Gevrey class

In mathematics, the Gevrey classes on a domain ${\displaystyle \Omega \subseteq \mathbb {R} ^{n}}$, introduced by Maurice Gevrey,^[1] are spaces of functions 'between' the space of analytic functions ${\displaystyle C^{\omega }(\Omega )}$ and the space of smooth (infinitely differentiable) functions ${\displaystyle C^{\infty }(\Omega )}$. In particular, for ${\displaystyle \sigma \geq 1}$, the Gevrey class ${\displaystyle G^{\sigma }(\Omega )}$, consists of those smooth functions ${\displaystyle g\in C^{\infty }(\Omega )}$ such that for every compact subset ${\displaystyle K\Subset \Omega }$ there exists a constant ${\displaystyle C}$, depending only on ${\displaystyle g,K}$, such that^[2]

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

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

When ${\displaystyle \sigma =1}$, ${\displaystyle G^{\sigma }(\Omega )}$ coincides with the class of analytic functions ${\displaystyle C^{\omega }(\Omega )}$, but for ${\displaystyle \sigma >1}$ there are compactly supported functions in the class that are not identically zero (an impossibility in ${\displaystyle C^{\omega }}$). It is in this sense that they interpolate between ${\displaystyle C^{\omega }}$ and ${\displaystyle C^{\infty }}$. 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 ${\displaystyle G^{2}(\Omega )}$.^[2]

Application

Gevrey functions are used in control engineering for trajectory planning.^{[3] [4]} A typical example is the function

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

with

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

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

See also

• Denjoy–Carleman theorem