Gevrey class

In mathematics, the Gevrey classes on a domain $\Omega \subseteq \mathbb {R} ^{n}$, introduced by Maurice Gevrey, are spaces of functions 'between' the space of analytic functions $C^{\omega }(\Omega )$ and the space of smooth (infinitely differentiable) functions $C^{\infty }(\Omega )$. In particular, for $\sigma \geq 1$, the Gevrey class $G^{\sigma }(\Omega )$, consists of those smooth functions $g\in C^{\infty }(\Omega )$ such that for every compact subset $K\Subset \Omega$ there exists a constant $C$, depending only on $g,K$, such that

$\sup _{x\in K}|D^{\alpha }g(x)|\leq C^{|\alpha |+1}|\alpha !|^{\sigma }\quad \forall \alpha \in \mathbb {Z} _{\geq 0}^{n}$

Where $D^{\alpha }$ denotes the partial derivative of order $\alpha$ (see multi-index notation).

When $\sigma =1$, $G^{\sigma }(\Omega )$ coincides with the class of analytic functions $C^{\omega }(\Omega )$, but for $\sigma >1$ there are compactly supported functions in the class that are not identically zero (an impossibility in $C^{\omega }$). It is in this sense that they interpolate between $C^{\omega }$ and $C^{\infty }$. The Gevrey classes find application in discussing the smoothness of solutions to certain partial differential equations: Gevrey originally formulated the definition while investigating the homogeneous heat equation, whose solutions are in $G^{2}(\Omega )$.

Application

Gevrey functions are used in control engineering for trajectory planning. A typical example is the function

$\Phi _{\omega ,T}(t)={\begin{cases}0&t\leq 0,\\1&t\geq T,\\{\frac {\int _{0}^{t}\Omega _{\omega ,T}(\tau )d\tau }{\int _{0}^{T}\Omega _{\omega ,T}(\tau )d\tau }}&t\in (0,T)\end{cases}}$

with

$\Omega _{\omega ,T}(t)={\begin{cases}0&t\notin [0,T],\\\exp \left({\frac {-1}{\left([1-{\frac {t}{T}}]~{\frac {t}{T}}\right)^{\omega }}}\right)&t\in (0,T)\end{cases}}$

and Gevrey order $\alpha =1+{\frac {1}{\omega }}.$

See also

• Denjoy–Carleman theorem