Gevrey class

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

[math]\displaystyle{ \sup_{x \in K} |D^\alpha g(x)| \le C^{|\alpha|+1}|\alpha!|^\sigma \quad \forall \alpha \in \Z_{\geq 0}^n }[/math]

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

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

Application

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

[math]\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} }[/math]

with

[math]\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} }[/math]

and Gevrey order [math]\displaystyle{ \alpha = 1 + \frac{1}{\omega}. }[/math]

See also

• Denjoy–Carleman theorem