Ultrapolynomial

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

Definition

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

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.