We demonstrate that beams originating from Fresnel diraction patterns are self-accelerating in free space. In addition to accelerating and self-healing, they also exhibit parabolic deceleration property, which is in stark dierence to other accelerating beams. We nd that the trajectory of Fresnel paraxial accelerating beams is similar to that of nonparaxial Weber beams. Decelerating and accelerating regions are separated by a critical propagation distance, at which no acceleration is present. During deceleration, the Fresnel diraction beams undergo self-smoothing, in which oscillations of the diracted waves gradually smooth out and are completely gone at the