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We study estimation and prediction of Gaussian processes with space-time covariance models belonging to the Dynamical Generalized Wendland family (DGW) (Porcu et al., 2020, Statistica Sinica), under fixed-domain asymptotics. Such a class is nonseparable, has dynamical compact supports, and parameterizes differentiability at the origin similarly to the space-time Matérn class (Ip and Li, 2017, Statistica Sinica). The results of the paper are classified into two parts. In the first part, wedoi:10.5705/ss.202020.0010 fatcat:7gzlsznuife6dd5rnaqmmdmdle