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

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. S2CID 123420666.
• 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). arXiv:1711.05628. doi:10.1007/s13324-021-00545-w. S2CID 201107206.

• Mathematical analysis