A new class of locally supported radial basis functions on the (unit) sphere is introduced by forming an infinite number of convolutions of ''isotropic finite elements''. The resulting up functions show useful properties: They are locally supported and are infinitely often differentiable. The main properties of these kernels are studied in detail. In particular, the development of a multiresolution analysis within the reference space of square--integrable functions over the sphere is given. Altogether, the paper presents a mathematically significant and numerically efficient introduction to multiscale approximation by locally supported radial basis functions on the sphere.