Английская Википедия:Gevrey class

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

<math>\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>D^\alpha</math> denotes the partial derivative of order <math>\alpha</math> (see multi-index notation).

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

Application

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

<math> \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> \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> \alpha = 1 + \frac{1}{\omega}.</math>

See also

• Denjoy–Carleman theorem