This paper is a first of a series aiming at revisiting technical aspects of the volume averaging theory. Here, we discuss the choice of the spatial averaging operator for periodic and quasiperiodic structures. We show that spatial averaging must be defined in terms of a convolution and analyze the properties of a variety of kernels, with a particular focus on the smoothness of average fields, the ability to attenuate geometrical fluctuations, Taylor series expansions, averaging of periodic fields and resilience to perturbations of periodicity. We conclude with a set of recommendations regarding kernels to use in the volume averaging theory.