Data defined on spherical domains occurs in various applications, such as surface modeling, omnidirectional imaging, and the analysis of keypoints in volumetric data. The theory of spherical signals lacks important concepts like the Gaussian function, which is permanently used in planar image processing. We propose a definition of a spherical Gaussian function as the Green's function of the spherical diffusion process. This allows to introduce a linear scale space on the sphere. We apply this new filter to the smoothing of 3D object surfaces.