In this paper, we construct some interesting Gevrey functions of order α for every α > 1 with compact support by a clever use of the Bray-Mandelbrojt iterative process. We then apply these results to the moment method, which will enable us to derive some upper bounds for the cost of fast boundary controls for a class of linear equations of parabolic or dispersive type that partially improve the existing results proved in [P. Lissy, On the Cost of Fast Controls for Some Families of Dispersive or Parabolic Equations in One Space Dimension SIAM J. Control Optim., 52(4), 2651-2676]. However this construction fails to improve the results of [G. Tenenbaum and M. Tucsnak, New blow-up rates of fast controls for the Schrödinger and heat equations, Journal of Differential Equations, 243 (2007), 70-100] in the precise case of the usual heat and Schrödinger equation.