Ultrapolynomial

Template:Multiple issues 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 $ℝ$ or $ℂ$) equipped with a norm (typically the absolute value). Then a function $P:K^d\to K$ of the form $P(x)=\sum _{\alpha \in {\mathbb{N}}^d}{c}_{\alpha }{x}^{\alpha }$ 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 $\alpha \in {\mathbb{N}}^d$, for some $L>0$ and $C>0$ (resp. for every $L>0$ and some $C(L)>0$).

References

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