Gevrey class

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

$\underset{x\in K}{\sup }|D^{\alpha }g(x)|\le C^{|\alpha |+1}|\alpha !{|}^{\sigma }\mathrm{\forall }\alpha \in {\mathbb{Z}}_{\ge 0}^n$

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

When $\sigma =1$, $G^{\sigma }(\mathrm{\Omega })$ coincides with the class of analytic functions $C^{\omega }(\mathrm{\Omega })$, 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 $C^{\mathrm{\infty }}$. 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 $G^2(\mathrm{\Omega })$.

Application

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

${\mathrm{\Phi }}_{\omega ,T}(t)=\{\begin{array}{cc}0& t\le 0,\\ 1& t\ge T,\\ \frac{\underset{0}{\overset{t}{\int }}{\mathrm{\Omega }}_{\omega ,T}(\tau )d\tau }{\underset{0}{\overset{T}{\int }}{\mathrm{\Omega }}_{\omega ,T}(\tau )d\tau }& t\in (0,T)\end{array}$

with

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

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

See also

• Denjoy–Carleman theorem