A few remarks on time-frequency analysis of Gevrey, analytic and   ultra-analytic functions
  We give a brief survey of recent results concerning almost diagonalization of pseudodifferential operators via Gabor frames. Moreover, we show new connections between symbols with Gevrey, analytic or ultra-analityc regularity and time-frequency analysis of the corresponding pseudodifferential operators. 
Elena Cordero, Fabio Nicola and Luigi Rodin
math.AP math.FA