October 2007 |volume= 18 |issue= 10 |doi=10.1080/10652460701445658|s2cid= 123420666 }}
{{cite journal |last1=Pilipović |first1=Stevan |last2=Pilipović |first2=Bojan |last3=Prangoski |first3=Jasson |arxiv=1711.05628 |title=Infinite order $$\Psi $$DOs: Composition with entire functions, new Shubin-Sobolev spaces, and index theorem |journal=Analysis and Mathematical Physics |year=2021 |volume=11 |issue=3 |doi=10.1007/s13324-021-00545-w |s2cid=201107206 }}

Category:Mathematical analysis