Ultrapolynomial

The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.

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 $\left|c_{\alpha }\right|\leq CL^{\left|\alpha \right|}/M_{\alpha }$ 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

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.